Struct Float 

pub struct Float(/* private fields */);

A floating-point number.

Floats are currently experimental. They are missing many important functions. However, the functions that are currently implemented are thoroughly tested and documented, with the exception of string conversion functions. The current string conversions are incomplete and will be changed in the future to match MPFR’s behavior.

Floats are similar to the primitive floats defined by the IEEE 754 standard. They include NaN, positive and $-\infty$, and positive and negative zero. There is only one NaN; there is no concept of a NaN payload.

All the finite Floats are dyadic rationals (rational numbers whose denominator is a power of 2). A finite Float consists of several fields:

• a sign, which denotes whether the Float is positive or negative;
• a significand, which is a Natural number whose value is equal to the Float’s absolute value multiplied by a power of 2;
• an exponent, which is one more than the floor of the base-2 logarithm of the Float’s absolute value;
• and finally, a precision, which is greater than zero and indicates the number of significant bits. It is common to think of a Float as an approximation to some real number, and the precision indicates how good the approximation is intended to be.

Floats inherit some odd behavior from the IEEE 754 standard regarding comparison. A NaN is not equal to any Float, including itself. Positive and negative zero compare as equal, despite being two distinct values. Additionally, (and this is not IEEE 754’s fault), Floats with different precisions compare as equal if they represent the same numeric value.

In many cases, the above behavior is unsatisfactory, so the ComparableFloat and ComparableFloat wrappers are provided. See their documentation for a description of their comparison behavior.

In documentation, we will use the ‘$=$’ sign to mean that two Floats are identical, writing things like $-\text{NaN}=\text{NaN}$ and $-(0.0) = -0.0$.

The Float type is designed to be very similar to the mpfr_t type in MPFR, and all Malachite functions produce exactly the same result as their counterparts in MPFR, unless otherwise noted.

Here are the structural difference between Float and mpfr_t:

• Float can only represent a single NaN value, with no sign or payload.
• Only finite, nonzero Floats have a significand, precision, and exponent. For other Floats, these concepts are undefined. In particular, unlike mpfr_t zeros, Float zeros do not have a precision.
• The types of mpfr_t components are configuration- and platform-dependent. The types of Float components are platform-independent, although the Limb type is configuration-dependent: it is u64 by default, but may be changed to u32 using the --32_bit_limbs compiler flag. The type of the exponent is always i32 and the type of the precision is always u64. The Limb type only has a visible effect on the functions that extract the raw significand. All other functions have the same interface when compiled with either Limb type.

Floats whose precision is 64 bits or less can be represented without any memory allocation. (Unless Malachite is compiled with 32_bit_limbs, in which case the limit is 32).

Implementations

impl Float

pub const fn abs_negative_zero(self) -> Self

If self is negative zero, returns positive zero; otherwise, returns self, taking self by value.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

assert_eq!(
    ComparableFloat(Float::NAN.abs_negative_zero()),
    ComparableFloat(Float::NAN)
);
assert_eq!(Float::INFINITY.abs_negative_zero(), Float::INFINITY);
assert_eq!(
    Float::NEGATIVE_INFINITY.abs_negative_zero(),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    ComparableFloat(Float::ZERO.abs_negative_zero()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(
    ComparableFloat(Float::NEGATIVE_ZERO.abs_negative_zero()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(Float::ONE.abs_negative_zero(), Float::ONE);
assert_eq!(Float::NEGATIVE_ONE.abs_negative_zero(), Float::NEGATIVE_ONE);

pub fn abs_negative_zero_ref(&self) -> Self

If self is negative zero, returns positive zero; otherwise, returns self, taking self by reference.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

assert_eq!(
    ComparableFloat(Float::NAN.abs_negative_zero_ref()),
    ComparableFloat(Float::NAN)
);
assert_eq!(Float::INFINITY.abs_negative_zero_ref(), Float::INFINITY);
assert_eq!(
    Float::NEGATIVE_INFINITY.abs_negative_zero_ref(),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    ComparableFloat(Float::ZERO.abs_negative_zero_ref()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(
    ComparableFloat(Float::NEGATIVE_ZERO.abs_negative_zero_ref()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(Float::ONE.abs_negative_zero_ref(), Float::ONE);
assert_eq!(
    Float::NEGATIVE_ONE.abs_negative_zero_ref(),
    Float::NEGATIVE_ONE
);

pub const fn abs_negative_zero_assign(&mut self)

If self is negative zero, replaces it with positive zero; otherwise, does nothing.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

let mut x = Float::NAN;
x.abs_negative_zero_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::NAN));

let mut x = Float::INFINITY;
x.abs_negative_zero_assign();
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x.abs_negative_zero_assign();
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::ZERO;
x.abs_negative_zero_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::ZERO));

let mut x = Float::NEGATIVE_ZERO;
x.abs_negative_zero_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::ZERO));

let mut x = Float::ONE;
x.abs_negative_zero_assign();
assert_eq!(x, Float::ONE);

let mut x = Float::NEGATIVE_ONE;
x.abs_negative_zero_assign();
assert_eq!(x, Float::NEGATIVE_ONE);

impl Float

pub fn add_prec_round( self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,-\infty,p,m)=f(-\infty,\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p,m)=0.0$
• $f(-0.0,-0.0,p,m)=-0.0$
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,-x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_prec instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::add_round instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_prec_round(Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round(Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "6.0");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round(Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round(Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round(Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "5.85988");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round(Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec_round_val_ref( self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,-\infty,p,m)=f(-\infty,\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p,m)=0.0$
• $f(-0.0,-0.0,p,m)=-0.0$
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,-x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_prec_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::add_round_val_ref instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "6.0");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "5.85988");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec_round_ref_val( &self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,-\infty,p,m)=f(-\infty,\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p,m)=0.0$
• $f(-0.0,-0.0,p,m)=-0.0$
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,-x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_prec_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::add_round_ref_val instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_prec_round_val_ref(&Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_ref_val(Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "6.0");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round_ref_val(Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_ref_val(Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_ref_val(Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "5.85988");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round_ref_val(Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec_round_ref_ref( &self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,-\infty,p,m)=f(-\infty,\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p,m)=0.0$
• $f(-0.0,-0.0,p,m)=-0.0$
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,p,m)=f(-0.0,0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,-x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_prec_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::add_round_ref_ref instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_prec_round_ref_ref(&Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_ref_ref(&Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "6.0");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round_ref_ref(&Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_ref_ref(&Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_round_ref_ref(&Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "5.85988");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_prec_round_ref_ref(&Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec(self, other: Self, prec: u64) -> (Self, Ordering)

Adds two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,-\infty,p)=f(-\infty,\infty,p)=\text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p)=0.0$
• $f(-0.0,-0.0,p)=-0.0$
• $f(0.0,-0.0,p)=f(-0.0,0.0,p)=0.0$
• $f(x,-x,p)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_round instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_prec(Float::from(E), 5);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec(Float::from(E), 20);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec_val_ref(self, other: &Self, prec: u64) -> (Self, Ordering)

Adds two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,-\infty,p)=f(-\infty,\infty,p)=\text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p)=0.0$
• $f(-0.0,-0.0,p)=-0.0$
• $f(0.0,-0.0,p)=f(-0.0,0.0,p)=0.0$
• $f(x,-x,p)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_round_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_prec_val_ref(&Float::from(E), 5);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_prec_val_ref(&Float::from(E), 20);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec_ref_val(&self, other: Self, prec: u64) -> (Self, Ordering)

Adds two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,-\infty,p)=f(-\infty,\infty,p)=\text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p)=0.0$
• $f(-0.0,-0.0,p)=-0.0$
• $f(0.0,-0.0,p)=f(-0.0,0.0,p)=0.0$
• $f(x,-x,p)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_round_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = (&Float::from(PI)).add_prec_ref_val(Float::from(E), 5);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = (&Float::from(PI)).add_prec_ref_val(Float::from(E), 20);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_prec_ref_ref(&self, other: &Self, prec: u64) -> (Self, Ordering)

Adds two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,-\infty,p)=f(-\infty,\infty,p)=\text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,p)=0.0$
• $f(-0.0,-0.0,p)=-0.0$
• $f(0.0,-0.0,p)=f(-0.0,0.0,p)=0.0$
• $f(x,-x,p)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_round_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = (&Float::from(PI)).add_prec_ref_ref(&Float::from(E), 5);
assert_eq!(sum.to_string(), "5.8");
assert_eq!(o, Less);

let (sum, o) = (&Float::from(PI)).add_prec_ref_ref(&Float::from(E), 20);
assert_eq!(sum.to_string(), "5.85987");
assert_eq!(o, Less);

pub fn add_round(self, other: Self, rm: RoundingMode) -> (Self, Ordering)

Adds two Floats, rounding the result with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,-\infty,m)=f(-\infty,\infty,m)= \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,m)=0.0$
• $f(-0.0,-0.0,m)=-0.0$
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=-0.0$ if $m$ is Floor
• $f(0.0,x,m)=f(x,0.0,m)=f(-0.0,x,m)=f(x,-0.0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_round(Float::from(E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_round(Float::from(E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_round(Float::from(E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn add_round_val_ref( self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result with the specified rounding mode. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,-\infty,m)=f(-\infty,\infty,m)= \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,m)=0.0$
• $f(-0.0,-0.0,m)=-0.0$
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=-0.0$ if $m$ is Floor
• $f(0.0,x,m)=f(x,0.0,m)=f(-0.0,x,m)=f(x,-0.0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_round_val_ref(&Float::from(E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_round_val_ref(&Float::from(E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_round_val_ref(&Float::from(E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn add_round_ref_val( &self, other: Self, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result with the specified rounding mode. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,-\infty,m)=f(-\infty,\infty,m)= \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,m)=0.0$
• $f(-0.0,-0.0,m)=-0.0$
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=-0.0$ if $m$ is Floor
• $f(0.0,x,m)=f(x,0.0,m)=f(-0.0,x,m)=f(x,-0.0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is self.significant_bits().

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = (&Float::from(PI)).add_round_ref_val(Float::from(E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = (&Float::from(PI)).add_round_ref_val(Float::from(E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = (&Float::from(PI)).add_round_ref_val(Float::from(E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn add_round_ref_ref( &self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Adds two Floats, rounding the result with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,-\infty,m)=f(-\infty,\infty,m)= \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0,m)=0.0$
• $f(-0.0,-0.0,m)=-0.0$
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,-0.0,m)=f(-0.0,0.0,m)=-0.0$ if $m$ is Floor
• $f(0.0,x,m)=f(x,0.0,m)=f(-0.0,x,m)=f(x,-0.0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_round_ref_ref(&Float::from(E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_round_ref_ref(&Float::from(E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_round_ref_ref(&Float::from(E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn add_prec_round_assign( &mut self, other: Self, prec: u64, rm: RoundingMode, ) -> Ordering

Adds a Float to a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::add_prec_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::add_round_assign instead. If both of these things are true, consider using += instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.add_prec_round_assign(Float::from(E), 5, Floor), Less);
assert_eq!(x.to_string(), "5.8");

let mut x = Float::from(PI);
assert_eq!(x.add_prec_round_assign(Float::from(E), 5, Ceiling), Greater);
assert_eq!(x.to_string(), "6.0");

let mut x = Float::from(PI);
assert_eq!(x.add_prec_round_assign(Float::from(E), 5, Nearest), Less);
assert_eq!(x.to_string(), "5.8");

let mut x = Float::from(PI);
assert_eq!(x.add_prec_round_assign(Float::from(E), 20, Floor), Less);
assert_eq!(x.to_string(), "5.85987");

let mut x = Float::from(PI);
assert_eq!(
    x.add_prec_round_assign(Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "5.85988");

let mut x = Float::from(PI);
assert_eq!(x.add_prec_round_assign(Float::from(E), 20, Nearest), Less);
assert_eq!(x.to_string(), "5.85987");

pub fn add_prec_round_assign_ref( &mut self, other: &Self, prec: u64, rm: RoundingMode, ) -> Ordering

Adds a Float to a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::add_prec_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::add_round_assign_ref instead. If both of these things are true, consider using += instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.add_prec_round_assign_ref(&Float::from(E), 5, Floor), Less);
assert_eq!(x.to_string(), "5.8");

let mut x = Float::from(PI);
assert_eq!(
    x.add_prec_round_assign_ref(&Float::from(E), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "6.0");

let mut x = Float::from(PI);
assert_eq!(
    x.add_prec_round_assign_ref(&Float::from(E), 5, Nearest),
    Less
);
assert_eq!(x.to_string(), "5.8");

let mut x = Float::from(PI);
assert_eq!(
    x.add_prec_round_assign_ref(&Float::from(E), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "5.85987");

let mut x = Float::from(PI);
assert_eq!(
    x.add_prec_round_assign_ref(&Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "5.85988");

let mut x = Float::from(PI);
assert_eq!(
    x.add_prec_round_assign_ref(&Float::from(E), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "5.85987");

pub fn add_prec_assign(&mut self, other: Self, prec: u64) -> Ordering

Adds a Float to a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using += instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.add_prec_assign(Float::from(E), 5), Less);
assert_eq!(x.to_string(), "5.8");

let mut x = Float::from(PI);
assert_eq!(x.add_prec_assign(Float::from(E), 20), Less);
assert_eq!(x.to_string(), "5.85987");

pub fn add_prec_assign_ref(&mut self, other: &Self, prec: u64) -> Ordering

Adds a Float to a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_round_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using += instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.add_prec_assign_ref(&Float::from(E), 5), Less);
assert_eq!(x.to_string(), "5.8");

let mut x = Float::from(PI);
assert_eq!(x.add_prec_assign_ref(&Float::from(E), 20), Less);
assert_eq!(x.to_string(), "5.85987");

pub fn add_round_assign(&mut self, other: Self, rm: RoundingMode) -> Ordering

Adds a Float to a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::add_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::add_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using += instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.add_round_assign(Float::from(E), Floor), Less);
assert_eq!(x.to_string(), "5.859874482048838");

let mut x = Float::from(PI);
assert_eq!(x.add_round_assign(Float::from(E), Ceiling), Greater);
assert_eq!(x.to_string(), "5.859874482048839");

let mut x = Float::from(PI);
assert_eq!(x.add_round_assign(Float::from(E), Nearest), Less);
assert_eq!(x.to_string(), "5.859874482048838");

pub fn add_round_assign_ref( &mut self, other: &Self, rm: RoundingMode, ) -> Ordering

Adds a Float to a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::add_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::add_prec_round_assign_ref instead. If you know you’ll be using the Nearest rounding mode, consider using += instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.add_round_assign_ref(&Float::from(E), Floor), Less);
assert_eq!(x.to_string(), "5.859874482048838");

let mut x = Float::from(PI);
assert_eq!(x.add_round_assign_ref(&Float::from(E), Ceiling), Greater);
assert_eq!(x.to_string(), "5.859874482048839");

let mut x = Float::from(PI);
assert_eq!(x.add_round_assign_ref(&Float::from(E), Nearest), Less);
assert_eq!(x.to_string(), "5.859874482048838");

pub fn add_rational_prec_round( self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,-x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_rational_prec instead. If you know that your target precision is the precision of the Float input, consider using Float::add_rational_round instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).add_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Floor);
assert_eq!(sum.to_string(), "3.4");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).add_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Ceiling);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Nearest);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Floor);
assert_eq!(sum.to_string(), "3.474922");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).add_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Ceiling);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Nearest);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

pub fn add_rational_prec_round_val_ref( self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,-x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_rational_prec_val_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::add_rational_round_val_ref instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(sum.to_string(), "3.4");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(sum.to_string(), "3.474922");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

pub fn add_rational_prec_round_ref_val( &self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,-x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_rational_prec_ref_val instead. If you know that your target precision is the precision of the Float input, consider using Float::add_rational_round_ref_val instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(sum.to_string(), "3.4");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(sum.to_string(), "3.474922");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

pub fn add_rational_prec_round_ref_ref( &self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,-x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::add_rational_prec_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::add_rational_round_ref_ref instead. If both of these things are true, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(sum.to_string(), "3.4");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(sum.to_string(), "3.5");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(sum.to_string(), "3.474922");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(sum.to_string(), "3.474926");
assert_eq!(o, Greater);

pub fn add_rational_prec(self, other: Rational, prec: u64) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,-x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_round instead. If you know that your target precision is the precision of the Float input, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec(Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "4.8");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec(Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "4.641594");
assert_eq!(o, Greater);

pub fn add_rational_prec_val_ref( self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,-x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_round_val_ref instead. If you know that your target precision is the precision of the Float input, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec_val_ref(&Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "4.8");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_val_ref(&Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "4.641594");
assert_eq!(o, Greater);

pub fn add_rational_prec_ref_val( &self, other: Rational, prec: u64, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,-x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_round_ref_val instead. If you know that your target precision is the precision of the Float input, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec_ref_val(Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "4.8");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_ref_val(Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "4.641594");
assert_eq!(o, Greater);

pub fn add_rational_prec_ref_ref( &self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,-x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_round_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_prec_ref_ref(&Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "4.8");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).add_rational_prec_ref_ref(&Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "4.641594");
assert_eq!(o, Greater);

pub fn add_rational_round( self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(0.0,x,m)=f(x,0,m)=f(-0.0,x,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_rational_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).add_rational_round(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).add_rational_round(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "3.474925986923129");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_round(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

pub fn add_rational_round_val_ref( self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(0.0,x,m)=f(x,0,m)=f(-0.0,x,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_rational_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).add_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).add_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "3.474925986923129");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

pub fn add_rational_round_ref_val( &self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result with the specified rounding mode. The Float is taken by reference and the Float by value. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(0.0,x,m)=f(x,0,m)=f(-0.0,x,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_rational_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).add_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).add_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "3.474925986923129");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

pub fn add_rational_round_ref_ref( &self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Adds a Float and a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(0.0,x,m)=f(x,0,m)=f(-0.0,x,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,-x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::add_rational_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using + instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).add_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).add_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "3.474925986923129");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).add_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "3.474925986923125");
assert_eq!(o, Less);

pub fn add_rational_prec_round_assign( &mut self, other: Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Adds a Rational to a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::add_rational_prec_assign instead. If you know that your target precision is the precision of the Float input, consider using Float::add_rational_round_assign instead. If both of these things are true, consider using += instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "3.4");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "3.5");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Greater
);
assert_eq!(x.to_string(), "3.5");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "3.474922");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "3.474926");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Greater
);
assert_eq!(x.to_string(), "3.474926");

This is mpfr_add_q from gmp_op.c, MPFR 4.2.0.

pub fn add_rational_prec_round_assign_ref( &mut self, other: &Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Adds a Rational to a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::add_rational_prec_assign_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::add_rational_round_assign_ref instead. If both of these things are true, consider using += instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact addition.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "3.4");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "3.5");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Greater
);
assert_eq!(x.to_string(), "3.5");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "3.474922");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "3.474926");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Greater
);
assert_eq!(x.to_string(), "3.474926");

pub fn add_rational_prec_assign( &mut self, other: Rational, prec: u64, ) -> Ordering

Adds a Rational to a Float in place, rounding the result to the nearest value of the specified precision. The Rational is taken by value. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using += instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_assign(Rational::exact_from(1.5), 5),
    Greater
);
assert_eq!(x.to_string(), "4.8");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_assign(Rational::exact_from(1.5), 20),
    Greater
);
assert_eq!(x.to_string(), "4.641594");

pub fn add_rational_prec_assign_ref( &mut self, other: &Rational, prec: u64, ) -> Ordering

Adds a Rational to a Float in place, rounding the result to the nearest value of the specified precision. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::add_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_round_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using += instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_assign_ref(&Rational::exact_from(1.5), 5),
    Greater
);
assert_eq!(x.to_string(), "4.8");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_prec_assign_ref(&Rational::exact_from(1.5), 20),
    Greater
);
assert_eq!(x.to_string(), "4.641594");

pub fn add_rational_round_assign( &mut self, other: Rational, rm: RoundingMode, ) -> Ordering

Adds a Rational to a Float in place, rounding the result with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::add_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::add_rational_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using += instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_round_assign(Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "3.474925986923125");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_round_assign(Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "3.474925986923129");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_round_assign(Rational::from_unsigneds(1u8, 3), Nearest),
    Less
);
assert_eq!(x.to_string(), "3.474925986923125");

pub fn add_rational_round_assign_ref( &mut self, other: &Rational, rm: RoundingMode, ) -> Ordering

Adds a Rational to a Float in place, rounding the result with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded sum is less than, equal to, or greater than the exact sum. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x+y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::add_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::add_rational_prec_round_assign_ref instead. If you know you’ll be using the Nearest rounding mode, consider using += instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "3.474925986923125");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "3.474925986923129");

let mut x = Float::from(PI);
assert_eq!(
    x.add_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Nearest),
    Less
);
assert_eq!(x.to_string(), "3.474925986923125");

impl Float

pub fn div_prec_round( self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p,m)=\infty$ if $x>0.0$
• $f(x,0.0,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_prec instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::div_round instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_round(Float::from(E), 5, Floor);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round(Float::from(E), 5, Ceiling);
assert_eq!(quotient.to_string(), "1.19");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round(Float::from(E), 5, Nearest);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round(Float::from(E), 20, Floor);
assert_eq!(quotient.to_string(), "1.155725");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round(Float::from(E), 20, Ceiling);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round(Float::from(E), 20, Nearest);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec_round_val_ref( self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is are taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p,m)=\infty$ if $x>0.0$
• $f(x,0.0,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_prec_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::div_round_val_ref instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_round_val_ref(&Float::from(E), 5, Floor);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_val_ref(&Float::from(E), 5, Ceiling);
assert_eq!(quotient.to_string(), "1.19");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round_val_ref(&Float::from(E), 5, Nearest);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_val_ref(&Float::from(E), 20, Floor);
assert_eq!(quotient.to_string(), "1.155725");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_val_ref(&Float::from(E), 20, Ceiling);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round_val_ref(&Float::from(E), 20, Nearest);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec_round_ref_val( &self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is are taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p,m)=\infty$ if $x>0.0$
• $f(x,0.0,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_prec_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::div_round_ref_val instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_round_ref_val(Float::from(E), 5, Floor);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_ref_val(Float::from(E), 5, Ceiling);
assert_eq!(quotient.to_string(), "1.19");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round_ref_val(Float::from(E), 5, Nearest);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_ref_val(Float::from(E), 20, Floor);
assert_eq!(quotient.to_string(), "1.155725");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_ref_val(Float::from(E), 20, Ceiling);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round_ref_val(Float::from(E), 20, Nearest);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec_round_ref_ref( &self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p,m)=\infty$ if $x>0.0$
• $f(x,0.0,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_prec_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::div_round_ref_ref instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_round_ref_ref(&Float::from(E), 5, Floor);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_ref_ref(&Float::from(E), 5, Ceiling);
assert_eq!(quotient.to_string(), "1.19");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round_ref_ref(&Float::from(E), 5, Nearest);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_ref_ref(&Float::from(E), 20, Floor);
assert_eq!(quotient.to_string(), "1.155725");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_round_ref_ref(&Float::from(E), 20, Ceiling);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_prec_round_ref_ref(&Float::from(E), 20, Nearest);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec(self, other: Self, prec: u64) -> (Self, Ordering)

Divides two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p)=\infty$ if $x>0.0$
• $f(x,0.0,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p)=-\infty$ if $x>0.0$
• $f(x,-0.0,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_round instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec(Float::from(E), 5);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec(Float::from(E), 20);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec_val_ref(self, other: &Self, prec: u64) -> (Self, Ordering)

Divides two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p)=\infty$ if $x>0.0$
• $f(x,0.0,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p)=-\infty$ if $x>0.0$
• $f(x,-0.0,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_round_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_val_ref(&Float::from(E), 5);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_val_ref(&Float::from(E), 20);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec_ref_val(&self, other: Self, prec: u64) -> (Self, Ordering)

Divides two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p)=\infty$ if $x>0.0$
• $f(x,0.0,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p)=-\infty$ if $x>0.0$
• $f(x,-0.0,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_round_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_ref_val(Float::from(E), 5);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_ref_val(Float::from(E), 20);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_prec_ref_ref(&self, other: &Self, prec: u64) -> (Self, Ordering)

Divides two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,p)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,p)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,p)=\infty$ if $x>0.0$
• $f(x,0.0,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,p)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,p)=-\infty$ if $x>0.0$
• $f(x,-0.0,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,p)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,p)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_round_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_prec_ref_ref(&Float::from(E), 5);
assert_eq!(quotient.to_string(), "1.12");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_prec_ref_ref(&Float::from(E), 20);
assert_eq!(quotient.to_string(), "1.155727");
assert_eq!(o, Greater);

pub fn div_round(self, other: Self, rm: RoundingMode) -> (Self, Ordering)

Divides two Floats, rounding the result with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,m)=\infty$ if $x>0.0$
• $f(x,0.0,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_round(Float::from(E), Floor);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_round(Float::from(E), Ceiling);
assert_eq!(quotient.to_string(), "1.155727349790922");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_round(Float::from(E), Nearest);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

pub fn div_round_val_ref( self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result with the specified rounding mode. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,m)=\infty$ if $x>0.0$
• $f(x,0.0,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_round_val_ref(&Float::from(E), Floor);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_round_val_ref(&Float::from(E), Ceiling);
assert_eq!(quotient.to_string(), "1.155727349790922");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_round_val_ref(&Float::from(E), Nearest);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

pub fn div_round_ref_val( &self, other: Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result with the specified rounding mode. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,m)=\infty$ if $x>0.0$
• $f(x,0.0,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_round_ref_val(Float::from(E), Floor);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_round_ref_val(Float::from(E), Ceiling);
assert_eq!(quotient.to_string(), "1.155727349790922");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_round_ref_val(Float::from(E), Nearest);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

pub fn div_round_ref_ref( &self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides two Floats, rounding the result with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm\infty,p,m)=f(\pm0.0,\pm0.0,p,m) = \text{NaN}$
• $f(\infty,x,m)=\infty$ if $0.0<x<\infty$
• $f(\infty,x,m)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0,m)=\infty$ if $x>0.0$
• $f(x,0.0,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x,m)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0,m)=-\infty$ if $x>0.0$
• $f(x,-0.0,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x,m)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x,m)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_round_ref_ref(&Float::from(E), Floor);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_round_ref_ref(&Float::from(E), Ceiling);
assert_eq!(quotient.to_string(), "1.155727349790922");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_round_ref_ref(&Float::from(E), Nearest);
assert_eq!(quotient.to_string(), "1.1557273497909217");
assert_eq!(o, Less);

pub fn div_prec_round_assign( &mut self, other: Self, prec: u64, rm: RoundingMode, ) -> Ordering

Divides a Float by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::div_prec_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::div_round_assign instead. If both of these things are true, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign(Float::from(E), 5, Floor),
    Less
);
assert_eq!(quotient.to_string(), "1.12");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign(Float::from(E), 5, Ceiling),
    Greater
);
assert_eq!(quotient.to_string(), "1.19");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign(Float::from(E), 5, Nearest),
    Less
);
assert_eq!(quotient.to_string(), "1.12");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign(Float::from(E), 20, Floor),
    Less
);
assert_eq!(quotient.to_string(), "1.155725");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign(Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(quotient.to_string(), "1.155727");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign(Float::from(E), 20, Nearest),
    Greater
);
assert_eq!(quotient.to_string(), "1.155727");

pub fn div_prec_round_assign_ref( &mut self, other: &Self, prec: u64, rm: RoundingMode, ) -> Ordering

Divides a Float by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::div_prec_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::div_round_assign_ref instead. If both of these things are true, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign_ref(&Float::from(E), 5, Floor),
    Less
);
assert_eq!(quotient.to_string(), "1.12");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign_ref(&Float::from(E), 5, Ceiling),
    Greater
);
assert_eq!(quotient.to_string(), "1.19");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign_ref(&Float::from(E), 5, Nearest),
    Less
);
assert_eq!(quotient.to_string(), "1.12");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign_ref(&Float::from(E), 20, Floor),
    Less
);
assert_eq!(quotient.to_string(), "1.155725");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign_ref(&Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(quotient.to_string(), "1.155727");

let mut quotient = Float::from(PI);
assert_eq!(
    quotient.div_prec_round_assign_ref(&Float::from(E), 20, Nearest),
    Greater
);
assert_eq!(quotient.to_string(), "1.155727");

pub fn div_prec_assign(&mut self, other: Self, prec: u64) -> Ordering

Divides a Float by a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.div_prec_assign(Float::from(E), 5), Less);
assert_eq!(x.to_string(), "1.12");

let mut x = Float::from(PI);
assert_eq!(x.div_prec_assign(Float::from(E), 20), Greater);
assert_eq!(x.to_string(), "1.155727");

pub fn div_prec_assign_ref(&mut self, other: &Self, prec: u64) -> Ordering

Divides a Float by a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_round_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.div_prec_assign_ref(&Float::from(E), 5), Less);
assert_eq!(x.to_string(), "1.12");

let mut x = Float::from(PI);
assert_eq!(x.div_prec_assign_ref(&Float::from(E), 20), Greater);
assert_eq!(x.to_string(), "1.155727");

pub fn div_round_assign(&mut self, other: Self, rm: RoundingMode) -> Ordering

Divides a Float by a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::div_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::div_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.div_round_assign(Float::from(E), Floor), Less);
assert_eq!(x.to_string(), "1.1557273497909217");

let mut x = Float::from(PI);
assert_eq!(x.div_round_assign(Float::from(E), Ceiling), Greater);
assert_eq!(x.to_string(), "1.155727349790922");

let mut x = Float::from(PI);
assert_eq!(x.div_round_assign(Float::from(E), Nearest), Less);
assert_eq!(x.to_string(), "1.1557273497909217");

pub fn div_round_assign_ref( &mut self, other: &Self, rm: RoundingMode, ) -> Ordering

Divides a Float by a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::div_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::div_prec_round_assign_ref instead. If you know you’ll be using the Nearest rounding mode, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.div_round_assign_ref(&Float::from(E), Floor), Less);
assert_eq!(x.to_string(), "1.1557273497909217");

let mut x = Float::from(PI);
assert_eq!(x.div_round_assign_ref(&Float::from(E), Ceiling), Greater);
assert_eq!(x.to_string(), "1.155727349790922");

let mut x = Float::from(PI);
assert_eq!(x.div_round_assign_ref(&Float::from(E), Nearest), Less);
assert_eq!(x.to_string(), "1.1557273497909217");

pub fn div_rational_prec_round( self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=f(\pm0.0,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x\geq 0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x>0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x>0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_rational_prec instead. If you know that your target precision is the precision of the Float input, consider using Float::div_rational_round instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Floor);
assert_eq!(quotient.to_string(), "9.0");
assert_eq!(o, Less);

let (quotient, o) =
    Float::from(PI).div_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Ceiling);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Nearest);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Floor);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

let (quotient, o) =
    Float::from(PI).div_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Ceiling);
assert_eq!(quotient.to_string(), "9.42479");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Nearest);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

pub fn div_rational_prec_round_val_ref( self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=f(\pm0.0,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x\geq 0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x>0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x>0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_rational_prec_val_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::div_rational_round_val_ref instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(quotient.to_string(), "9.0");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(quotient.to_string(), "9.42479");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

pub fn div_rational_prec_round_ref_val( &self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=f(\pm0.0,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x\geq 0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x>0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x>0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_rational_prec_ref_val instead. If you know that your target precision is the precision of the Float input, consider using Float::div_rational_round_ref_val instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(quotient.to_string(), "9.0");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(quotient.to_string(), "9.42479");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

pub fn div_rational_prec_round_ref_ref( &self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=f(\pm0.0,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x\geq 0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x>0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x>0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::div_rational_prec_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::div_rational_round_ref_ref instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(quotient.to_string(), "9.0");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(quotient.to_string(), "9.5");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(quotient.to_string(), "9.42479");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(quotient.to_string(), "9.42477");
assert_eq!(o, Less);

pub fn div_rational_prec(self, other: Rational, prec: u64) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=f(\pm0.0,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x\geq 0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x>0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x>0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_round instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_rational_prec(Rational::exact_from(1.5), 5);
assert_eq!(quotient.to_string(), "2.1");
assert_eq!(o, Greater);

let (quotient, o) = Float::from(PI).div_rational_prec(Rational::exact_from(1.5), 20);
assert_eq!(quotient.to_string(), "2.094395");
assert_eq!(o, Less);

pub fn div_rational_prec_val_ref( self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=f(\pm0.0,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x\geq 0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x>0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x>0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_round_val_ref instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_prec_val_ref(&Rational::exact_from(1.5), 5);
assert_eq!(quotient.to_string(), "2.1");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_prec_val_ref(&Rational::exact_from(1.5), 20);
assert_eq!(quotient.to_string(), "2.094395");
assert_eq!(o, Less);

pub fn div_rational_prec_ref_val( &self, other: Rational, prec: u64, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=f(\pm0.0,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x\geq 0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x>0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x>0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_round_ref_val instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::from(PI).div_rational_prec_ref_val(Rational::exact_from(1.5), 5);
assert_eq!(quotient.to_string(), "2.1");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_prec_ref_val(Rational::exact_from(1.5), 20);
assert_eq!(quotient.to_string(), "2.094395");
assert_eq!(o, Less);

pub fn div_rational_prec_ref_ref( &self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=f(\pm0.0,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x\geq 0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x\geq 0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x>0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x>0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_round_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_prec_ref_ref(&Rational::exact_from(1.5), 5);
assert_eq!(quotient.to_string(), "2.1");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_prec_ref_ref(&Rational::exact_from(1.5), 20);
assert_eq!(quotient.to_string(), "2.094395");
assert_eq!(o, Less);

pub fn div_rational_round( self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=f(\pm0.0,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x\geq 0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x>0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x>0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_rational_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_round(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

let (quotient, o) =
    Float::from(PI).div_rational_round(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(quotient.to_string(), "9.42477796076939");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_round(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

pub fn div_rational_round_val_ref( self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=f(\pm0.0,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x\geq 0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x>0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x>0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_rational_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

let (quotient, o) =
    Float::from(PI).div_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(quotient.to_string(), "9.42477796076939");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

pub fn div_rational_round_ref_val( &self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=f(\pm0.0,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x\geq 0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x>0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x>0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_rational_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

let (quotient, o) =
    Float::from(PI).div_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(quotient.to_string(), "9.42477796076939");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

pub fn div_rational_round_ref_ref( &self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Float by a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=f(\pm0.0,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x\geq 0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x\geq 0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x>0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x>0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::div_rational_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::from(PI).div_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

let (quotient, o) =
    Float::from(PI).div_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(quotient.to_string(), "9.42477796076939");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::from(PI).div_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(quotient.to_string(), "9.42477796076938");
assert_eq!(o, Less);

pub fn div_rational_prec_round_assign( &mut self, other: Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Divides a Float by a Rational in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::div_rational_prec_assign instead. If you know that your target precision is the precision of the Float input, consider using Float::div_rational_round_assign instead. If both of these things are true, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "9.0");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "9.5");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Greater
);
assert_eq!(x.to_string(), "9.5");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "9.42477");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "9.42479");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "9.42477");

pub fn div_rational_prec_round_assign_ref( &mut self, other: &Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Divides a Float by a Rational in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::div_rational_prec_assign_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::div_rational_round_assign_ref instead. If both of these things are true, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "9.0");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "9.5");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Greater
);
assert_eq!(x.to_string(), "9.5");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "9.42477");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "9.42479");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "9.42477");

pub fn div_rational_prec_assign( &mut self, other: Rational, prec: u64, ) -> Ordering

Divides a Float by a Rational in place, rounding the result to the nearest value of the specified precision. The Rational is taken by value. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_assign(Rational::exact_from(1.5), 5),
    Greater
);
assert_eq!(x.to_string(), "2.1");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_assign(Rational::exact_from(1.5), 20),
    Less
);
assert_eq!(x.to_string(), "2.094395");

pub fn div_rational_prec_assign_ref( &mut self, other: &Rational, prec: u64, ) -> Ordering

Divides a Float by a Rational in place, rounding the result to the nearest value of the specified precision. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::div_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_assign_ref(&Rational::exact_from(1.5), 5),
    Greater
);
assert_eq!(x.to_string(), "2.1");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_prec_assign_ref(&Rational::exact_from(1.5), 20),
    Less
);
assert_eq!(x.to_string(), "2.094395");

pub fn div_rational_round_assign( &mut self, other: Rational, rm: RoundingMode, ) -> Ordering

Divides a Float by a Rational in place, rounding the result with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::div_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::div_rational_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_round_assign(Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "9.42477796076938");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_round_assign(Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "9.42477796076939");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_round_assign(Rational::from_unsigneds(1u8, 3), Nearest),
    Less
);
assert_eq!(x.to_string(), "9.42477796076938");

pub fn div_rational_round_assign_ref( &mut self, other: &Rational, rm: RoundingMode, ) -> Ordering

Divides a Float by a Rational in place, rounding the result with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::div_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::div_rational_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using /= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "9.42477796076938");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "9.42477796076939");

let mut x = Float::from(PI);
assert_eq!(
    x.div_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Nearest),
    Less
);
assert_eq!(x.to_string(), "9.42477796076938");

pub fn rational_div_float_prec_round( x: Rational, y: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the specified precision and with the specified rounding mode. The Rational and the Float are both taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::rational_div_float_prec instead. If you know that your target precision is the precision of the Float input, consider using Float::rational_div_float_round instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_prec_round(Rational::from(3), Float::from(PI), 5, Floor);
assert_eq!(quotient.to_string(), "0.94");
assert_eq!(o, Less);

let (quotient, o) =
    Float::rational_div_float_prec_round(Rational::from(3), Float::from(PI), 5, Ceiling);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_prec_round(Rational::from(3), Float::from(PI), 5, Nearest);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_prec_round(Rational::from(3), Float::from(PI), 20, Floor);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

let (quotient, o) =
    Float::rational_div_float_prec_round(Rational::from(3), Float::from(PI), 20, Ceiling);
assert_eq!(quotient.to_string(), "0.95493");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_prec_round(Rational::from(3), Float::from(PI), 20, Nearest);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec_round_val_ref( x: Rational, y: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by value and the Float by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::rational_div_float_prec_val_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::rational_div_float_round_val_ref instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::rational_div_float_prec_round_val_ref(
    Rational::from(3),
    &Float::from(PI),
    5,
    Floor,
);
assert_eq!(quotient.to_string(), "0.94");
assert_eq!(o, Less);

let (quotient, o) = Float::rational_div_float_prec_round_val_ref(
    Rational::from(3),
    &Float::from(PI),
    5,
    Ceiling,
);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_val_ref(
    Rational::from(3),
    &Float::from(PI),
    5,
    Nearest,
);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_val_ref(
    Rational::from(3),
    &Float::from(PI),
    20,
    Floor,
);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

let (quotient, o) = Float::rational_div_float_prec_round_val_ref(
    Rational::from(3),
    &Float::from(PI),
    20,
    Ceiling,
);
assert_eq!(quotient.to_string(), "0.95493");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_val_ref(
    Rational::from(3),
    &Float::from(PI),
    20,
    Nearest,
);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec_round_ref_val( x: &Rational, y: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by reference and the Float by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::rational_div_float_prec_ref_val instead. If you know that your target precision is the precision of the Float input, consider using Float::rational_div_float_round_ref_val instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::rational_div_float_prec_round_ref_val(
    &Rational::from(3),
    Float::from(PI),
    5,
    Floor,
);
assert_eq!(quotient.to_string(), "0.94");
assert_eq!(o, Less);

let (quotient, o) = Float::rational_div_float_prec_round_ref_val(
    &Rational::from(3),
    Float::from(PI),
    5,
    Ceiling,
);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_ref_val(
    &Rational::from(3),
    Float::from(PI),
    5,
    Nearest,
);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_ref_val(
    &Rational::from(3),
    Float::from(PI),
    20,
    Floor,
);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

let (quotient, o) = Float::rational_div_float_prec_round_ref_val(
    &Rational::from(3),
    Float::from(PI),
    20,
    Ceiling,
);
assert_eq!(quotient.to_string(), "0.95493");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_ref_val(
    &Rational::from(3),
    Float::from(PI),
    20,
    Nearest,
);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec_round_ref_ref( x: &Rational, y: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the specified precision and with the specified rounding mode. The Rational and the Float are both taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::rational_div_float_prec_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::rational_div_float_round_ref_ref instead. If both of these things are true, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact division.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::rational_div_float_prec_round_ref_ref(
    &Rational::from(3),
    &Float::from(PI),
    5,
    Floor,
);
assert_eq!(quotient.to_string(), "0.94");
assert_eq!(o, Less);

let (quotient, o) = Float::rational_div_float_prec_round_ref_ref(
    &Rational::from(3),
    &Float::from(PI),
    5,
    Ceiling,
);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_ref_ref(
    &Rational::from(3),
    &Float::from(PI),
    5,
    Nearest,
);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_ref_ref(
    &Rational::from(3),
    &Float::from(PI),
    20,
    Floor,
);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

let (quotient, o) = Float::rational_div_float_prec_round_ref_ref(
    &Rational::from(3),
    &Float::from(PI),
    20,
    Ceiling,
);
assert_eq!(quotient.to_string(), "0.95493");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec_round_ref_ref(
    &Rational::from(3),
    &Float::from(PI),
    20,
    Nearest,
);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec( x: Rational, y: Self, prec: u64, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the nearest value of the specified precision. The Rational and the Float are both are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p)=f(0,\pm0.0,p)=\text{NaN}$
• $f(x,\infty,x,p)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p)=0.0$ if $x>0$
• $f(0,x,p)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::rational_div_float_prec_round instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) = Float::rational_div_float_prec(Rational::from(3), Float::from(PI), 5);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) = Float::rational_div_float_prec(Rational::from(3), Float::from(PI), 20);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec_val_ref( x: Rational, y: &Self, prec: u64, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the nearest value of the specified precision. The Rational is taken by value and the Float by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p)=f(0,\pm0.0,p)=\text{NaN}$
• $f(x,\infty,x,p)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p)=0.0$ if $x>0$
• $f(0,x,p)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::rational_div_float_prec_round_val_ref instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_prec_val_ref(Rational::from(3), &Float::from(PI), 5);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_prec_val_ref(Rational::from(3), &Float::from(PI), 20);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec_ref_val( x: &Rational, y: Self, prec: u64, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the nearest value of the specified precision. The Rational is taken by reference and the Float by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p)=f(0,\pm0.0,p)=\text{NaN}$
• $f(x,\infty,x,p)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p)=0.0$ if $x>0$
• $f(0,x,p)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::rational_div_float_prec_round_ref_val instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_prec_ref_val(&Rational::from(3), Float::from(PI), 5);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_prec_ref_val(&Rational::from(3), Float::from(PI), 20);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_prec_ref_ref( x: &Rational, y: &Self, prec: u64, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result to the nearest value of the specified precision. The Rational and the Float are both are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(x,\text{NaN},p)=f(0,\pm0.0,p)=\text{NaN}$
• $f(x,\infty,x,p)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p)=0.0$ if $x>0$
• $f(0,x,p)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::rational_div_float_prec_round_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_prec_ref_ref(&Rational::from(3), &Float::from(PI), 5);
assert_eq!(quotient.to_string(), "0.97");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_prec_ref_ref(&Rational::from(3), &Float::from(PI), 20);
assert_eq!(quotient.to_string(), "0.954929");
assert_eq!(o, Less);

pub fn rational_div_float_round( x: Rational, y: Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result with the specified rounding mode. The Rational and the Float are both are taken by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(x,\text{NaN},m)=f(0,\pm0.0,m)=\text{NaN}$
• $f(x,\infty,x,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,m)=0.0$ if $x>0$
• $f(0,x,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::rational_div_float_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_round(Rational::from(3), Float::from(PI), Floor);
assert_eq!(quotient.to_string(), "0.9549296585513716");
assert_eq!(o, Less);

let (quotient, o) =
    Float::rational_div_float_round(Rational::from(3), Float::from(PI), Ceiling);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_round(Rational::from(3), Float::from(PI), Nearest);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

pub fn rational_div_float_round_val_ref( x: Rational, y: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result with the specified rounding mode. The Rational is taken by value and the Float by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(x,\text{NaN},m)=f(0,\pm0.0,m)=\text{NaN}$
• $f(x,\infty,x,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,m)=0.0$ if $x>0$
• $f(0,x,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::rational_div_float_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_round_val_ref(Rational::from(3), &Float::from(PI), Floor);
assert_eq!(quotient.to_string(), "0.9549296585513716");
assert_eq!(o, Less);

let (quotient, o) =
    Float::rational_div_float_round_val_ref(Rational::from(3), &Float::from(PI), Ceiling);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_round_val_ref(Rational::from(3), &Float::from(PI), Nearest);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

pub fn rational_div_float_round_ref_val( x: &Rational, y: Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result with the specified rounding mode. The Rational is taken by reference and the Float by value. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(x,\text{NaN},m)=f(0,\pm0.0,m)=\text{NaN}$
• $f(x,\infty,x,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,m)=0.0$ if $x>0$
• $f(0,x,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::rational_div_float_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_round_ref_val(&Rational::from(3), Float::from(PI), Floor);
assert_eq!(quotient.to_string(), "0.9549296585513716");
assert_eq!(o, Less);

let (quotient, o) =
    Float::rational_div_float_round_ref_val(&Rational::from(3), Float::from(PI), Ceiling);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_round_ref_val(&Rational::from(3), Float::from(PI), Nearest);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

pub fn rational_div_float_round_ref_ref( x: &Rational, y: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Divides a Rational by a Float, rounding the result with the specified rounding mode. The Rational and the Float are both are taken by reference. An Ordering is also returned, indicating whether the rounded quotient is less than, equal to, or greater than the exact quotient. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x/y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(x,\text{NaN},m)=f(0,\pm0.0,m)=\text{NaN}$
• $f(x,\infty,x,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,m)=0.0$ if $x>0$
• $f(0,x,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::rational_div_float_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using / instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), y.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (quotient, o) =
    Float::rational_div_float_round_ref_ref(&Rational::from(3), &Float::from(PI), Floor);
assert_eq!(quotient.to_string(), "0.9549296585513716");
assert_eq!(o, Less);

let (quotient, o) =
    Float::rational_div_float_round_ref_ref(&Rational::from(3), &Float::from(PI), Ceiling);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

let (quotient, o) =
    Float::rational_div_float_round_ref_ref(&Rational::from(3), &Float::from(PI), Nearest);
assert_eq!(quotient.to_string(), "0.9549296585513725");
assert_eq!(o, Greater);

impl Float

pub fn mul_prec_round( self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm0.0,p,m)=f(\pm0.0,\pm\infty,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x>0.0$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_prec instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::mul_round instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_round(Float::from(E), 5, Floor);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round(Float::from(E), 5, Ceiling);
assert_eq!(product.to_string(), "9.0");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round(Float::from(E), 5, Nearest);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round(Float::from(E), 20, Floor);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round(Float::from(E), 20, Ceiling);
assert_eq!(product.to_string(), "8.53975");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round(Float::from(E), 20, Nearest);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec_round_val_ref( self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is are taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm0.0,p,m)=f(\pm0.0,\pm\infty,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x>0.0$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_prec_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::mul_round_val_ref instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_round_val_ref(&Float::from(E), 5, Floor);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_val_ref(&Float::from(E), 5, Ceiling);
assert_eq!(product.to_string(), "9.0");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round_val_ref(&Float::from(E), 5, Nearest);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_val_ref(&Float::from(E), 20, Floor);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_val_ref(&Float::from(E), 20, Ceiling);
assert_eq!(product.to_string(), "8.53975");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round_val_ref(&Float::from(E), 20, Nearest);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec_round_ref_val( &self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is are taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm0.0,p,m)=f(\pm0.0,\pm\infty,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x>0.0$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_prec_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::mul_round_ref_val instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_round_ref_val(Float::from(E), 5, Floor);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_ref_val(Float::from(E), 5, Ceiling);
assert_eq!(product.to_string(), "9.0");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round_ref_val(Float::from(E), 5, Nearest);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_ref_val(Float::from(E), 20, Floor);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_ref_val(Float::from(E), 20, Ceiling);
assert_eq!(product.to_string(), "8.53975");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round_ref_val(Float::from(E), 20, Nearest);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec_round_ref_ref( &self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\pm\infty,\pm0.0,p,m)=f(\pm0.0,\pm\infty,p,m) = \text{NaN}$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=\infty$ if $x>0.0$
• $f(\infty,x,p,m)=f(x,\infty,p,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,p,m)=f(x,-\infty,p,m)=\infty$ if $x<0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p,m)=f(x,0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p,m)=f(x,-0.0,p,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_prec_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::mul_round_ref_ref instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_round_ref_ref(&Float::from(E), 5, Floor);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_ref_ref(&Float::from(E), 5, Ceiling);
assert_eq!(product.to_string(), "9.0");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round_ref_ref(&Float::from(E), 5, Nearest);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_ref_ref(&Float::from(E), 20, Floor);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_round_ref_ref(&Float::from(E), 20, Ceiling);
assert_eq!(product.to_string(), "8.53975");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_prec_round_ref_ref(&Float::from(E), 20, Nearest);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec(self, other: Self, prec: u64) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm0.0,p)=f(\pm0.0,\pm\infty,p) = \text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x>0.0$
• $f(\infty,x,p)=f(x,\infty,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x>0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=f(x,0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p)=f(x,0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_round instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec(Float::from(E), 5);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec(Float::from(E), 20);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec_val_ref(self, other: &Self, prec: u64) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm0.0,p)=f(\pm0.0,\pm\infty,p) = \text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x>0.0$
• $f(\infty,x,p)=f(x,\infty,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x>0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=f(x,0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p)=f(x,0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_round_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_val_ref(&Float::from(E), 5);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_val_ref(&Float::from(E), 20);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec_ref_val(&self, other: Self, prec: u64) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm0.0,p)=f(\pm0.0,\pm\infty,p) = \text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x>0.0$
• $f(\infty,x,p)=f(x,\infty,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x>0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=f(x,0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p)=f(x,0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_round_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_ref_val(Float::from(E), 5);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_ref_val(Float::from(E), 20);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_prec_ref_ref(&self, other: &Self, prec: u64) -> (Self, Ordering)

Multiplies two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\pm\infty,\pm0.0,p)=f(\pm0.0,\pm\infty,p) = \text{NaN}$
• $f(\infty,x,p)=f(x,\infty,p)=\infty$ if $x>0.0$
• $f(\infty,x,p)=f(x,\infty,p)=-\infty$ if $x<0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=-\infty$ if $x>0.0$
• $f(-\infty,x,p)=f(x,-\infty,p)=\infty$ if $x<0.0$
• $f(0.0,x,p)=f(x,0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,p)=f(x,0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,p)=f(x,-0.0,p)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_round_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using * instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_prec_ref_ref(&Float::from(E), 5);
assert_eq!(product.to_string(), "8.5");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_prec_ref_ref(&Float::from(E), 20);
assert_eq!(product.to_string(), "8.53973");
assert_eq!(o, Less);

pub fn mul_round(self, other: Self, rm: RoundingMode) -> (Self, Ordering)

Multiplies two Floats, rounding the result with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\pm\infty,\pm0.0,m)=f(\pm0.0,\pm\infty,m) = \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x>0.0$
• $f(\infty,x,m)=f(x,\infty,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=f(x,0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,m)=f(x,0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_round(Float::from(E), Floor);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_round(Float::from(E), Ceiling);
assert_eq!(product.to_string(), "8.539734222673568");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_round(Float::from(E), Nearest);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

pub fn mul_round_val_ref( self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result with the specified rounding mode. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\pm\infty,\pm0.0,m)=f(\pm0.0,\pm\infty,m) = \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x>0.0$
• $f(\infty,x,m)=f(x,\infty,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=f(x,0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,m)=f(x,0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_round_val_ref(&Float::from(E), Floor);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_round_val_ref(&Float::from(E), Ceiling);
assert_eq!(product.to_string(), "8.539734222673568");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_round_val_ref(&Float::from(E), Nearest);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

pub fn mul_round_ref_val( &self, other: Self, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result with the specified rounding mode. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\pm\infty,\pm0.0,m)=f(\pm0.0,\pm\infty,m) = \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x>0.0$
• $f(\infty,x,m)=f(x,\infty,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=f(x,0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,m)=f(x,0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_round_ref_val(Float::from(E), Floor);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_round_ref_val(Float::from(E), Ceiling);
assert_eq!(product.to_string(), "8.539734222673568");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_round_ref_val(Float::from(E), Nearest);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

pub fn mul_round_ref_ref( &self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies two Floats, rounding the result with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\pm\infty,\pm0.0,m)=f(\pm0.0,\pm\infty,m) = \text{NaN}$
• $f(\infty,x,m)=f(x,\infty,m)=\infty$ if $x>0.0$
• $f(\infty,x,m)=f(x,\infty,m)=-\infty$ if $x<0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=-\infty$ if $x>0.0$
• $f(-\infty,x,m)=f(x,-\infty,m)=\infty$ if $x<0.0$
• $f(0.0,x,m)=f(x,0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x,m)=f(x,0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x,m)=f(x,-0.0,m)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_round_ref_ref(&Float::from(E), Floor);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_round_ref_ref(&Float::from(E), Ceiling);
assert_eq!(product.to_string(), "8.539734222673568");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_round_ref_ref(&Float::from(E), Nearest);
assert_eq!(product.to_string(), "8.539734222673566");
assert_eq!(o, Less);

pub fn mul_prec_round_assign( &mut self, other: Self, prec: u64, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::mul_prec_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::mul_round_assign instead. If both of these things are true, consider using *= instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign(Float::from(E), 5, Floor),
    Less
);
assert_eq!(product.to_string(), "8.5");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign(Float::from(E), 5, Ceiling),
    Greater
);
assert_eq!(product.to_string(), "9.0");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign(Float::from(E), 5, Nearest),
    Less
);
assert_eq!(product.to_string(), "8.5");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign(Float::from(E), 20, Floor),
    Less
);
assert_eq!(product.to_string(), "8.53973");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign(Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(product.to_string(), "8.53975");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign(Float::from(E), 20, Nearest),
    Less
);
assert_eq!(product.to_string(), "8.53973");

pub fn mul_prec_round_assign_ref( &mut self, other: &Self, prec: u64, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::mul_prec_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::mul_round_assign_ref instead. If both of these things are true, consider using *= instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign_ref(&Float::from(E), 5, Floor),
    Less
);
assert_eq!(product.to_string(), "8.5");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign_ref(&Float::from(E), 5, Ceiling),
    Greater
);
assert_eq!(product.to_string(), "9.0");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign_ref(&Float::from(E), 5, Nearest),
    Less
);
assert_eq!(product.to_string(), "8.5");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign_ref(&Float::from(E), 20, Floor),
    Less
);
assert_eq!(product.to_string(), "8.53973");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign_ref(&Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(product.to_string(), "8.53975");

let mut product = Float::from(PI);
assert_eq!(
    product.mul_prec_round_assign_ref(&Float::from(E), 20, Nearest),
    Less
);
assert_eq!(product.to_string(), "8.53973");

pub fn mul_prec_assign(&mut self, other: Self, prec: u64) -> Ordering

Multiplies a Float by a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using *= instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.mul_prec_assign(Float::from(E), 5), Less);
assert_eq!(x.to_string(), "8.5");

let mut x = Float::from(PI);
assert_eq!(x.mul_prec_assign(Float::from(E), 20), Less);
assert_eq!(x.to_string(), "8.53973");

pub fn mul_prec_assign_ref(&mut self, other: &Self, prec: u64) -> Ordering

Multiplies a Float by a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_round_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using *= instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.mul_prec_assign_ref(&Float::from(E), 5), Less);
assert_eq!(x.to_string(), "8.5");

let mut x = Float::from(PI);
assert_eq!(x.mul_prec_assign_ref(&Float::from(E), 20), Less);
assert_eq!(x.to_string(), "8.53973");

pub fn mul_round_assign(&mut self, other: Self, rm: RoundingMode) -> Ordering

Multiplies a Float by a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::mul_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::mul_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.mul_round_assign(Float::from(E), Floor), Less);
assert_eq!(x.to_string(), "8.539734222673566");

let mut x = Float::from(PI);
assert_eq!(x.mul_round_assign(Float::from(E), Ceiling), Greater);
assert_eq!(x.to_string(), "8.539734222673568");

let mut x = Float::from(PI);
assert_eq!(x.mul_round_assign(Float::from(E), Nearest), Less);
assert_eq!(x.to_string(), "8.539734222673566");

pub fn mul_round_assign_ref( &mut self, other: &Self, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::mul_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::mul_prec_round_assign_ref instead. If you know you’ll be using the Nearest rounding mode, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.mul_round_assign_ref(&Float::from(E), Floor), Less);
assert_eq!(x.to_string(), "8.539734222673566");

let mut x = Float::from(PI);
assert_eq!(x.mul_round_assign_ref(&Float::from(E), Ceiling), Greater);
assert_eq!(x.to_string(), "8.539734222673568");

let mut x = Float::from(PI);
assert_eq!(x.mul_round_assign_ref(&Float::from(E), Nearest), Less);
assert_eq!(x.to_string(), "8.539734222673566");

pub fn mul_rational_prec_round( self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x>0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x>0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x\geq0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_rational_prec instead. If you know that your target precision is the precision of the Float input, consider using Float::mul_rational_round instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) =
    Float::from(PI).mul_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Floor);
assert_eq!(product.to_string(), "1.0");
assert_eq!(o, Less);

let (product, o) =
    Float::from(PI).mul_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Ceiling);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Nearest);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Floor);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

let (product, o) =
    Float::from(PI).mul_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Ceiling);
assert_eq!(product.to_string(), "1.047199");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Nearest);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

pub fn mul_rational_prec_round_val_ref( self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x>0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x>0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x\geq0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_rational_prec_val_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::mul_rational_round_val_ref instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(product.to_string(), "1.0");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(product.to_string(), "1.047199");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

pub fn mul_rational_prec_round_ref_val( &self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x>0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x>0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x\geq0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_rational_prec_ref_val instead. If you know that your target precision is the precision of the Float input, consider using Float::mul_rational_round_ref_val instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(product.to_string(), "1.0");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(product.to_string(), "1.047199");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

pub fn mul_rational_prec_round_ref_ref( &self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(\pm\infty,0,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x>0$
• $f(\infty,x,p,m)=-\infty$ if $x<0$
• $f(-\infty,x,p,m)=-\infty$ if $x>0$
• $f(-\infty,x,p,m)=\infty$ if $x<0$
• $f(0.0,x,p,m)=0.0$ if $x\geq0$
• $f(0.0,x,p,m)=-0.0$ if $x<0$
• $f(-0.0,x,p,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,p,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::mul_rational_prec_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::mul_rational_round_ref_ref instead. If both of these things are true, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(product.to_string(), "1.0");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(product.to_string(), "1.06");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(product.to_string(), "1.047199");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(product.to_string(), "1.047197");
assert_eq!(o, Less);

pub fn mul_rational_prec(self, other: Rational, prec: u64) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x>0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x>0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x\geq0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x\geq0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_round instead. If you know that your target precision is the precision of the Float input, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec(Rational::exact_from(1.5), 5);
assert_eq!(product.to_string(), "4.8");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec(Rational::exact_from(1.5), 20);
assert_eq!(product.to_string(), "4.712387");
assert_eq!(o, Less);

pub fn mul_rational_prec_val_ref( self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x>0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x>0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x\geq0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x\geq0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_round_val_ref instead. If you know that your target precision is the precision of the Float input, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec_val_ref(&Rational::exact_from(1.5), 5);
assert_eq!(product.to_string(), "4.8");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_prec_val_ref(&Rational::exact_from(1.5), 20);
assert_eq!(product.to_string(), "4.712387");
assert_eq!(o, Less);

pub fn mul_rational_prec_ref_val( &self, other: Rational, prec: u64, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x>0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x>0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x\geq0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x\geq0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_round_ref_val instead. If you know that your target precision is the precision of the Float input, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec_ref_val(Rational::exact_from(1.5), 5);
assert_eq!(product.to_string(), "4.8");
assert_eq!(o, Greater);

let (product, o) = Float::from(PI).mul_rational_prec_ref_val(Rational::exact_from(1.5), 20);
assert_eq!(product.to_string(), "4.712387");
assert_eq!(o, Less);

pub fn mul_rational_prec_ref_ref( &self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(\pm\infty,0,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x>0$
• $f(\infty,x,p)=-\infty$ if $x<0$
• $f(-\infty,x,p)=-\infty$ if $x>0$
• $f(-\infty,x,p)=\infty$ if $x<0$
• $f(0.0,x,p)=0.0$ if $x\geq0$
• $f(0.0,x,p)=-0.0$ if $x<0$
• $f(-0.0,x,p)=-0.0$ if $x\geq0$
• $f(-0.0,x,p)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_round_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) = Float::from(PI).mul_rational_prec_ref_ref(&Rational::exact_from(1.5), 5);
assert_eq!(product.to_string(), "4.8");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_prec_ref_ref(&Rational::exact_from(1.5), 20);
assert_eq!(product.to_string(), "4.712387");
assert_eq!(o, Less);

pub fn mul_rational_round( self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x>0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x>0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x\geq0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_rational_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) =
    Float::from(PI).mul_rational_round(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(product.to_string(), "1.047197551196597");
assert_eq!(o, Less);

let (product, o) =
    Float::from(PI).mul_rational_round(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_round(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

pub fn mul_rational_round_val_ref( self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x>0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x>0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x\geq0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_rational_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) =
    Float::from(PI).mul_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(product.to_string(), "1.047197551196597");
assert_eq!(o, Less);

let (product, o) =
    Float::from(PI).mul_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

pub fn mul_rational_round_ref_val( &self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x>0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x>0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x\geq0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_rational_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) =
    Float::from(PI).mul_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(product.to_string(), "1.047197551196597");
assert_eq!(o, Less);

let (product, o) =
    Float::from(PI).mul_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

pub fn mul_rational_round_ref_ref( &self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Multiplies a Float by a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=f(\pm\infty,0,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x>0$
• $f(\infty,x,m)=-\infty$ if $x<0$
• $f(-\infty,x,m)=-\infty$ if $x>0$
• $f(-\infty,x,m)=\infty$ if $x<0$
• $f(0.0,x,m)=0.0$ if $x\geq0$
• $f(0.0,x,m)=-0.0$ if $x<0$
• $f(-0.0,x,m)=-0.0$ if $x\geq0$
• $f(-0.0,x,m)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::mul_rational_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using * instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (product, o) =
    Float::from(PI).mul_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(product.to_string(), "1.047197551196597");
assert_eq!(o, Less);

let (product, o) =
    Float::from(PI).mul_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

let (product, o) =
    Float::from(PI).mul_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(product.to_string(), "1.047197551196598");
assert_eq!(o, Greater);

pub fn mul_rational_prec_round_assign( &mut self, other: Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Rational in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::mul_rational_prec_assign instead. If you know that your target precision is the precision of the Float input, consider using Float::mul_rational_round_assign instead. If both of these things are true, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "1.0");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "1.06");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Greater
);
assert_eq!(x.to_string(), "1.06");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "1.047197");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "1.047199");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "1.047197");

pub fn mul_rational_prec_round_assign_ref( &mut self, other: &Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Rational in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::mul_rational_prec_assign_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::mul_rational_round_assign_ref instead. If both of these things are true, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact multiplication.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "1.0");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "1.06");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Greater
);
assert_eq!(x.to_string(), "1.06");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "1.047197");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "1.047199");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "1.047197");

pub fn mul_rational_prec_assign( &mut self, other: Rational, prec: u64, ) -> Ordering

Multiplies a Float by a Rational in place, rounding the result to the nearest value of the specified precision. The Rational is taken by value. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_assign(Rational::exact_from(1.5), 5),
    Greater
);
assert_eq!(x.to_string(), "4.8");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_assign(Rational::exact_from(1.5), 20),
    Less
);
assert_eq!(x.to_string(), "4.712387");

pub fn mul_rational_prec_assign_ref( &mut self, other: &Rational, prec: u64, ) -> Ordering

Multiplies a Float by a Rational in place, rounding the result to the nearest value of the specified precision. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::mul_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_assign_ref(&Rational::exact_from(1.5), 5),
    Greater
);
assert_eq!(x.to_string(), "4.8");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_prec_assign_ref(&Rational::exact_from(1.5), 20),
    Less
);
assert_eq!(x.to_string(), "4.712387");

pub fn mul_rational_round_assign( &mut self, other: Rational, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Rational in place, rounding the result with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::mul_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::mul_rational_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_round_assign(Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "1.047197551196597");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_round_assign(Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "1.047197551196598");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_round_assign(Rational::from_unsigneds(1u8, 3), Nearest),
    Greater
);
assert_eq!(x.to_string(), "1.047197551196598");

pub fn mul_rational_round_assign_ref( &mut self, other: &Rational, rm: RoundingMode, ) -> Ordering

Multiplies a Float by a Rational in place, rounding the result with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded product is less than, equal to, or greater than the exact product. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $xy$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::mul_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::mul_rational_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using *= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "1.047197551196597");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "1.047197551196598");

let mut x = Float::from(PI);
assert_eq!(
    x.mul_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Nearest),
    Greater
);
assert_eq!(x.to_string(), "1.047197551196598");

impl Float

pub fn power_of_2_prec_round( pow: i64, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Raises 2 to an integer power, returning a Float with the specified precision and with the specified rounding mode. An Ordering is also returned, indicating whether the returned power is less than, equal to, or greater than the exact power. The ordering is usually Equal, but is Less or Greater if overflow or underflow occurs.

$f(k) = 2^k$, and the result has precision prec.

• If pow is greater than $2^{30}-2$ and rm is Floor or Down, the largest representable Float with the given precision is returned.
• If pow is greater than $2^{30}-2$ and rm is Ceiling or Up, or Nearest, $\infty$ is returned.
• If pow is less than $-2^{30}$ and rm is Floor, Down, or Nearest, positive zero is returned.
• If pow is less than $-2^{30}$ and rm is Ceiling or Up, the smallest positive Float is returned.

If you want the behavior of Nearest (that is, returning $\infty$ on overflow and positive zero on underflow), you can use Float::power_of_2_prec instead.

If you need a Float with precision 1, then the PowerOf2 implementation may be used instead.

Panics

Panics if prec is zero, or if rm is exact and pow is greater than $2^{30}-2$ or less than $-2^{30}$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (p, o) = Float::power_of_2_prec_round(0, 1, Nearest);
assert_eq!(p.to_string(), "1.0");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec_round(0, 100, Nearest);
assert_eq!(p.to_string(), "1.0");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec_round(100, 1, Nearest);
assert_eq!(p.to_string(), "1.0e30");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec_round(100, 100, Nearest);
assert_eq!(p.to_string(), "1267650600228229401496703205376.0");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec_round(-100, 1, Nearest);
assert_eq!(p.to_string(), "8.0e-31");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec_round(-100, 100, Nearest);
assert_eq!(p.to_string(), "7.88860905221011805411728565283e-31");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec_round(i64::power_of_2(30) - 1, 10, Floor);
assert_eq!(p.to_string(), "too_big");
assert_eq!(o, Less);

let (p, o) = Float::power_of_2_prec_round(i64::power_of_2(30) - 1, 10, Ceiling);
assert_eq!(p.to_string(), "Infinity");
assert_eq!(o, Greater);

let (p, o) = Float::power_of_2_prec_round(-i64::power_of_2(30) - 1, 10, Floor);
assert_eq!(p.to_string(), "0.0");
assert_eq!(o, Less);

let (p, o) = Float::power_of_2_prec_round(-i64::power_of_2(30) - 1, 10, Ceiling);
assert_eq!(p.to_string(), "too_small");
assert_eq!(o, Greater);

pub fn power_of_2_prec(pow: i64, prec: u64) -> (Self, Ordering)

Raises 2 to an integer power, returning a Float with the specified precision. An Ordering is also returned, indicating whether the returned power is less than, equal to, or greater than the exact power. The ordering is usually Equal, but is Greater in the case of overflow and Less in the case of underflow.

$f(k) = 2^k$, and the result has precision prec.

If pow is greater than $2^{30}-2$, $\infty$ is returned. If pow is less than $-2^{30}$, positive zero is returned. If you want different overflow and underflow behavior, try using Float::power_of_2_prec_round instead.

If you need a Float with precision 1, then the PowerOf2 implementation may be used instead.

Panics

Panics if prec is zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (p, o) = Float::power_of_2_prec(0, 1);
assert_eq!(p.to_string(), "1.0");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec(0, 100);
assert_eq!(p.to_string(), "1.0");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec(100, 1);
assert_eq!(p.to_string(), "1.0e30");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec(100, 100);
assert_eq!(p.to_string(), "1267650600228229401496703205376.0");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec(-100, 1);
assert_eq!(p.to_string(), "8.0e-31");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec(-100, 100);
assert_eq!(p.to_string(), "7.88860905221011805411728565283e-31");
assert_eq!(o, Equal);

let (p, o) = Float::power_of_2_prec(i64::power_of_2(30) - 1, 10);
assert_eq!(p.to_string(), "Infinity");
assert_eq!(o, Greater);

let (p, o) = Float::power_of_2_prec(-i64::power_of_2(30) - 1, 10);
assert_eq!(p.to_string(), "0.0");
assert_eq!(o, Less);

impl Float

pub fn reciprocal_prec_round( self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Takes the reciprocal of a Float, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by value. An Ordering is also returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,p,m) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p+1}$.
• If $1/x$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p,m)=\text{NaN}$
• $f(\infty,p,m)=0.0$
• $f(-\infty,p,m)=-0.0$
• $f(0.0,p,m)=\infty$
• $f(-0.0,p,m)=-\infty$

Overflow:

• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.

This function cannot underflow.

If you know you’ll be using Nearest, consider using Float::reciprocal_prec instead. If you know that your target precision is the precision of the input, consider using Float::reciprocal_round instead. If both of these things are true, consider using Float::reciprocal instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact reciprocation.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round(5, Floor);
assert_eq!(reciprocal.to_string(), "0.31");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round(5, Ceiling);
assert_eq!(reciprocal.to_string(), "0.33");
assert_eq!(o, Greater);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round(5, Nearest);
assert_eq!(reciprocal.to_string(), "0.31");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round(20, Floor);
assert_eq!(reciprocal.to_string(), "0.3183098");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round(20, Ceiling);
assert_eq!(reciprocal.to_string(), "0.3183103");
assert_eq!(o, Greater);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round(20, Nearest);
assert_eq!(reciprocal.to_string(), "0.3183098");
assert_eq!(o, Less);

pub fn reciprocal_prec_round_ref( &self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Takes the reciprocal of a Float, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by reference. An Ordering is also returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,p,m) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p+1}$.
• If $1/x$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p,m)=\text{NaN}$
• $f(\infty,p,m)=0.0$
• $f(-\infty,p,m)=-0.0$
• $f(0.0,p,m)=\infty$
• $f(-0.0,p,m)=-\infty$

Overflow:

• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.

This function cannot underflow.

If you know you’ll be using Nearest, consider using Float::reciprocal_prec_ref instead. If you know that your target precision is the precision of the input, consider using Float::reciprocal_round_ref instead. If both of these things are true, consider using (&Float)::reciprocal() instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact reciprocation.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round_ref(5, Floor);
assert_eq!(reciprocal.to_string(), "0.31");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round_ref(5, Ceiling);
assert_eq!(reciprocal.to_string(), "0.33");
assert_eq!(o, Greater);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round_ref(5, Nearest);
assert_eq!(reciprocal.to_string(), "0.31");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round_ref(20, Floor);
assert_eq!(reciprocal.to_string(), "0.3183098");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round_ref(20, Ceiling);
assert_eq!(reciprocal.to_string(), "0.3183103");
assert_eq!(o, Greater);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_round_ref(20, Nearest);
assert_eq!(reciprocal.to_string(), "0.3183098");
assert_eq!(o, Less);

pub fn reciprocal_prec(self, prec: u64) -> (Self, Ordering)

Takes the reciprocal of a Float, rounding the result to the nearest value of the specified precision. The Float is taken by value. An Ordering is also returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the reciprocal is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,p) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p)=\text{NaN}$
• $f(\infty,p)=0.0$
• $f(-\infty,p)=-0.0$
• $f(0.0,p)=\infty$
• $f(-0.0,p)=-\infty$

Overflow:

• If $f(x,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.

This function cannot underflow.

If you want to use a rounding mode other than Nearest, consider using Float::reciprocal_prec_round instead. If you know that your target precision is the precision of the input, consider using Float::reciprocal instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (reciprocal, o) = Float::from(PI).reciprocal_prec(5);
assert_eq!(reciprocal.to_string(), "0.31");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec(20);
assert_eq!(reciprocal.to_string(), "0.3183098");
assert_eq!(o, Less);

pub fn reciprocal_prec_ref(&self, prec: u64) -> (Self, Ordering)

Takes the reciprocal of a Float, rounding the result to the nearest value of the specified precision. The Float is taken by reference. An Ordering is also returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the reciprocal is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,p) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p)=\text{NaN}$
• $f(\infty,p)=0.0$
• $f(-\infty,p)=-0.0$
• $f(0.0,p)=\infty$
• $f(-0.0,p)=-\infty$

Overflow:

• If $f(x,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.

This function cannot underflow.

If you want to use a rounding mode other than Nearest, consider using Float::reciprocal_prec_round_ref instead. If you know that your target precision is the precision of the input, consider using (&Float)::reciprocal() instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (reciprocal, o) = Float::from(PI).reciprocal_prec_ref(5);
assert_eq!(reciprocal.to_string(), "0.31");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_prec_ref(20);
assert_eq!(reciprocal.to_string(), "0.3183098");
assert_eq!(o, Less);

pub fn reciprocal_round(self, rm: RoundingMode) -> (Self, Ordering)

Takes the reciprocal of a Float, rounding the result with the specified rounding mode. The Float is taken by value. An Ordering is also returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p+1}$, where $p$ is the precision of the input.
• If $1/x$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$, where $p$ is the precision of the input.

If the output has a precision, it is the precision of the input.

Special cases:

• $f(\text{NaN},m)=\text{NaN}$
• $f(\infty,m)=0.0$
• $f(-\infty,m)=-0.0$
• $f(0.0,m)=\infty$
• $f(-0.0,m)=-\infty$

Overflow:

• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.

This function cannot underflow.

If you want to specify an output precision, consider using Float::reciprocal_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using Float::reciprocal instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact but the precision of the input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (reciprocal, o) = Float::from(PI).reciprocal_round(Floor);
assert_eq!(reciprocal.to_string(), "0.3183098861837905");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_round(Ceiling);
assert_eq!(reciprocal.to_string(), "0.318309886183791");
assert_eq!(o, Greater);

let (reciprocal, o) = Float::from(PI).reciprocal_round(Nearest);
assert_eq!(reciprocal.to_string(), "0.3183098861837905");
assert_eq!(o, Less);

pub fn reciprocal_round_ref(&self, rm: RoundingMode) -> (Self, Ordering)

Takes the reciprocal of a Float, rounding the result with the specified rounding mode. The Float is taken by reference. An Ordering is also returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p+1}$, where $p$ is the precision of the input.
• If $1/x$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$, where $p$ is the precision of the input.

If the output has a precision, it is the precision of the input.

Special cases:

• $f(\text{NaN},m)=\text{NaN}$
• $f(\infty,m)=0.0$
• $f(-\infty,m)=-0.0$
• $f(0.0,m)=\infty$
• $f(-0.0,m)=-\infty$

Overflow:

• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.

This function cannot underflow.

If you want to specify an output precision, consider using Float::reciprocal_prec_round_ref instead. If you know you’ll be using the Nearest rounding mode, consider using (&Float)::reciprocal() instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact but the precision of the input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (reciprocal, o) = Float::from(PI).reciprocal_round_ref(Floor);
assert_eq!(reciprocal.to_string(), "0.3183098861837905");
assert_eq!(o, Less);

let (reciprocal, o) = Float::from(PI).reciprocal_round_ref(Ceiling);
assert_eq!(reciprocal.to_string(), "0.318309886183791");
assert_eq!(o, Greater);

let (reciprocal, o) = Float::from(PI).reciprocal_round_ref(Nearest);
assert_eq!(reciprocal.to_string(), "0.3183098861837905");
assert_eq!(o, Less);

pub fn reciprocal_prec_round_assign( &mut self, prec: u64, rm: RoundingMode, ) -> Ordering

Takes the reciprocal of a Float in place, rounding the result to the specified precision and with the specified rounding mode. An Ordering is returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $1/x$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::reciprocal_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::reciprocal_prec_assign instead. If you know that your target precision is the precision of the input, consider using Float::reciprocal_round_assign instead. If both of these things are true, consider using Float::reciprocal_assign instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact reciprocation;

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_round_assign(5, Floor), Less);
assert_eq!(x.to_string(), "0.31");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_round_assign(5, Ceiling), Greater);
assert_eq!(x.to_string(), "0.33");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_round_assign(5, Nearest), Less);
assert_eq!(x.to_string(), "0.31");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_round_assign(20, Floor), Less);
assert_eq!(x.to_string(), "0.3183098");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_round_assign(20, Ceiling), Greater);
assert_eq!(x.to_string(), "0.3183103");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_round_assign(20, Nearest), Less);
assert_eq!(x.to_string(), "0.3183098");

pub fn reciprocal_prec_assign(&mut self, prec: u64) -> Ordering

Takes the reciprocal of a Float in place, rounding the result to the nearest value of the specified precision. An Ordering is returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the reciprocal is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::reciprocal_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::reciprocal_prec_round_assign instead. If you know that your target precision is the precision of the input, consider using Float::reciprocal instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_assign(5), Less);
assert_eq!(x.to_string(), "0.31");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_prec_assign(20), Less);
assert_eq!(x.to_string(), "0.3183098");

pub fn reciprocal_round_assign(&mut self, rm: RoundingMode) -> Ordering

Takes the reciprocal of a Float in place, rounding the result with the specified rounding mode. An Ordering is returned, indicating whether the rounded reciprocal is less than, equal to, or greater than the exact reciprocal. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input. See RoundingMode for a description of the possible rounding modes.

$$ x \gets 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $1/x$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the precision of the input.

See the Float::reciprocal_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::reciprocal_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using Float::reciprocal_assign instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact but the precision of the input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_round_assign(Floor), Less);
assert_eq!(x.to_string(), "0.3183098861837905");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_round_assign(Ceiling), Greater);
assert_eq!(x.to_string(), "0.318309886183791");

let mut x = Float::from(PI);
assert_eq!(x.reciprocal_round_assign(Nearest), Less);
assert_eq!(x.to_string(), "0.3183098861837905");

impl Float

pub fn shl_prec_round<T: PrimitiveInt>( self, bits: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and precision, and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p,m) = x2^k. $$

• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is prec.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::shl_prec instead. If you know that your target precision is the precision of the input, consider using Float::shl_round instead. If both of these things are true, or you don’t care about overflow or underflow behavior, consider using << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact and the result overflows or underflows, or cannot be expressed exactly with the specified precision.

Examples

See here.

pub fn shl_prec_round_ref<T: PrimitiveInt>( &self, bits: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and precision, and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p,m) = x2^k. $$

• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is prec.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::shl_prec_ref instead. If you know that your target precision is the precision of the input, consider using Float::shl_round instead. If both of these things are true, or you don’t care about overflow or underflow behavior, consider using << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact and the result overflows or underflows, or cannot be expressed exactly with the specified precision.

Examples

See here.

pub fn shl_prec_round_assign<T: PrimitiveInt>( &mut self, bits: T, prec: u64, rm: RoundingMode, ) -> Ordering

Left-shifts a Float (multiplies it by a power of 2) in place, rounding the result with the specified rounding mode and precision.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ x \gets x2^k. $$

• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is prec.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::shl_prec_assign instead. If you know that your target precision is the precision of the input, consider using Float::shl_round_assign instead. If both of these things are true, or you don’t care about overflow or underflow behavior, consider using <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact and the result overflows or underflows, or cannot be expressed exactly with the specified precision.

Examples

See here.

pub fn shl_prec<T: PrimitiveInt>(self, bits: T, prec: u64) -> (Self, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified precision, and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p) = x2^k. $$

• If $f(x,k,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you know that your target precision is the precision of the input, or you don’t care about overflow or underflow behavior, consider using << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

pub fn shl_prec_ref<T: PrimitiveInt>( &self, bits: T, prec: u64, ) -> (Self, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified precision, and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p) = x2^k. $$

• If $f(x,k,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you know that your target precision is the precision of the input, or you don’t care about overflow or underflow behavior, consider using << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

pub fn shl_prec_assign<T: PrimitiveInt>( &mut self, bits: T, prec: u64, ) -> Ordering

Left-shifts a Float (multiplies it by a power of 2) in place, rounding the result with the specified precision.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ x \gets x2^k. $$

• If $f(x,k,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you know that your target precision is the precision of the input, or you don’t care about overflow or underflow behavior, consider using <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

impl Float

pub fn shr_prec_round<T: PrimitiveInt>( self, bits: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and precision, and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p,m) = x/2^k. $$

• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is prec.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::shr_prec instead. If you know that your target precision is the precision of the input, consider using Float::shr_round instead. If both of these things are true, or you don’t care about overflow or underflow behavior, consider using >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact and the result overflows or underflows, or cannot be expressed exactly with the specified precision.

Examples

See here.

pub fn shr_prec_round_ref<T: PrimitiveInt>( &self, bits: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and precision, and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p,m) = x/2^k. $$

• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is prec.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::shr_prec_ref instead. If you know that your target precision is the precision of the input, consider using Float::shr_round instead. If both of these things are true, or you don’t care about overflow or underflow behavior, consider using >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact and the result overflows or underflows, or cannot be expressed exactly with the specified precision.

Examples

See here.

pub fn shr_prec_round_assign<T: PrimitiveInt>( &mut self, bits: T, prec: u64, rm: RoundingMode, ) -> Ordering

Right-shifts a Float (divides it by a power of 2) in place, rounding the result with the specified rounding mode and precision.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ x \gets x/2^k. $$

• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is prec.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::shr_prec_assign instead. If you know that your target precision is the precision of the input, consider using Float::shr_round_assign instead. If both of these things are true, or you don’t care about overflow or underflow behavior, consider using >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact and the result overflows or underflows, or cannot be expressed exactly with the specified precision.

Examples

See here.

pub fn shr_prec<T: PrimitiveInt>(self, bits: T, prec: u64) -> (Self, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified precision, and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p) = x/2^k. $$

• If $f(x,k,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you know that your target precision is the precision of the input, or you don’t care about overflow or underflow behavior, consider using >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

pub fn shr_prec_ref<T: PrimitiveInt>( &self, bits: T, prec: u64, ) -> (Self, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified precision, and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ f(x,k,p) = x/2^k. $$

• If $f(x,k,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you know that your target precision is the precision of the input, or you don’t care about overflow or underflow behavior, consider using >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

pub fn shr_prec_assign<T: PrimitiveInt>( &mut self, bits: T, prec: u64, ) -> Ordering

Right-shifts a Float (divides it by a power of 2) in place, rounding the result with the specified precision.

NaN, infinities, and zeros are unchanged. If the output has a precision, it is prec.

$$ x \gets x/2^k. $$

• If $f(x,k,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you know that your target precision is the precision of the input, or you don’t care about overflow or underflow behavior, consider using >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

impl Float

pub fn square_prec_round(self, prec: u64, rm: RoundingMode) -> (Self, Ordering)

Squares a Float, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by value. An Ordering is also returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,p,m) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p+1}$.
• If $x^2$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p,m)=\text{NaN}$
• $f(\pm\infty,p,m)=\infty$
• $f(\pm0.0,p,m)=0.0$

Overflow and underflow:

• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::square_prec instead. If you know that your target precision is the precision of the input, consider using Float::square_round instead. If both of these things are true, consider using Float::square instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is self.significant_bits(), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact squaring.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (square, o) = Float::from(PI).square_prec_round(5, Floor);
assert_eq!(square.to_string(), "9.5");
assert_eq!(o, Less);

let (square, o) = Float::from(PI).square_prec_round(5, Ceiling);
assert_eq!(square.to_string(), "10.0");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_round(5, Nearest);
assert_eq!(square.to_string(), "10.0");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_round(20, Floor);
assert_eq!(square.to_string(), "9.8696");
assert_eq!(o, Less);

let (square, o) = Float::from(PI).square_prec_round(20, Ceiling);
assert_eq!(square.to_string(), "9.86961");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_round(20, Nearest);
assert_eq!(square.to_string(), "9.8696");
assert_eq!(o, Less);

pub fn square_prec_round_ref( &self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Squares a Float, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by reference. An Ordering is also returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,p,m) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p+1}$.
• If $x^2$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p,m)=\text{NaN}$
• $f(\pm\infty,p,m)=\infty$
• $f(\pm0.0,p,m)=0.0$

Overflow and underflow:

• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::square_prec_ref instead. If you know that your target precision is the precision of the input, consider using Float::square_round_ref instead. If both of these things are true, consider using (&Float).square()instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is self.significant_bits(), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact squaring.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (square, o) = Float::from(PI).square_prec_round_ref(5, Floor);
assert_eq!(square.to_string(), "9.5");
assert_eq!(o, Less);

let (square, o) = Float::from(PI).square_prec_round_ref(5, Ceiling);
assert_eq!(square.to_string(), "10.0");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_round_ref(5, Nearest);
assert_eq!(square.to_string(), "10.0");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_round_ref(20, Floor);
assert_eq!(square.to_string(), "9.8696");
assert_eq!(o, Less);

let (square, o) = Float::from(PI).square_prec_round_ref(20, Ceiling);
assert_eq!(square.to_string(), "9.86961");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_round_ref(20, Nearest);
assert_eq!(square.to_string(), "9.8696");
assert_eq!(o, Less);

pub fn square_prec(self, prec: u64) -> (Self, Ordering)

Squares a Float, rounding the result to the nearest value of the specified precision. The Float is taken by value. An Ordering is also returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the square is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,p) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p)=\text{NaN}$
• $f(\pm\infty,p)=\infty$
• $f(\pm0.0,p)=0.0$

Overflow and underflow:

• If $f(x,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::square_prec_round instead. If you know that your target precision is the precision of the input, consider using Float::square instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is self.significant_bits(), and $m$ is prec.

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (square, o) = Float::from(PI).square_prec(5);
assert_eq!(square.to_string(), "10.0");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec(20);
assert_eq!(square.to_string(), "9.8696");
assert_eq!(o, Less);

pub fn square_prec_ref(&self, prec: u64) -> (Self, Ordering)

Squares a Float, rounding the result to the nearest value of the specified precision. The Float is taken by reference. An Ordering is also returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the square is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,p) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},p)=\text{NaN}$
• $f(\pm\infty,p)=\infty$
• $f(\pm0.0,p)=0.0$

Overflow and underflow:

• If $f(x,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::square_prec_round_ref instead. If you know that your target precision is the precision of the input, consider using (&Float).square() instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is self.significant_bits(), and $m$ is prec.

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (square, o) = Float::from(PI).square_prec_ref(5);
assert_eq!(square.to_string(), "10.0");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_prec_ref(20);
assert_eq!(square.to_string(), "9.8696");
assert_eq!(o, Less);

pub fn square_round(self, rm: RoundingMode) -> (Self, Ordering)

Squares a Float, rounding the result with the specified rounding mode. The Float is taken by value. An Ordering is also returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p+1}$, where $p$ is the precision of the input.
• If $x^2$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$, where $p$ is the precision of the input.

If the output has a precision, it is the precision of the input.

Special cases:

• $f(\text{NaN},m)=\text{NaN}$
• $f(\pm\infty,m)=\infty$
• $f(\pm0.0,m)=0.0$

Overflow and underflow:

• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::square_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using Float::square instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact but the precision of the input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (square, o) = Float::from(PI).square_round(Floor);
assert_eq!(square.to_string(), "9.86960440108935");
assert_eq!(o, Less);

let (square, o) = Float::from(PI).square_round(Ceiling);
assert_eq!(square.to_string(), "9.86960440108936");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_round(Nearest);
assert_eq!(square.to_string(), "9.86960440108936");
assert_eq!(o, Greater);

pub fn square_round_ref(&self, rm: RoundingMode) -> (Self, Ordering)

Squares a Float, rounding the result with the specified rounding mode. The Float is taken by reference. An Ordering is also returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p+1}$, where $p$ is the precision of the input.
• If $x^2$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$, where $p$ is the precision of the input.

If the output has a precision, it is the precision of the input.

Special cases:

• $f(\text{NaN},m)=\text{NaN}$
• $f(\pm\infty,m)=\infty$
• $f(\pm0.0,m)=0.0$

Overflow and underflow:

• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::square_prec_round_ref instead. If you know you’ll be using the Nearest rounding mode, consider using (&Float).square() instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact but the precision of the input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (square, o) = Float::from(PI).square_round_ref(Floor);
assert_eq!(square.to_string(), "9.86960440108935");
assert_eq!(o, Less);

let (square, o) = Float::from(PI).square_round_ref(Ceiling);
assert_eq!(square.to_string(), "9.86960440108936");
assert_eq!(o, Greater);

let (square, o) = Float::from(PI).square_round_ref(Nearest);
assert_eq!(square.to_string(), "9.86960440108936");
assert_eq!(o, Greater);

pub fn square_prec_round_assign( &mut self, prec: u64, rm: RoundingMode, ) -> Ordering

Squares a Float in place, rounding the result to the specified precision and with the specified rounding mode. An Ordering is returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p+1}$.
• If $x^2$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::square_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::square_prec_assign instead. If you know that your target precision is the precision of the input, consider using Float::square_round_assign instead. If both of these things are true, consider using Float::square_assign instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is self.significant_bits(), and $m$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact squaring;

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.square_prec_round_assign(5, Floor), Less);
assert_eq!(x.to_string(), "9.5");

let mut x = Float::from(PI);
assert_eq!(x.square_prec_round_assign(5, Ceiling), Greater);
assert_eq!(x.to_string(), "10.0");

let mut x = Float::from(PI);
assert_eq!(x.square_prec_round_assign(5, Nearest), Greater);
assert_eq!(x.to_string(), "10.0");

let mut x = Float::from(PI);
assert_eq!(x.square_prec_round_assign(20, Floor), Less);
assert_eq!(x.to_string(), "9.8696");

let mut x = Float::from(PI);
assert_eq!(x.square_prec_round_assign(20, Ceiling), Greater);
assert_eq!(x.to_string(), "9.86961");

let mut x = Float::from(PI);
assert_eq!(x.square_prec_round_assign(20, Nearest), Less);
assert_eq!(x.to_string(), "9.8696");

pub fn square_prec_assign(&mut self, prec: u64) -> Ordering

Squares a Float in place, rounding the result to the nearest value of the specified precision. An Ordering is returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the square is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::square_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::square_prec_round_assign instead. If you know that your target precision is the precision of the input, consider using Float::square instead.

Worst-case complexity

$T(n, m) = O(n \log n \log\log n + m)$

$M(n, m) = O(n \log n + m)$

where $T$ is time, $M$ is additional memory, $n$ is self.significant_bits(), and $m$ is prec.

Examples

use core::f64::consts::PI;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.square_prec_assign(5), Greater);
assert_eq!(x.to_string(), "10.0");

let mut x = Float::from(PI);
assert_eq!(x.square_prec_assign(20), Less);
assert_eq!(x.to_string(), "9.8696");

pub fn square_round_assign(&mut self, rm: RoundingMode) -> Ordering

Squares a Float in place, rounding the result with the specified rounding mode. An Ordering is returned, indicating whether the rounded square is less than, equal to, or greater than the exact square. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x^2$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the precision of the input.

See the Float::square_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::square_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using Float::square_assign instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if rm is Exact but the precision of the input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.square_round_assign(Floor), Less);
assert_eq!(x.to_string(), "9.86960440108935");

let mut x = Float::from(PI);
assert_eq!(x.square_round_assign(Ceiling), Greater);
assert_eq!(x.to_string(), "9.86960440108936");

let mut x = Float::from(PI);
assert_eq!(x.square_round_assign(Nearest), Greater);
assert_eq!(x.to_string(), "9.86960440108936");

impl Float

pub fn sub_prec_round( self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,\infty,p,m)=f(-\infty,-\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p,m)=0.0$
• $f(-0.0,0.0,p,m)=-0.0$
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_prec instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::sub_round instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_round(Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round(Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "0.44");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round(Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round(Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round(Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "0.4233112");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round(Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec_round_val_ref( self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,\infty,p,m)=f(-\infty,-\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p,m)=0.0$
• $f(-0.0,0.0,p,m)=-0.0$
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_prec_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::sub_round_val_ref instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_round_val_ref(&Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_val_ref(&Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "0.44");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round_val_ref(&Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_val_ref(&Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_val_ref(&Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "0.4233112");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round_val_ref(&Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec_round_ref_val( &self, other: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the specified precision and with the specified rounding mode. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,\infty,p,m)=f(-\infty,-\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p,m)=0.0$
• $f(-0.0,0.0,p,m)=-0.0$
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_prec_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::sub_round_ref_val instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_round_ref_val(Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_ref_val(Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "0.44");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round_ref_val(Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_ref_val(Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_ref_val(Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "0.4233112");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round_ref_val(Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec_round_ref_ref( &self, other: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the specified precision and with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=f(x,\text{NaN},p,m)=f(\infty,\infty,p,m)=f(-\infty,-\infty,p,m)= \text{NaN}$
• $f(\infty,x,p,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p,m)=0.0$
• $f(-0.0,0.0,p,m)=-0.0$
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p,m)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_prec_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::sub_round_ref_ref instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_round_ref_ref(&Float::from(E), 5, Floor);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_ref_ref(&Float::from(E), 5, Ceiling);
assert_eq!(sum.to_string(), "0.44");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round_ref_ref(&Float::from(E), 5, Nearest);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_ref_ref(&Float::from(E), 20, Floor);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_round_ref_ref(&Float::from(E), 20, Ceiling);
assert_eq!(sum.to_string(), "0.4233112");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_prec_round_ref_ref(&Float::from(E), 20, Nearest);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec(self, other: Self, prec: u64) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,\infty,p)=f(-\infty,-\infty,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p)=0.0$
• $f(-0.0,0.0,p)=-0.0$
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_round instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec(Float::from(E), 5);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec(Float::from(E), 20);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec_val_ref(self, other: &Self, prec: u64) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by value and the second by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,\infty,p)=f(-\infty,-\infty,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p)=0.0$
• $f(-0.0,0.0,p)=-0.0$
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_round_val_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_val_ref(&Float::from(E), 5);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_val_ref(&Float::from(E), 20);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec_ref_val(&self, other: Self, prec: u64) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the nearest value of the specified precision. The first Float is taken by reference and the second by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,\infty,p)=f(-\infty,-\infty,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p)=0.0$
• $f(-0.0,0.0,p)=-0.0$
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_round_ref_val instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_ref_val(Float::from(E), 5);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_ref_val(Float::from(E), 20);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_prec_ref_ref(&self, other: &Self, prec: u64) -> (Self, Ordering)

Subtracts two Floats, rounding the result to the nearest value of the specified precision. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=f(x,\text{NaN},p)=f(\infty,\infty,p)=f(-\infty,-\infty,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,p)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,p)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,p)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,p)=0.0$
• $f(-0.0,0.0,p)=-0.0$
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,p)=f(-0.0,-0.0,p,m)=-0.0$ if $m$ is Floor
• $f(x,x,p)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,p)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_round_ref_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_prec_ref_ref(&Float::from(E), 5);
assert_eq!(sum.to_string(), "0.42");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_prec_ref_ref(&Float::from(E), 20);
assert_eq!(sum.to_string(), "0.4233108");
assert_eq!(o, Less);

pub fn sub_round(self, other: Self, rm: RoundingMode) -> (Self, Ordering)

Subtracts two Floats, rounding the result with the specified rounding mode. Both Floats are taken by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,\infty,m)=f(-\infty,-\infty,m)= \text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,m)=0.0$
• $f(-0.0,0.0,m)=-0.0$
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=-0.0$ if $m$ is Floor
• $f(x,x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_round(Float::from(-E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_round(Float::from(-E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_round(Float::from(-E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn sub_round_val_ref( self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,\infty,m)=f(-\infty,-\infty,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,m)=0.0$
• $f(-0.0,0.0,m)=-0.0$
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=-0.0$ if $m$ is Floor
• $f(x,x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_round_val_ref(&Float::from(-E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_round_val_ref(&Float::from(-E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_round_val_ref(&Float::from(-E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn sub_round_ref_val( &self, other: Self, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,\infty,m)=f(-\infty,-\infty,m)= \text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,m)=0.0$
• $f(-0.0,0.0,m)=-0.0$
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=-0.0$ if $m$ is Floor
• $f(x,x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is self.significant_bits().

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_round_ref_val(Float::from(-E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_round_ref_val(Float::from(-E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_round_ref_val(Float::from(-E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn sub_round_ref_ref( &self, other: &Self, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts two Floats, rounding the result with the specified rounding mode. Both Floats are taken by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

Special cases:

• $f(\text{NaN},x,m)=f(x,\text{NaN},m)=f(\infty,\infty,m)=f(-\infty,-\infty,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0,m)=0.0$
• $f(-0.0,0.0,m)=-0.0$
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=0.0$ if $m$ is not Floor
• $f(0.0,0.0,m)=f(-0.0,-0.0,m)=-0.0$ if $m$ is Floor
• $f(x,x,m)=0.0$ if $x$ is finite and nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is finite and nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_round_ref_ref(&Float::from(-E), Floor);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_round_ref_ref(&Float::from(-E), Ceiling);
assert_eq!(sum.to_string(), "5.859874482048839");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_round_ref_ref(&Float::from(-E), Nearest);
assert_eq!(sum.to_string(), "5.859874482048838");
assert_eq!(o, Less);

pub fn sub_prec_round_assign( &mut self, other: Self, prec: u64, rm: RoundingMode, ) -> Ordering

Subtracts a Float by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::sub_prec_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::sub_round_assign instead. If both of these things are true, consider using -= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_round_assign(Float::from(E), 5, Floor), Less);
assert_eq!(x.to_string(), "0.42");

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_round_assign(Float::from(E), 5, Ceiling), Greater);
assert_eq!(x.to_string(), "0.44");

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_round_assign(Float::from(E), 5, Nearest), Less);
assert_eq!(x.to_string(), "0.42");

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_round_assign(Float::from(E), 20, Floor), Less);
assert_eq!(x.to_string(), "0.4233108");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_prec_round_assign(Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "0.4233112");

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_round_assign(Float::from(E), 20, Nearest), Less);
assert_eq!(x.to_string(), "0.4233108");

pub fn sub_prec_round_assign_ref( &mut self, other: &Self, prec: u64, rm: RoundingMode, ) -> Ordering

Subtracts a Float by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::sub_prec_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using Float::sub_round_assign instead. If both of these things are true, consider using -= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_round_assign_ref(&Float::from(E), 5, Floor), Less);
assert_eq!(x.to_string(), "0.42");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_prec_round_assign_ref(&Float::from(E), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "0.44");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_prec_round_assign_ref(&Float::from(E), 5, Nearest),
    Less
);
assert_eq!(x.to_string(), "0.42");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_prec_round_assign_ref(&Float::from(E), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "0.4233108");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_prec_round_assign_ref(&Float::from(E), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "0.4233112");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_prec_round_assign_ref(&Float::from(E), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "0.4233108");

pub fn sub_prec_assign(&mut self, other: Self, prec: u64) -> Ordering

Subtracts a Float by a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using -= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_assign(Float::from(E), 5), Less);
assert_eq!(x.to_string(), "0.42");

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_assign(Float::from(E), 20), Less);
assert_eq!(x.to_string(), "0.4233108");

pub fn sub_prec_assign_ref(&mut self, other: &Self, prec: u64) -> Ordering

Subtracts a Float by a Float in place, rounding the result to the nearest value of the specified precision. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_round_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using -= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use core::f64::consts::{E, PI};
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_assign_ref(&Float::from(E), 5), Less);
assert_eq!(x.to_string(), "0.42");

let mut x = Float::from(PI);
assert_eq!(x.sub_prec_assign_ref(&Float::from(E), 20), Less);
assert_eq!(x.to_string(), "0.4233108");

pub fn sub_round_assign(&mut self, other: Self, rm: RoundingMode) -> Ordering

Subtracts a Float by a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by value. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::sub_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::sub_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using -= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.sub_round_assign(Float::from(-E), Floor), Less);
assert_eq!(x.to_string(), "5.859874482048838");

let mut x = Float::from(PI);
assert_eq!(x.sub_round_assign(Float::from(-E), Ceiling), Greater);
assert_eq!(x.to_string(), "5.859874482048839");

let mut x = Float::from(PI);
assert_eq!(x.sub_round_assign(Float::from(-E), Nearest), Less);
assert_eq!(x.to_string(), "5.859874482048838");

pub fn sub_round_assign_ref( &mut self, other: &Self, rm: RoundingMode, ) -> Ordering

Subtracts a Float by a Float in place, rounding the result with the specified rounding mode. The Float on the right-hand side is taken by reference. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the maximum of the precision of the inputs. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the maximum precision of the inputs.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

If the output has a precision, it is the maximum of the precisions of the inputs.

See the Float::sub_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::sub_prec_round_assign_ref instead. If you know you’ll be using the Nearest rounding mode, consider using -= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Panics

Panics if rm is Exact but the maximum precision of the inputs is not high enough to represent the output.

Examples

use core::f64::consts::{E, PI};
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(x.sub_round_assign_ref(&Float::from(-E), Floor), Less);
assert_eq!(x.to_string(), "5.859874482048838");

let mut x = Float::from(PI);
assert_eq!(x.sub_round_assign_ref(&Float::from(-E), Ceiling), Greater);
assert_eq!(x.to_string(), "5.859874482048839");

let mut x = Float::from(PI);
assert_eq!(x.sub_round_assign_ref(&Float::from(-E), Nearest), Less);
assert_eq!(x.to_string(), "5.859874482048838");

pub fn sub_rational_prec_round( self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_rational_prec instead. If you know that your target precision is the precision of the Float input, consider using Float::sub_rational_round instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).sub_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Floor);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Ceiling);
assert_eq!(sum.to_string(), "2.9");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).sub_rational_prec_round(Rational::from_unsigneds(1u8, 3), 5, Nearest);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Floor);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Ceiling);
assert_eq!(sum.to_string(), "2.808262");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).sub_rational_prec_round(Rational::from_unsigneds(1u8, 3), 20, Nearest);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

pub fn sub_rational_prec_round_val_ref( self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_rational_prec_val_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::sub_rational_round_val_ref instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(sum.to_string(), "2.9");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(sum.to_string(), "2.808262");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_rational_prec_round_val_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

pub fn sub_rational_prec_round_ref_val( &self, other: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_rational_prec_ref_val instead. If you know that your target precision is the precision of the Float input, consider using Float::sub_rational_round_ref_val instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(sum.to_string(), "2.9");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(sum.to_string(), "2.808262");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_val(
    Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

pub fn sub_rational_prec_round_ref_ref( &self, other: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the specified precision and with the specified rounding mode. The Float and the Rational are both taken by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,p,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p,m)=\text{NaN}$
• $f(\infty,x,p,m)=\infty$
• $f(-\infty,x,p,m)=-\infty$
• $f(0.0,0,p,m)=0.0$
• $f(-0.0,0,p,m)=-0.0$
• $f(x,x,p,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,p,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,p,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,p,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you know you’ll be using Nearest, consider using Float::sub_rational_prec_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::sub_rational_round_ref_ref instead. If both of these things are true, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Floor,
);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Ceiling,
);
assert_eq!(sum.to_string(), "2.9");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    5,
    Nearest,
);
assert_eq!(sum.to_string(), "2.8");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Floor,
);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Ceiling,
);
assert_eq!(sum.to_string(), "2.808262");
assert_eq!(o, Greater);

let (sum, o) = Float::from(PI).sub_rational_prec_round_ref_ref(
    &Rational::from_unsigneds(1u8, 3),
    20,
    Nearest,
);
assert_eq!(sum.to_string(), "2.808258");
assert_eq!(o, Less);

pub fn sub_rational_prec(self, other: Rational, prec: u64) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_round instead. If you know that your target precision is the precision of the Float input, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec(Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "1.62");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec(Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "1.641592");
assert_eq!(o, Less);

pub fn sub_rational_prec_val_ref( self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_round_val_ref instead. If you know that your target precision is the precision of the Float input, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec_val_ref(&Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "1.62");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_val_ref(&Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "1.641592");
assert_eq!(o, Less);

pub fn sub_rational_prec_ref_val( &self, other: Rational, prec: u64, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_round_ref_val instead. If you know that your target precision is the precision of the Float input, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec_ref_val(Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "1.62");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_ref_val(Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "1.641592");
assert_eq!(o, Less);

pub fn sub_rational_prec_ref_ref( &self, other: &Rational, prec: u64, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result to the nearest value of the specified precision. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y,p) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

Special cases:

• $f(\text{NaN},x,p)=\text{NaN}$
• $f(\infty,x,p)=\infty$
• $f(-\infty,x,p)=-\infty$
• $f(0.0,0,p)=0.0$
• $f(-0.0,0,p)=-0.0$
• $f(x,x,p)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y,p)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y,p)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,p)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_round_ref_ref instead. If you know that your target precision is the precision of the Float input, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_prec_ref_ref(&Rational::exact_from(1.5), 5);
assert_eq!(sum.to_string(), "1.62");
assert_eq!(o, Less);

let (sum, o) = Float::from(PI).sub_rational_prec_ref_ref(&Rational::exact_from(1.5), 20);
assert_eq!(sum.to_string(), "1.641592");
assert_eq!(o, Less);

pub fn sub_rational_round( self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(x,0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x,m)=f(-0.0,x,m)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_rational_prec_round instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) = Float::from(PI).sub_rational_round(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "2.808259320256457");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_round(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).sub_rational_round(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

pub fn sub_rational_round_val_ref( self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result with the specified rounding mode. The Float is taken by value and the Rational by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(x,0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x,m)=f(-0.0,x,m)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_rational_prec_round_val_ref instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).sub_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "2.808259320256457");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).sub_rational_round_val_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

pub fn sub_rational_round_ref_val( &self, other: Rational, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result with the specified rounding mode. The Float is taken by reference and the Rational by value. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(x,0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x,m)=f(-0.0,x,m)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_rational_prec_round_ref_val instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).sub_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "2.808259320256457");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).sub_rational_round_ref_val(Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

pub fn sub_rational_round_ref_ref( &self, other: &Rational, rm: RoundingMode, ) -> (Self, Ordering)

Subtracts a Float by a Rational, rounding the result with the specified rounding mode. The Float and the Rational are both are taken by reference. An Ordering is also returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function returns a NaN it also returns Equal.

The precision of the output is the precision of the Float input. See RoundingMode for a description of the possible rounding modes.

$$ f(x,y,m) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the Float input.

Special cases:

• $f(\text{NaN},x,m)=\text{NaN}$
• $f(\infty,x,m)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x,m)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0,m)=0.0$
• $f(-0.0,0,m)=-0.0$
• $f(x,0,m)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x,m)=f(-0.0,x,m)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x,m)=0.0$ if $x$ is nonzero and $m$ is not Floor
• $f(x,x,m)=-0.0$ if $x$ is nonzero and $m$ is Floor

Overflow and underflow:

• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,y,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,y,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,y,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you want to specify an output precision, consider using Float::sub_rational_prec_round_ref_ref instead. If you know you’ll be using the Nearest rounding mode, consider using - instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the Float input is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (sum, o) =
    Float::from(PI).sub_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Floor);
assert_eq!(sum.to_string(), "2.808259320256457");
assert_eq!(o, Less);

let (sum, o) =
    Float::from(PI).sub_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Ceiling);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

let (sum, o) =
    Float::from(PI).sub_rational_round_ref_ref(&Rational::from_unsigneds(1u8, 3), Nearest);
assert_eq!(sum.to_string(), "2.808259320256461");
assert_eq!(o, Greater);

pub fn sub_rational_prec_round_assign( &mut self, other: Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Subtracts a Rational by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::sub_rational_prec_assign instead. If you know that your target precision is the precision of the Float input, consider using Float::sub_rational_round_assign instead. If both of these things are true, consider using -= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "2.8");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "2.9");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Less
);
assert_eq!(x.to_string(), "2.8");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "2.808258");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "2.808262");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign(Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "2.808258");

This is mpfr_sub_q from gmp_op.c, MPFR 4.2.0.

pub fn sub_rational_prec_round_assign_ref( &mut self, other: &Rational, prec: u64, rm: RoundingMode, ) -> Ordering

Subtracts a Rational by a Float in place, rounding the result to the specified precision and with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_rational_prec_round documentation for information on special cases, overflow, and underflow.

If you know you’ll be using Nearest, consider using Float::sub_rational_prec_assign_ref instead. If you know that your target precision is the precision of the Float input, consider using Float::sub_rational_round_assign_ref instead. If both of these things are true, consider using -= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Panics

Panics if rm is Exact but prec is too small for an exact subtraction.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Floor),
    Less
);
assert_eq!(x.to_string(), "2.8");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "2.9");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 5, Nearest),
    Less
);
assert_eq!(x.to_string(), "2.8");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Floor),
    Less
);
assert_eq!(x.to_string(), "2.808258");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Ceiling),
    Greater
);
assert_eq!(x.to_string(), "2.808262");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_round_assign_ref(&Rational::from_unsigneds(1u8, 3), 20, Nearest),
    Less
);
assert_eq!(x.to_string(), "2.808258");

pub fn sub_rational_prec_assign( &mut self, other: Rational, prec: u64, ) -> Ordering

Subtracts a Rational by a Float in place, rounding the result to the nearest value of the specified precision. The Rational is taken by value. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_rational_prec documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_round_assign instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using -= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_assign(Rational::exact_from(1.5), 5),
    Less
);
assert_eq!(x.to_string(), "1.62");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_assign(Rational::exact_from(1.5), 20),
    Less
);
assert_eq!(x.to_string(), "1.641592");

pub fn sub_rational_prec_assign_ref( &mut self, other: &Rational, prec: u64, ) -> Ordering

Subtracts a Rational by a Float in place, rounding the result to the nearest value of the specified precision. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$.

If the output has a precision, it is prec.

See the Float::sub_rational_prec_val_ref documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_round_assign_ref instead. If you know that your target precision is the maximum of the precisions of the two inputs, consider using -= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(other.significant_bits(), prec).

Examples

use core::f64::consts::PI;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_assign_ref(&Rational::exact_from(1.5), 5),
    Less
);
assert_eq!(x.to_string(), "1.62");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_prec_assign_ref(&Rational::exact_from(1.5), 20),
    Less
);
assert_eq!(x.to_string(), "1.641592");

pub fn sub_rational_round_assign( &mut self, other: Rational, rm: RoundingMode, ) -> Ordering

Subtracts a Rational by a Float in place, rounding the result with the specified rounding mode. The Rational is taken by value. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::sub_rational_round documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::sub_rational_prec_round_assign instead. If you know you’ll be using the Nearest rounding mode, consider using -= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_round_assign(Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "2.808259320256457");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_round_assign(Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "2.808259320256461");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_round_assign(Rational::from_unsigneds(1u8, 3), Nearest),
    Greater
);
assert_eq!(x.to_string(), "2.808259320256461");

pub fn sub_rational_round_assign_ref( &mut self, other: &Rational, rm: RoundingMode, ) -> Ordering

Subtracts a Rational by a Float in place, rounding the result with the specified rounding mode. The Rational is taken by reference. An Ordering is returned, indicating whether the rounded difference is less than, equal to, or greater than the exact difference. Although NaNs are not comparable to any Float, whenever this function sets the Float to NaN it also returns Equal.

The precision of the output is the precision of the input Float. See RoundingMode for a description of the possible rounding modes.

$$ x \gets x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, and $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p+1}$, where $p$ is the precision of the input Float.
• If $x-y$ is finite and nonzero, and $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

If the output has a precision, it is the precision of the input Float.

See the Float::sub_rational_round_val_ref documentation for information on special cases, overflow, and underflow.

If you want to specify an output precision, consider using Float::sub_rational_prec_round_assign_ref instead. If you know you’ll be using the Nearest rounding mode, consider using -= instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Panics

Panics if rm is Exact but the precision of the input Float is not high enough to represent the output.

Examples

use core::f64::consts::PI;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Floor),
    Less
);
assert_eq!(x.to_string(), "2.808259320256457");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Ceiling),
    Greater
);
assert_eq!(x.to_string(), "2.808259320256461");

let mut x = Float::from(PI);
assert_eq!(
    x.sub_rational_round_assign_ref(&Rational::from_unsigneds(1u8, 3), Nearest),
    Greater
);
assert_eq!(x.to_string(), "2.808259320256461");

impl Float

pub const fn is_nan(&self) -> bool

Determines whether a Float is NaN.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{NaN, One};
use malachite_float::Float;

assert_eq!(Float::NAN.is_nan(), true);
assert_eq!(Float::ONE.is_nan(), false);

pub const fn is_finite(&self) -> bool

Determines whether a Float is finite.

NaN is not finite.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, One};
use malachite_float::Float;

assert_eq!(Float::NAN.is_finite(), false);
assert_eq!(Float::INFINITY.is_finite(), false);
assert_eq!(Float::ONE.is_finite(), true);

pub const fn is_infinite(&self) -> bool

Determines whether a Float is infinite.

NaN is not infinite.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, One};
use malachite_float::Float;

assert_eq!(Float::NAN.is_infinite(), false);
assert_eq!(Float::INFINITY.is_infinite(), true);
assert_eq!(Float::ONE.is_infinite(), false);

pub const fn is_positive_zero(&self) -> bool

Determines whether a Float is positive zero.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.is_positive_zero(), false);
assert_eq!(Float::INFINITY.is_positive_zero(), false);
assert_eq!(Float::ONE.is_positive_zero(), false);
assert_eq!(Float::ZERO.is_positive_zero(), true);
assert_eq!(Float::NEGATIVE_ZERO.is_positive_zero(), false);

pub const fn is_negative_zero(&self) -> bool

Determines whether a Float is negative zero.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.is_negative_zero(), false);
assert_eq!(Float::INFINITY.is_negative_zero(), false);
assert_eq!(Float::ONE.is_negative_zero(), false);
assert_eq!(Float::ZERO.is_negative_zero(), false);
assert_eq!(Float::NEGATIVE_ZERO.is_negative_zero(), true);

pub const fn is_zero(&self) -> bool

Determines whether a Float is zero (positive or negative).

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.is_zero(), false);
assert_eq!(Float::INFINITY.is_zero(), false);
assert_eq!(Float::ONE.is_zero(), false);
assert_eq!(Float::ZERO.is_zero(), true);
assert_eq!(Float::NEGATIVE_ZERO.is_zero(), true);

pub const fn is_normal(&self) -> bool

Determines whether a Float is normal, that is, finite and nonzero.

There is no notion of subnormal Floats.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.is_normal(), false);
assert_eq!(Float::INFINITY.is_normal(), false);
assert_eq!(Float::ZERO.is_normal(), false);
assert_eq!(Float::NEGATIVE_ZERO.is_normal(), false);
assert_eq!(Float::ONE.is_normal(), true);

pub const fn is_sign_positive(&self) -> bool

Determines whether a Float’s sign is positive.

A NaN has no sign, so this function returns false when given a NaN.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::Float;

assert_eq!(Float::NAN.is_sign_positive(), false);
assert_eq!(Float::INFINITY.is_sign_positive(), true);
assert_eq!(Float::NEGATIVE_INFINITY.is_sign_positive(), false);
assert_eq!(Float::ZERO.is_sign_positive(), true);
assert_eq!(Float::NEGATIVE_ZERO.is_sign_positive(), false);
assert_eq!(Float::ONE.is_sign_positive(), true);
assert_eq!(Float::NEGATIVE_ONE.is_sign_positive(), false);

pub const fn is_sign_negative(&self) -> bool

Determines whether a Float’s sign is negative.

A NaN has no sign, so this function returns false when given a NaN.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::Float;

assert_eq!(Float::NAN.is_sign_negative(), false);
assert_eq!(Float::INFINITY.is_sign_negative(), false);
assert_eq!(Float::NEGATIVE_INFINITY.is_sign_negative(), true);
assert_eq!(Float::ZERO.is_sign_negative(), false);
assert_eq!(Float::NEGATIVE_ZERO.is_sign_negative(), true);
assert_eq!(Float::ONE.is_sign_negative(), false);
assert_eq!(Float::NEGATIVE_ONE.is_sign_negative(), true);

pub const fn classify(&self) -> FpCategory

Classifies a Float into one of several categories.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::Float;
use std::num::FpCategory;

assert_eq!(Float::NAN.classify(), FpCategory::Nan);
assert_eq!(Float::INFINITY.classify(), FpCategory::Infinite);
assert_eq!(Float::NEGATIVE_INFINITY.classify(), FpCategory::Infinite);
assert_eq!(Float::ZERO.classify(), FpCategory::Zero);
assert_eq!(Float::NEGATIVE_ZERO.classify(), FpCategory::Zero);
assert_eq!(Float::ONE.classify(), FpCategory::Normal);
assert_eq!(Float::NEGATIVE_ONE.classify(), FpCategory::Normal);

pub fn into_non_nan(self) -> Option<Self>

Turns a NaN into a None and wraps any non-NaN Float with a Some. The Float is taken by value.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.into_non_nan(), None);
assert_eq!(Float::INFINITY.into_non_nan(), Some(Float::INFINITY));
assert_eq!(Float::ZERO.into_non_nan(), Some(Float::ZERO));
assert_eq!(
    Float::NEGATIVE_ZERO.into_non_nan(),
    Some(Float::NEGATIVE_ZERO)
);
assert_eq!(Float::ONE.into_non_nan(), Some(Float::ONE));

pub fn to_non_nan(&self) -> Option<Self>

Turns a NaN into a None and wraps any non-NaN Float with a Some. The Float is taken by reference.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.to_non_nan(), None);
assert_eq!(Float::INFINITY.to_non_nan(), Some(Float::INFINITY));
assert_eq!(Float::ZERO.to_non_nan(), Some(Float::ZERO));
assert_eq!(
    Float::NEGATIVE_ZERO.to_non_nan(),
    Some(Float::NEGATIVE_ZERO)
);
assert_eq!(Float::ONE.to_non_nan(), Some(Float::ONE));

pub fn into_finite(self) -> Option<Self>

Turns any Float that’s NaN or infinite into a None and wraps any finite Float with a Some. The Float is taken by value.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.into_finite(), None);
assert_eq!(Float::INFINITY.into_finite(), None);
assert_eq!(Float::ZERO.into_finite(), Some(Float::ZERO));
assert_eq!(
    Float::NEGATIVE_ZERO.into_finite(),
    Some(Float::NEGATIVE_ZERO)
);
assert_eq!(Float::ONE.into_finite(), Some(Float::ONE));

pub fn to_finite(&self) -> Option<Self>

Turns any Float that’s NaN or infinite into a None and wraps any finite Float with a Some. The Float is taken by reference.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeZero, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.to_finite(), None);
assert_eq!(Float::INFINITY.to_finite(), None);
assert_eq!(Float::ZERO.to_finite(), Some(Float::ZERO));
assert_eq!(Float::NEGATIVE_ZERO.to_finite(), Some(Float::NEGATIVE_ZERO));
assert_eq!(Float::ONE.to_finite(), Some(Float::ONE));

impl Float

pub fn complexity(&self) -> u64

Determines a Float’s complexity. The complexity is defined as follows:

$$ f(\text{NaN}) = f(\pm\infty) = f(\pm 0.0) = 1, $$

and, if $x$ is finite and nonzero,

$$ f(x) = \max(|\lfloor \log_2 x\rfloor|, p), $$

where $p$ is the precision of $x$.

Informally, the complexity is proportional to the number of characters you would need to write the Float out without using exponents.

See also the Float implementation of SignificantBits.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::num::basic::traits::{NaN, One};
use malachite_float::Float;

assert_eq!(Float::NAN.complexity(), 1);
assert_eq!(Float::ONE.complexity(), 1);
assert_eq!(Float::one_prec(100).complexity(), 100);
assert_eq!(Float::from(std::f64::consts::PI).complexity(), 50);
assert_eq!(Float::power_of_2(100u64).complexity(), 100);
assert_eq!(Float::power_of_2(-100i64).complexity(), 100);

impl Float

pub const MIN_POSITIVE: Self

The minimum representable positive value, or $2^{-2^{30}}$, with precision 1.

pub fn min_positive_value_prec(prec: u64) -> Self

Returns the minimum representable positive value, or $2^{-2^{30}}$, with the given precision.

$$ f(p) = 2^{-2^{30}}, $$

and the output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;

assert_eq!(Float::min_positive_value_prec(1).to_string(), "too_small");
assert_eq!(Float::min_positive_value_prec(10).to_string(), "too_small");
assert_eq!(Float::min_positive_value_prec(100).to_string(), "too_small");

assert_eq!(Float::min_positive_value_prec(1).get_prec(), Some(1));
assert_eq!(Float::min_positive_value_prec(10).get_prec(), Some(10));
assert_eq!(Float::min_positive_value_prec(100).get_prec(), Some(100));

pub fn max_finite_value_with_prec(prec: u64) -> Self

There is no maximum finite Float, but there is one for any given precision. This function returns that Float.

$$ f(p) = (1-(1/2)^p)2^{2^{30}-1}, $$ where $p$ is the prec. The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;

assert_eq!(Float::max_finite_value_with_prec(1).to_string(), "too_big");
assert_eq!(Float::max_finite_value_with_prec(10).to_string(), "too_big");
assert_eq!(
    Float::max_finite_value_with_prec(100).to_string(),
    "too_big"
);

assert_eq!(Float::max_finite_value_with_prec(1).get_prec(), Some(1));
assert_eq!(Float::max_finite_value_with_prec(10).get_prec(), Some(10));
assert_eq!(Float::max_finite_value_with_prec(100).get_prec(), Some(100));

pub fn one_prec(prec: u64) -> Self

Returns the number 1, with the given precision.

$$ f(p) = 1, $$

and the output has precision $p$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;

assert_eq!(Float::one_prec(1), 1);
assert_eq!(Float::one_prec(10), 1);
assert_eq!(Float::one_prec(100), 1);

assert_eq!(Float::one_prec(1).get_prec(), Some(1));
assert_eq!(Float::one_prec(10).get_prec(), Some(10));
assert_eq!(Float::one_prec(100).get_prec(), Some(100));

pub fn two_prec(prec: u64) -> Self

Returns the number 2, with the given precision.

$$ f(p) = 2, $$

and the output has precision $p$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;

assert_eq!(Float::two_prec(1), 2);
assert_eq!(Float::two_prec(10), 2);
assert_eq!(Float::two_prec(100), 2);

assert_eq!(Float::two_prec(1).get_prec(), Some(1));
assert_eq!(Float::two_prec(10).get_prec(), Some(10));
assert_eq!(Float::two_prec(100).get_prec(), Some(100));

pub fn negative_one_prec(prec: u64) -> Self

Returns the number $-1$, with the given precision.

$$ f(p) = -1, $$

and the output has precision $p$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;

assert_eq!(Float::negative_one_prec(1), -1);
assert_eq!(Float::negative_one_prec(10), -1);
assert_eq!(Float::negative_one_prec(100), -1);

assert_eq!(Float::negative_one_prec(1).get_prec(), Some(1));
assert_eq!(Float::negative_one_prec(10).get_prec(), Some(10));
assert_eq!(Float::negative_one_prec(100).get_prec(), Some(100));

pub fn one_half_prec(prec: u64) -> Self

Returns the number 0.5, with the given precision.

$$ f(p) = 0.5, $$

and the output has precision $p$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;

assert_eq!(Float::one_half_prec(1), 0.5);
assert_eq!(Float::one_half_prec(10), 0.5);
assert_eq!(Float::one_half_prec(100), 0.5);

assert_eq!(Float::one_half_prec(1).get_prec(), Some(1));
assert_eq!(Float::one_half_prec(10).get_prec(), Some(10));
assert_eq!(Float::one_half_prec(100).get_prec(), Some(100));

impl Float

pub fn to_significand(&self) -> Option<Natural>

Gets the significand of a Float, taking the Float by value.

The significand is the smallest positive integer which is some power of 2 times the Float, and whose number of significant bits is a multiple of the limb width. If the Float is NaN, infinite, or zero, then None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

#[cfg(not(feature = "32_bit_limbs"))]
use malachite_base::num::arithmetic::traits::PowerOf2;
#[cfg(not(feature = "32_bit_limbs"))]
use malachite_base::num::basic::traits::One;
use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_float::Float;
#[cfg(not(feature = "32_bit_limbs"))]
use malachite_nz::natural::Natural;

assert_eq!(Float::NAN.to_significand(), None);
assert_eq!(Float::INFINITY.to_significand(), None);
assert_eq!(Float::ZERO.to_significand(), None);

#[cfg(not(feature = "32_bit_limbs"))]
{
    assert_eq!(Float::ONE.to_significand(), Some(Natural::power_of_2(63)));
    assert_eq!(
        Float::from(std::f64::consts::PI).to_significand().unwrap(),
        14488038916154245120u64
    );
}

pub fn into_significand(self) -> Option<Natural>

Gets the significand of a Float, taking the Float by reference.

The significand is the smallest positive integer which is some power of 2 times the Float, and whose number of significant bits is a multiple of the limb width. If the Float is NaN, infinite, or zero, then None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

#[cfg(not(feature = "32_bit_limbs"))]
use malachite_base::num::arithmetic::traits::PowerOf2;
#[cfg(not(feature = "32_bit_limbs"))]
use malachite_base::num::basic::traits::One;
use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_float::Float;
#[cfg(not(feature = "32_bit_limbs"))]
use malachite_nz::natural::Natural;

assert_eq!(Float::NAN.into_significand(), None);
assert_eq!(Float::INFINITY.into_significand(), None);
assert_eq!(Float::ZERO.into_significand(), None);

#[cfg(not(feature = "32_bit_limbs"))]
{
    assert_eq!(Float::ONE.into_significand(), Some(Natural::power_of_2(63)));
    assert_eq!(
        Float::from(std::f64::consts::PI)
            .into_significand()
            .unwrap(),
        14488038916154245120u64
    );
}

pub const fn significand_ref(&self) -> Option<&Natural>

Returns a reference to the significand of a Float.

The significand is the smallest positive integer which is some power of 2 times the Float, and whose number of significant bits is a multiple of the limb width. If the Float is NaN, infinite, or zero, then None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

#[cfg(not(feature = "32_bit_limbs"))]
use malachite_base::num::arithmetic::traits::PowerOf2;
#[cfg(not(feature = "32_bit_limbs"))]
use malachite_base::num::basic::traits::One;
use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_float::Float;
#[cfg(not(feature = "32_bit_limbs"))]
use malachite_nz::natural::Natural;

assert_eq!(Float::NAN.significand_ref(), None);
assert_eq!(Float::INFINITY.significand_ref(), None);
assert_eq!(Float::ZERO.significand_ref(), None);

#[cfg(not(feature = "32_bit_limbs"))]
{
    assert_eq!(
        *Float::ONE.significand_ref().unwrap(),
        Natural::power_of_2(63)
    );
    assert_eq!(
        *Float::from(std::f64::consts::PI).significand_ref().unwrap(),
        14488038916154245120u64
    );
}

pub const fn get_exponent(&self) -> Option<i32>

Returns a Float’s exponent.

$$ f(\text{NaN}) = f(\pm\infty) = f(\pm 0.0) = \text{None}, $$

and, if $x$ is finite and nonzero,

$$ f(x) = \operatorname{Some}(\lfloor \log_2 x \rfloor + 1). $$

The output is in the range $[-(2^{30}-1), 2^{30}-1]$.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::num::basic::traits::{Infinity, NaN, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.get_exponent(), None);
assert_eq!(Float::INFINITY.get_exponent(), None);
assert_eq!(Float::ZERO.get_exponent(), None);

assert_eq!(Float::ONE.get_exponent(), Some(1));
assert_eq!(Float::from(std::f64::consts::PI).get_exponent(), Some(2));
assert_eq!(Float::power_of_2(100u64).get_exponent(), Some(101));
assert_eq!(Float::power_of_2(-100i64).get_exponent(), Some(-99));

pub const fn get_prec(&self) -> Option<u64>

Returns a Float’s precision. The precision is a positive integer denoting how many of the Float’s bits are significant.

Only Floats that are finite and nonzero have a precision. For other Floats, None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.get_prec(), None);
assert_eq!(Float::INFINITY.get_prec(), None);
assert_eq!(Float::ZERO.get_prec(), None);

assert_eq!(Float::ONE.get_prec(), Some(1));
assert_eq!(Float::one_prec(100).get_prec(), Some(100));
assert_eq!(Float::from(std::f64::consts::PI).get_prec(), Some(50));

pub fn get_min_prec(&self) -> Option<u64>

Returns the minimum precision necessary to represent the given Float’s value.

For example, Float:one_prec(100) has a precision of 100, but its minimum precision is 1, because that’s all that’s necessary to represent the value 1.

The minimum precision is always less than or equal to the actual precision.

Only Floats that are finite and nonzero have a minimum precision. For other Floats, None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.get_min_prec(), None);
assert_eq!(Float::INFINITY.get_min_prec(), None);
assert_eq!(Float::ZERO.get_min_prec(), None);

assert_eq!(Float::ONE.get_min_prec(), Some(1));
assert_eq!(Float::one_prec(100).get_min_prec(), Some(1));
assert_eq!(Float::from(std::f64::consts::PI).get_min_prec(), Some(50));

pub fn set_prec_round(&mut self, prec: u64, rm: RoundingMode) -> Ordering

Changes a Float’s precision. If the precision decreases, rounding may be necessary, and will use the provided RoundingMode.

Returns an Ordering, indicating whether the final value is less than, greater than, or equal to the original value.

If the Float originally had the maximum exponent, it is possible for this function to overflow. This is even possible if rm is Nearest, even though infinity is never nearer to the exact result than any finite Float is. This is to match the behavior of MPFR.

This function never underflows.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or if rm is Exact but setting the desired precision requires rounding.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let original_x = Float::from(1.0f64 / 3.0);
assert_eq!(original_x.to_string(), "0.33333333333333331");
assert_eq!(original_x.get_prec(), Some(53));

let mut x = original_x.clone();
assert_eq!(x.set_prec_round(100, Exact), Equal);
assert_eq!(x.to_string(), "0.3333333333333333148296162562474");
assert_eq!(x.get_prec(), Some(100));

let mut x = original_x.clone();
assert_eq!(x.set_prec_round(10, Floor), Less);
assert_eq!(x.to_string(), "0.333");
assert_eq!(x.get_prec(), Some(10));

let mut x = original_x.clone();
assert_eq!(x.set_prec_round(10, Ceiling), Greater);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));

pub fn set_prec(&mut self, p: u64) -> Ordering

Changes a Float’s precision. If the precision decreases, rounding may be necessary, and Nearest will be used.

Returns an Ordering, indicating whether the final value is less than, greater than, or equal to the original value.

If the Float originally had the maximum exponent, it is possible for this function to overflow, even though infinity is never nearer to the exact result than any finite Float is. This is to match the behavior of MPFR.

This function never underflows.

To use a different rounding mode, try Float::set_prec_round.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

let original_x = Float::from(1.0f64 / 3.0);
assert_eq!(original_x.to_string(), "0.33333333333333331");
assert_eq!(original_x.get_prec(), Some(53));

let mut x = original_x.clone();
assert_eq!(x.set_prec(100), Equal);
assert_eq!(x.to_string(), "0.3333333333333333148296162562474");
assert_eq!(x.get_prec(), Some(100));

let mut x = original_x.clone();
assert_eq!(x.set_prec(10), Greater);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));

pub fn from_float_prec_round( x: Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Creates a Float from another Float, possibly with a different precision. If the precision decreases, rounding may be necessary, and will use the provided RoundingMode. The input Float is taken by value.

Returns an Ordering, indicating whether the final value is less than, greater than, or equal to the original value.

If the input Float has the maximum exponent, it is possible for this function to overflow. This is even possible if rm is Nearest, even though infinity is never nearer to the exact result than any finite Float is. This is to match the behavior of MPFR.

This function never underflows.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or if rm is Exact but setting the desired precision requires rounding.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let original_x = Float::from(1.0f64 / 3.0);
assert_eq!(original_x.to_string(), "0.33333333333333331");
assert_eq!(original_x.get_prec(), Some(53));

let (x, o) = Float::from_float_prec_round(original_x.clone(), 100, Exact);
assert_eq!(x.to_string(), "0.3333333333333333148296162562474");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Equal);

let (x, o) = Float::from_float_prec_round(original_x.clone(), 10, Floor);
assert_eq!(x.to_string(), "0.333");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) = Float::from_float_prec_round(original_x.clone(), 10, Ceiling);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

pub fn from_float_prec_round_ref( x: &Self, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Creates a Float from another Float, possibly with a different precision. If the precision decreases, rounding may be necessary, and will use the provided RoundingMode. The input Float is taken by reference.

Returns an Ordering, indicating whether the final value is less than, greater than, or equal to the original value.

If the input Float has the maximum exponent, it is possible for this function to overflow. This is even possible if rm is Nearest, even though infinity is never nearer to the exact result than any finite Float is. This is to match the behavior of MPFR.

This function never underflows.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or if rm is Exact but setting the desired precision requires rounding.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let original_x = Float::from(1.0f64 / 3.0);
assert_eq!(original_x.to_string(), "0.33333333333333331");
assert_eq!(original_x.get_prec(), Some(53));

let (x, o) = Float::from_float_prec_round_ref(&original_x, 100, Exact);
assert_eq!(x.to_string(), "0.3333333333333333148296162562474");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Equal);

let (x, o) = Float::from_float_prec_round_ref(&original_x, 10, Floor);
assert_eq!(x.to_string(), "0.333");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) = Float::from_float_prec_round_ref(&original_x, 10, Ceiling);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

pub fn from_float_prec(x: Self, prec: u64) -> (Self, Ordering)

Creates a Float from another Float, possibly with a different precision. If the precision decreases, rounding may be necessary, and will use Nearest. The input Float is taken by value.

Returns an Ordering, indicating whether the final value is less than, greater than, or equal to the original value.

If the Float originally had the maximum exponent, it is possible for this function to overflow, even though infinity is never nearer to the exact result than any finite Float is. This is to match the behavior of MPFR.

This function never underflows.

To use a different rounding mode, try Float::from_float_prec_round.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

let original_x = Float::from(1.0f64 / 3.0);
assert_eq!(original_x.to_string(), "0.33333333333333331");
assert_eq!(original_x.get_prec(), Some(53));

let (x, o) = Float::from_float_prec(original_x.clone(), 100);
assert_eq!(x.to_string(), "0.3333333333333333148296162562474");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Equal);

let (x, o) = Float::from_float_prec(original_x.clone(), 10);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

pub fn from_float_prec_ref(x: &Self, prec: u64) -> (Self, Ordering)

Creates a Float from another Float, possibly with a different precision. If the precision decreases, rounding may be necessary, and will use Nearest. The input Float is taken by reference.

Returns an Ordering, indicating whether the final value is less than, greater than, or equal to the original value.

If the Float originally had the maximum exponent, it is possible for this function to overflow, even though infinity is never nearer to the exact result than any finite Float is. This is to match the behavior of MPFR.

This function never underflows.

To use a different rounding mode, try Float::from_float_prec_round_ref.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

let original_x = Float::from(1.0f64 / 3.0);
assert_eq!(original_x.to_string(), "0.33333333333333331");
assert_eq!(original_x.get_prec(), Some(53));

let (x, o) = Float::from_float_prec_ref(&original_x, 100);
assert_eq!(x.to_string(), "0.3333333333333333148296162562474");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Equal);

let (x, o) = Float::from_float_prec_ref(&original_x, 10);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

impl Float

pub fn ulp(&self) -> Option<Self>

Gets a Float’s ulp (unit in last place, or unit of least precision).

If the Float is positive, its ulp is the distance to the next-largest Float with the same precision; if it is negative, the next-smallest. (This definition works even if the Float is the largest in its binade. If the Float is the largest in its binade and has the maximum exponent, we can define its ulp to be the distance to the next-smallest Float with the same precision if positive, and to the next-largest Float with the same precision if negative.)

If the Float is NaN, infinite, or zero, then None is returned.

This function does not overflow or underflow, technically. But it is possible that a Float’s ulp is too small to represent, for example if the Float has the minimum exponent and its precision is greater than 1, or if the precision is extremely large in general. In such cases, None is returned.

$$ f(\text{NaN}) = f(\pm\infty) = f(\pm 0.0) = \text{None}, $$

and, if $x$ is finite and nonzero,

$$ f(x) = \operatorname{Some}(2^{\lfloor \log_2 x \rfloor-p+1}), $$ where $p$ is the precision of $x$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeOne, One, Zero};
use malachite_float::Float;

assert_eq!(Float::NAN.ulp(), None);
assert_eq!(Float::INFINITY.ulp(), None);
assert_eq!(Float::ZERO.ulp(), None);

let s = Float::ONE.ulp().map(|x| x.to_string());
assert_eq!(s.as_ref().map(|s| s.as_str()), Some("1.0"));

let s = Float::one_prec(100).ulp().map(|x| x.to_string());
assert_eq!(s.as_ref().map(|s| s.as_str()), Some("2.0e-30"));

let s = Float::from(std::f64::consts::PI)
    .ulp()
    .map(|x| x.to_string());
assert_eq!(s.as_ref().map(|s| s.as_str()), Some("4.0e-15"));

let s = Float::power_of_2(100u64).ulp().map(|x| x.to_string());
assert_eq!(s.as_ref().map(|s| s.as_str()), Some("1.0e30"));

let s = Float::power_of_2(-100i64).ulp().map(|x| x.to_string());
assert_eq!(s.as_ref().map(|s| s.as_str()), Some("8.0e-31"));

let s = Float::NEGATIVE_ONE.ulp().map(|x| x.to_string());
assert_eq!(s.as_ref().map(|s| s.as_str()), Some("1.0"));

pub fn increment(&mut self)

Increments a Float by its ulp. See Float::ulp for details.

If the Float is positive and is the largest Float in its binade with its precision, then

• If its exponent is not the maximum exponent, it will become the smallest Float in the next-higher binade, and its precision will increase by 1 (so that its ulp remains the same);
• If its exponent is the maximum exponent, it will become $\infty$.

If the Float is negative and is closer to zero than any other Float in its binade with its precision, then

• If its precision is 1, it will become negative zero.
• If its precision is greater than 1 and its exponent is not the minimum exponent, it will become the farthest-from-zero Float in the next-lower binade, and its precision will decrease by 1 (so that its ulp remains the same).
• If its precision is greater than 1 and its exponent is the minimum exponent, it will become negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is NaN, infinite, or zero.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::num::basic::traits::{NegativeOne, One};
use malachite_float::Float;

let mut x = Float::ONE;
assert_eq!(x.to_string(), "1.0");
x.increment();
assert_eq!(x.to_string(), "2.0");

let mut x = Float::one_prec(100);
assert_eq!(x.to_string(), "1.0");
x.increment();
assert_eq!(x.to_string(), "1.000000000000000000000000000002");

let mut x = Float::from(std::f64::consts::PI);
assert_eq!(x.to_string(), "3.141592653589793");
x.increment();
assert_eq!(x.to_string(), "3.141592653589797");

let mut x = Float::power_of_2(100u64);
assert_eq!(x.to_string(), "1.0e30");
x.increment();
assert_eq!(x.to_string(), "3.0e30");

let mut x = Float::power_of_2(-100i64);
assert_eq!(x.to_string(), "8.0e-31");
x.increment();
assert_eq!(x.to_string(), "1.6e-30");

let mut x = Float::NEGATIVE_ONE;
assert_eq!(x.to_string(), "-1.0");
x.increment();
assert_eq!(x.to_string(), "-0.0");

pub fn decrement(&mut self)

Decrements a Float by its ulp. See Float::ulp for details.

If the Float is negative and is the largest Float in its binade with its precision, then

• If its exponent is not the maximum exponent, it will become the closest-to-zero Float in the next-higher binade, and its precision will increase by 1 (so that its ulp remains the same);
• If its exponent is the maximum exponent, it will become $-\infty$.

If the Float is positive and is smaller than any other Float in its binade with its precision, then

• If its precision is 1, it will become positive zero.
• If its precision is greater than 1 and its exponent is not the minimum exponent, it will become the largest Float in the next-lower binade, and its precision will decrease by 1 (so that its ulp remains the same).
• If its precision is greater than 1 and its exponent is the minimum exponent, it will become positive zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is NaN, infinite, or zero.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::num::basic::traits::{NegativeOne, One};
use malachite_float::Float;

let mut x = Float::ONE;
assert_eq!(x.to_string(), "1.0");
x.decrement();
assert_eq!(x.to_string(), "0.0");

let mut x = Float::one_prec(100);
assert_eq!(x.to_string(), "1.0");
x.decrement();
assert_eq!(x.to_string(), "0.999999999999999999999999999998");

let mut x = Float::from(std::f64::consts::PI);
assert_eq!(x.to_string(), "3.141592653589793");
x.decrement();
assert_eq!(x.to_string(), "3.14159265358979");

let mut x = Float::power_of_2(100u64);
assert_eq!(x.to_string(), "1.0e30");
x.decrement();
assert_eq!(x.to_string(), "0.0");

let mut x = Float::power_of_2(-100i64);
assert_eq!(x.to_string(), "8.0e-31");
x.decrement();
assert_eq!(x.to_string(), "0.0");

let mut x = Float::NEGATIVE_ONE;
assert_eq!(x.to_string(), "-1.0");
x.decrement();
assert_eq!(x.to_string(), "-2.0");

impl Float

pub fn log_2_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering)

Returns an approximation to the natural logarithm of 2, with the given precision and rounded using the given RoundingMode. An Ordering is also returned, indicating whether the rounded value is less than or greater than the exact value of the constant. (Since the constant is irrational, the rounded value is never equal to the exact value.)

$$ L = \log 2. $$

The constant is irrational.

The output has precision prec.

Worst-case complexity

$T(n) = O(n (\log n)^2 \log\log n)$

$M(n) = O(n (\log n)^2)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or if rm is Exact.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (l2, o) = Float::log_2_prec_round(100, Floor);
assert_eq!(l2.to_string(), "0.693147180559945309417232121458");
assert_eq!(o, Less);

let (l2, o) = Float::log_2_prec_round(100, Ceiling);
assert_eq!(l2.to_string(), "0.693147180559945309417232121459");
assert_eq!(o, Greater);

This is mpfr_const_log2_internal from const_log2.c, MPFR 4.2.0.

pub fn log_2_prec(prec: u64) -> (Self, Ordering)

Returns an approximation to the natural logarithm of 2, with the given precision and rounded to the nearest Float of that precision. An Ordering is also returned, indicating whether the rounded value is less than or greater than the exact value of the constant. (Since the constant is irrational, the rounded value is never equal to the exact value.)

$$ L = \log 2. $$

The constant is irrational.

The output has precision prec.

Worst-case complexity

$T(n) = O(n (\log n)^2 \log\log n)$

$M(n) = O(n (\log n)^2)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

let (l2, o) = Float::log_2_prec(1);
assert_eq!(l2.to_string(), "0.5");
assert_eq!(o, Less);

let (l2, o) = Float::log_2_prec(10);
assert_eq!(l2.to_string(), "0.693");
assert_eq!(o, Greater);

let (l2, o) = Float::log_2_prec(100);
assert_eq!(l2.to_string(), "0.693147180559945309417232121458");
assert_eq!(o, Less);

impl Float

pub fn prime_constant_prec_round( prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Returns an approximation to the prime constant, with the given precision and rounded using the given RoundingMode. An Ordering is also returned, indicating whether the rounded value is less than or greater than the exact value of the constant. (Since the constant is irrational, the rounded value is never equal to the exact value.)

The prime constant is the real number whose $n$th bit is prime if and only if $n$ is prime. That is, $$ P = \sum_{p\ text{prime}}2^{-p}. $$

The constant is irrational.

The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or if rm is Exact.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (pc, o) = Float::prime_constant_prec_round(100, Floor);
assert_eq!(pc.to_string(), "0.4146825098511116602481096221542");
assert_eq!(o, Less);

let (pc, o) = Float::prime_constant_prec_round(100, Ceiling);
assert_eq!(pc.to_string(), "0.4146825098511116602481096221546");
assert_eq!(o, Greater);

pub fn prime_constant_prec(prec: u64) -> (Self, Ordering)

Returns an approximation to the prime constant, with the given precision and rounded to the nearest Float of that precision. An Ordering is also returned, indicating whether the rounded value is less than or greater than the exact value of the constant. (Since the constant is irrational, the rounded value is never equal to the exact value.)

The prime constant is the real number whose $n$th bit is prime if and only if $n$ is prime. That is, $$ P = \sum_{p\ text{prime}}2^{-p}. $$

The constant is irrational.

The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

let (pc, o) = Float::prime_constant_prec(1);
assert_eq!(pc.to_string(), "0.5");
assert_eq!(o, Greater);

let (pc, o) = Float::prime_constant_prec(10);
assert_eq!(pc.to_string(), "0.4146");
assert_eq!(o, Less);

let (pc, o) = Float::prime_constant_prec(100);
assert_eq!(pc.to_string(), "0.4146825098511116602481096221542");
assert_eq!(o, Less);

impl Float

pub fn thue_morse_constant_prec_round( prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Returns an approximation to the Thue-Morse constant, with the given precision and rounded using the given RoundingMode. An Ordering is also returned, indicating whether the rounded value is less than or greater than the exact value of the constant. (Since the constant is irrational, the rounded value is never equal to the exact value.)

The Thue-Morse constant is the real number whose bits are the Thue-Morse sequence. That is, $$ \tau = \sum_{k=0}^\infty\frac{t_n}{2^{n+1}}, $$ where $t_n$ is the Thue-Morse sequence.

An alternative expression, from https://mathworld.wolfram.com/Thue-MorseConstant.html, is $$ \tau = \frac{1}{4}\left[2-\prod_{k=0}^\infty\left(1-\frac{1}{2^{2^k}}\right)\right]. $$

The constant is irrational and transcendental.

The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or if rm is Exact.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

let (tmc, o) = Float::thue_morse_constant_prec_round(100, Floor);
assert_eq!(tmc.to_string(), "0.4124540336401075977833613682584");
assert_eq!(o, Less);

let (tmc, o) = Float::thue_morse_constant_prec_round(100, Ceiling);
assert_eq!(tmc.to_string(), "0.4124540336401075977833613682588");
assert_eq!(o, Greater);

pub fn thue_morse_constant_prec(prec: u64) -> (Self, Ordering)

Returns an approximation to the Thue-Morse constant, with the given precision and rounded to the nearest Float of that precision. An Ordering is also returned, indicating whether the rounded value is less than or greater than the exact value of the constant. (Since the constant is irrational, the rounded value is never equal to the exact value.)

The Thue-Morse constant is the real number whose bits are the Thue-Morse sequence. That is, $$ \tau = \sum_{k=0}^\infty\frac{t_n}{2^{n+1}}, $$ where $t_n$ is the Thue-Morse sequence.

An alternative expression, from https://mathworld.wolfram.com/Thue-MorseConstant.html, is $$ \tau = \frac{1}{4}\left[2-\prod_{k=0}^\infty\left(1-\frac{1}{2^{2^k}}\right)\right]. $$

The constant is irrational and transcendental.

The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

let (tmc, o) = Float::thue_morse_constant_prec(1);
assert_eq!(tmc.to_string(), "0.5");
assert_eq!(o, Greater);

let (tmc, o) = Float::thue_morse_constant_prec(10);
assert_eq!(tmc.to_string(), "0.4126");
assert_eq!(o, Greater);

let (tmc, o) = Float::thue_morse_constant_prec(100);
assert_eq!(tmc.to_string(), "0.4124540336401075977833613682584");
assert_eq!(o, Less);

impl Float

pub fn non_dyadic_from_bits_prec_round<I: Iterator<Item = bool>>( bits: I, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Returns an approximation of a real number, given the number’s bits. To avoid troublesome edge cases, the number should not be a dyadic rational (and the iterator of bits should therefore be infinite, and not eventually 0 or 1). Given this assumption, the rounding mode Exact should never be passed.

The approximation has precision prec and is rounded according to the provided rounding mode.

This function reads prec + z bits, or prec + z + 1 bits if rm is Nearest, where z is the number of leading false bits in bits.

This function always produces a value in the interval $[1/2,1]$. In particular, it never overflows or underflows.

$$ f((x_k),p,m) = C+\varepsilon, $$ where $$ C=\sum_{k=0}^\infty x_k 2^{-(k+1)}. $$

• If $m$ is not Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 C\rfloor-p+1}$.
• If $m$ is Nearest, then $|\varepsilon| < 2^{\lfloor\log_2 C\rfloor-p}$.

The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero or rm is Exact.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use std::cmp::Ordering::*;

// Produces 10100100010000100000...
struct Bits {
    b: bool,
    k: usize,
    j: usize,
}

impl Iterator for Bits {
    type Item = bool;

    fn next(&mut self) -> Option<bool> {
        Some(if self.b {
            self.b = false;
            self.j = self.k;
            true
        } else {
            self.j -= 1;
            if self.j == 0 {
                self.k += 1;
                self.b = true;
            }
            false
        })
    }
}

impl Bits {
    fn new() -> Bits {
        Bits {
            b: true,
            k: 1,
            j: 1,
        }
    }
}

let (c, o) = Float::non_dyadic_from_bits_prec_round(Bits::new(), 100, Floor);
assert_eq!(c.to_string(), "0.6416325606551538662938427702254");
assert_eq!(o, Less);

let (c, o) = Float::non_dyadic_from_bits_prec_round(Bits::new(), 100, Ceiling);
assert_eq!(c.to_string(), "0.641632560655153866293842770226");
assert_eq!(o, Greater);

pub fn non_dyadic_from_bits_prec<I: Iterator<Item = bool>>( bits: I, prec: u64, ) -> (Self, Ordering)

Returns an approximation of a real number, given the number’s bits. To avoid troublesome edge cases, the number should not be a dyadic rational (and the iterator of bits should therefore be infinite, and not eventually 0 or 1).

The approximation has precision prec and is rounded according to the Nearest rounding mode.

This function reads prec + z + 1 bits, where z is the number of leading false bits in bits.

This function always produces a value in the interval $[1/2,1]$. In particular, it never overflows or underflows.

$$ f((x_k),p,m) = C+\varepsilon, $$ where $$ C=\sum_{k=0}^\infty x_k 2^{-(k+1)} $$ and $|\varepsilon| < 2^{\lfloor\log_2 C\rfloor-p}$.

The output has precision prec.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_float::Float;
use std::cmp::Ordering::*;

// Produces 10100100010000100000...
struct Bits {
    b: bool,
    k: usize,
    j: usize,
}

impl Iterator for Bits {
    type Item = bool;

    fn next(&mut self) -> Option<bool> {
        Some(if self.b {
            self.b = false;
            self.j = self.k;
            true
        } else {
            self.j -= 1;
            if self.j == 0 {
                self.k += 1;
                self.b = true;
            }
            false
        })
    }
}

impl Bits {
    fn new() -> Bits {
        Bits {
            b: true,
            k: 1,
            j: 1,
        }
    }
}

let (c, o) = Float::non_dyadic_from_bits_prec(Bits::new(), 1);
assert_eq!(c.to_string(), "0.5");
assert_eq!(o, Less);

let (c, o) = Float::non_dyadic_from_bits_prec(Bits::new(), 10);
assert_eq!(c.to_string(), "0.642");
assert_eq!(o, Less);

let (c, o) = Float::non_dyadic_from_bits_prec(Bits::new(), 100);
assert_eq!(c.to_string(), "0.6416325606551538662938427702254");
assert_eq!(o, Less);

impl Float

pub fn from_integer_prec_round( x: Integer, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts an Integer to a Float, taking the Integer by value. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_integer_prec instead.

• If the Integer rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$ if rm is Ceiling, Up, or Nearest, and rounds down to $(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.
• If the Integer rounds to a value less than or equal to $-2^{2^{30}-1}$), this function overflows to $-\infty$ if rm is Ceiling, Up, or Nearest, and rounds up to $-(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero, or if rm is exact and the Integer cannot be exactly represented with the specified precision.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_nz::integer::Integer;
use std::cmp::Ordering::*;

let (x, o) = Float::from_integer_prec_round(Integer::ZERO, 10, Exact);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_round(Integer::from(123), 20, Exact);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_round(Integer::from(123), 4, Floor);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_integer_prec_round(Integer::from(123), 4, Ceiling);
assert_eq!(x.to_string(), "1.3e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

let (x, o) = Float::from_integer_prec_round(Integer::from(-123), 20, Exact);
assert_eq!(x.to_string(), "-123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_round(Integer::from(-123), 4, Floor);
assert_eq!(x.to_string(), "-1.3e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_integer_prec_round(Integer::from(-123), 4, Ceiling);
assert_eq!(x.to_string(), "-1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

pub fn from_integer_prec_round_ref( x: &Integer, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts an Integer to a Float, taking the Integer by reference. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_integer_prec_ref instead.

• If the Integer rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$ if rm is Ceiling, Up, or Nearest, and rounds down to $(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.
• If the Integer rounds to a value less than or equal to $-2^{2^{30}-1}$), this function overflows to $-\infty$ if rm is Ceiling, Up, or Nearest, and rounds up to $-(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero, or if rm is exact and the Integer cannot be exactly represented with the specified precision.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_nz::integer::Integer;
use std::cmp::Ordering::*;

let (x, o) = Float::from_integer_prec_round_ref(&Integer::ZERO, 10, Exact);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_round_ref(&Integer::from(123), 20, Exact);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_round_ref(&Integer::from(123), 4, Floor);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_integer_prec_round_ref(&Integer::from(123), 4, Ceiling);
assert_eq!(x.to_string(), "1.3e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

let (x, o) = Float::from_integer_prec_round_ref(&Integer::from(-123), 20, Exact);
assert_eq!(x.to_string(), "-123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_round_ref(&Integer::from(-123), 4, Floor);
assert_eq!(x.to_string(), "-1.3e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_integer_prec_round_ref(&Integer::from(-123), 4, Ceiling);
assert_eq!(x.to_string(), "-1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

pub fn from_integer_prec(x: Integer, prec: u64) -> (Self, Ordering)

Converts an Integer to a Float, taking the Integer by value. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you want the Float’s precision to be equal to the Integer’s number of significant bits, try just using Float::try_from instead.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_integer_prec_round.

• If the Integer rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$.
• If the Integer rounds to a value less than or equal to -$2^{2^{30}-1}$), this function overflows to $\infty$.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::integer::Integer;
use std::cmp::Ordering::*;

let (x, o) = Float::from_integer_prec(Integer::ZERO, 10);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec(Integer::from(123), 20);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec(Integer::from(123), 4);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_integer_prec(Integer::from(-123), 20);
assert_eq!(x.to_string(), "-123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec(Integer::from(-123), 4);
assert_eq!(x.to_string(), "-1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

pub fn from_integer_prec_ref(x: &Integer, prec: u64) -> (Self, Ordering)

Converts an Integer to a Float, taking the Integer by reference. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you want the Float’s precision to be equal to the Integer’s number of significant bits, try just using Float::try_from instead.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_integer_prec_round_ref.

• If the Integer rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$.
• If the Integer rounds to a value less than or equal to -$2^{2^{30}-1}$), this function overflows to $\infty$.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::integer::Integer;
use std::cmp::Ordering::*;

let (x, o) = Float::from_integer_prec_ref(&Integer::ZERO, 10);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_ref(&Integer::from(123), 20);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_ref(&Integer::from(123), 4);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_integer_prec_ref(&Integer::from(-123), 20);
assert_eq!(x.to_string(), "-123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_integer_prec_ref(&Integer::from(-123), 4);
assert_eq!(x.to_string(), "-1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

impl Float

pub fn from_natural_prec_round( x: Natural, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts a Natural to a Float, taking the Natural by value. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_natural_prec instead.

• If the Natural rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$ if rm is Ceiling, Up, or Nearest, and rounds down to $(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero, or if rm is exact and the Natural cannot be exactly represented with the specified precision.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use std::cmp::Ordering::*;

let (x, o) = Float::from_natural_prec_round(Natural::ZERO, 10, Exact);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec_round(Natural::from(123u32), 20, Exact);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec_round(Natural::from(123u32), 4, Floor);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_natural_prec_round(Natural::from(123u32), 4, Ceiling);
assert_eq!(x.to_string(), "1.3e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

pub fn from_natural_prec_round_ref( x: &Natural, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts a Natural to a Float, taking the Natural by reference. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_natural_prec_ref instead.

• If the Natural rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$ if rm is Ceiling, Up, or Nearest, and rounds down to $(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero, or if rm is exact and the Natural cannot be exactly represented with the specified precision.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use std::cmp::Ordering::*;

let (x, o) = Float::from_natural_prec_round_ref(&Natural::ZERO, 10, Exact);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec_round_ref(&Natural::from(123u32), 20, Exact);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec_round_ref(&Natural::from(123u32), 4, Floor);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

let (x, o) = Float::from_natural_prec_round_ref(&Natural::from(123u32), 4, Ceiling);
assert_eq!(x.to_string(), "1.3e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Greater);

pub fn from_natural_prec(x: Natural, prec: u64) -> (Self, Ordering)

Converts a Natural to a Float, taking the Natural by value. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you want the Float’s precision to be equal to the Natural’s number of significant bits, try just using Float::try_from instead.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_natural_prec_round.

• If the Natural rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use std::cmp::Ordering::*;

let (x, o) = Float::from_natural_prec(Natural::ZERO, 10);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec(Natural::from(123u32), 20);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec(Natural::from(123u32), 4);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

pub fn from_natural_prec_ref(x: &Natural, prec: u64) -> (Self, Ordering)

Converts a Natural to a Float, taking the Natural by reference. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you want the Float’s precision to be equal to the Natural’s number of significant bits, try just using Float::try_from instead.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_natural_prec_round_ref.

• If the Natural rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$.

Worst-case complexity

$T(m,n) = O(\max(m,n))$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, $m$ is n.significant_bits(), and $n$ is prec.

Panics

Panics if prec is zero.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use std::cmp::Ordering::*;

let (x, o) = Float::from_natural_prec_ref(&Natural::ZERO, 10);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec_ref(&Natural::from(123u32), 20);
assert_eq!(x.to_string(), "123.0");
assert_eq!(x.get_prec(), Some(20));
assert_eq!(o, Equal);

let (x, o) = Float::from_natural_prec_ref(&Natural::from(123u32), 4);
assert_eq!(x.to_string(), "1.2e2");
assert_eq!(x.get_prec(), Some(4));
assert_eq!(o, Less);

impl Float

pub fn from_primitive_float_prec_round<T: PrimitiveFloat>( x: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts a primitive float to a Float. If the Float is nonzero and finite, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value. (Although a NaN is not comparable to any Float, converting a NaN to a NaN will also return Equal, indicating an exact conversion.)

If you’re only using Nearest, try using Float::from_primitive_float_prec instead.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(prec, x.sci_exponent().abs()).

Panics

Panics if prec is zero, or if rm is exact and the primitive float cannot be exactly represented with the specified precision.

Examples

See here.

pub fn from_primitive_float_prec<T: PrimitiveFloat>( x: T, prec: u64, ) -> (Self, Ordering)

Converts a primitive float to a Float. If the Float is nonzero and finite, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value. (Although a NaN is not comparable to any Float, converting a NaN to a NaN will also return Equal, indicating an exact conversion.)

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_primitive_float_prec_round.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(prec, x.sci_exponent().abs()).

Panics

Panics if prec is zero.

Examples

See here.

impl Float

pub const fn const_from_unsigned_times_power_of_2(x: Limb, pow: i32) -> Self

Converts an unsigned primitive integer to a Float, after multiplying it by the specified power of 2.

The type of the integer is u64, unless the 32_bit_limbs feature is set, in which case the type is u32.

If the integer is nonzero, the precision of the Float is the minimum possible precision to represent the integer exactly.

If you don’t need to use this function in a const context, try just using from instead, followed by >> or <<.

$$ f(x,k) = x2^k. $$

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the result is too large or too small to be represented by a Float.

Examples

use malachite_float::Float;

assert_eq!(
    Float::const_from_unsigned_times_power_of_2(0, 0).to_string(),
    "0.0"
);
assert_eq!(
    Float::const_from_unsigned_times_power_of_2(123, 0).to_string(),
    "123.0"
);
assert_eq!(
    Float::const_from_unsigned_times_power_of_2(123, 1).to_string(),
    "246.0"
);
assert_eq!(
    Float::const_from_unsigned_times_power_of_2(123, -1).to_string(),
    "61.5"
);
#[cfg(not(feature = "32_bit_limbs"))]
{
    assert_eq!(
        Float::const_from_unsigned_times_power_of_2(884279719003555, -48).to_string(),
        "3.141592653589793"
    );
}

pub const fn const_from_unsigned(x: Limb) -> Self

Converts an unsigned primitive integer to a Float.

The type of the integer is u64, unless the 32_bit_limbs feature is set, in which case the type is u32.

If the integer is nonzero, the precision of the Float is the minimum possible precision to represent the integer exactly.

If you don’t need to use this function in a const context, try just using from instead; it will probably be slightly faster.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_float::Float;

assert_eq!(Float::const_from_unsigned(0).to_string(), "0.0");
assert_eq!(Float::const_from_unsigned(123).to_string(), "123.0");

pub const fn const_from_signed_times_power_of_2(x: SignedLimb, pow: i32) -> Self

Converts a signed primitive integer to a Float, after multiplying it by the specified power of 2.

The type of the integer is i64, unless the 32_bit_limbs feature is set, in which case the type is i32.

If the integer is nonzero, the precision of the Float is the minimum possible precision to represent the integer exactly.

If you don’t need to use this function in a const context, try just using from instead, followed by >> or <<.

$$ f(x,k) = x2^k. $$

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the result is too large or too small to be represented by a Float.

Examples

use malachite_float::Float;

assert_eq!(
    Float::const_from_signed_times_power_of_2(0, 0).to_string(),
    "0.0"
);
assert_eq!(
    Float::const_from_signed_times_power_of_2(123, 0).to_string(),
    "123.0"
);
assert_eq!(
    Float::const_from_signed_times_power_of_2(123, 1).to_string(),
    "246.0"
);
assert_eq!(
    Float::const_from_signed_times_power_of_2(123, -1).to_string(),
    "61.5"
);
assert_eq!(
    Float::const_from_signed_times_power_of_2(-123, 0).to_string(),
    "-123.0"
);
assert_eq!(
    Float::const_from_signed_times_power_of_2(-123, 1).to_string(),
    "-246.0"
);
assert_eq!(
    Float::const_from_signed_times_power_of_2(-123, -1).to_string(),
    "-61.5"
);
#[cfg(not(feature = "32_bit_limbs"))]
{
    assert_eq!(
        Float::const_from_signed_times_power_of_2(884279719003555, -48).to_string(),
        "3.141592653589793"
    );
    assert_eq!(
        Float::const_from_signed_times_power_of_2(-884279719003555, -48).to_string(),
        "-3.141592653589793"
    );
}

pub const fn const_from_signed(x: SignedLimb) -> Self

Converts a signed primitive integer to a Float.

The type of the integer is i64, unless the 32_bit_limbs feature is set, in which case the type is i32.

If the integer is nonzero, the precision of the Float is the minimum possible precision to represent the integer exactly.

If you don’t need to use this function in a const context, try just using from instead; it will probably be slightly faster.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_float::Float;

assert_eq!(Float::const_from_signed(0).to_string(), "0.0");
assert_eq!(Float::const_from_signed(123).to_string(), "123.0");
assert_eq!(Float::const_from_signed(-123).to_string(), "-123.0");

pub fn from_unsigned_prec_round<T: PrimitiveUnsigned>( x: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

where Natural: From<T>,

Converts a primitive unsigned integer to a Float. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_unsigned_prec instead.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero, or if rm is exact and the primitive integer cannot be exactly represented with the specified precision.

Examples

See here.

pub fn from_unsigned_prec<T: PrimitiveUnsigned>( x: T, prec: u64, ) -> (Self, Ordering)

where Natural: From<T>,

Converts an unsigned primitive integer to a Float. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you want the Float’s precision to be equal to the integer’s number of significant bits, try just using Float::from instead.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

See here.

pub fn from_signed_prec_round<T: PrimitiveSigned>( x: T, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

where Integer: From<T>,

Converts a primitive signed integer to a Float. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_signed_prec instead.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero, or if rm is exact and the primitive integer cannot be exactly represented with the specified precision.

Examples

See here.

pub fn from_signed_prec<T: PrimitiveSigned>(x: T, prec: u64) -> (Self, Ordering)

where Integer: From<T>,

Converts a signed primitive integer to a Float. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you want the Float’s precision to be equal to the integer’s number of significant bits, try just using Float::from instead.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is prec.

Panics

Panics if prec is zero.

Examples

See here.

impl Float

pub fn from_rational_prec_round( x: Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts a Rational to a Float, taking the Rational by value. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_rational_prec instead.

• If the Rational rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$ if rm is Ceiling, Up, or Nearest, and rounds down to $(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.
• If the Rational rounds to a value less than or equal to $-2^{2^{30}-1}$), this function overflows to $-\infty$ if rm is Floor, Up, or Nearest, and rounds up to $-(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.
• If the Rational rounds to a positive value less than $2^{-2^{30}}$), this function underflows to positive zero if rm is Floor or Down, rounds up to $2^{-2^{30}}$ if rm is Ceiling or Up, underflows to positive zero if rm is Nearest and the Rational rounds to a value less than or equal to $2^{-2^{30}-1}$, and rounds up to $2^{-2^{30}}$ if rm is Nearest and the Rational rounds to a value greater than $2^{-2^{30}-1}$.
• If the Rational rounds to a negative value greater than $-2^{-2^{30}}$), this function underflows to negative zero if rm is Ceiling or Down, rounds down to $-2^{-2^{30}}$ if rm is Floor or Up, underflows to negative zero if rm is Nearest and the Rational rounds to a value greater than or equal to $-2^{-2^{30}-1}$, and rounds down to $-2^{-2^{30}}$ if rm is Nearest and the Rational rounds to a value less than $-2^{-2^{30}-1}$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(n.significant_bits(), prec).

Panics

Panics if prec is zero, or if rm is exact and the Rational cannot be exactly represented with the specified precision.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (x, o) = Float::from_rational_prec_round(Rational::from_signeds(1, 3), 10, Floor);
assert_eq!(x.to_string(), "0.333");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) = Float::from_rational_prec_round(Rational::from_signeds(1, 3), 10, Ceiling);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec_round(Rational::from_signeds(1, 3), 10, Nearest);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec_round(Rational::from_signeds(-1, 3), 10, Floor);
assert_eq!(x.to_string(), "-0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) = Float::from_rational_prec_round(Rational::from_signeds(-1, 3), 10, Ceiling);
assert_eq!(x.to_string(), "-0.333");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec_round(Rational::from_signeds(-1, 3), 10, Nearest);
assert_eq!(x.to_string(), "-0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

pub fn from_rational_prec(x: Rational, prec: u64) -> (Self, Ordering)

Converts a Rational to a Float, taking the Rational by value. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Rational is dyadic (its denominator is a power of 2), then you can convert it to a Float using try_from instead. The precision of the resulting Float will be the number of significant bits of the Rational’s numerator.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_rational_prec_round.

• If the Rational rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$.
• If the Rational rounds to a value less than or equal to $-2^{2^{30}-1}$), this function overflows to $-\infty$.
• If the Rational rounds to a positive value less than $2^{-2^{30}}$), this function underflows to positive zero if the Rational rounds to a value less than or equal to $2^{-2^{30}-1}$ and rounds up to $2^{-2^{30}}$ if the Rational rounds to a value greater than $2^{-2^{30}-1}$.
• If the Rational rounds to a negative value greater than $2^{-2^{30}}$), this function underflows to negative zero if the Rational rounds to a value greater than or equal to $-2^{-2^{30}-1}$ and rounds down to $-2^{-2^{30}}$ if the Rational rounds to a value less than $-2^{-2^{30}-1}$.

Panics

Panics if prec is zero.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), prec).

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (x, o) = Float::from_rational_prec(Rational::ZERO, 10);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_rational_prec(Rational::from_signeds(1, 3), 10);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec(Rational::from_signeds(1, 3), 100);
assert_eq!(x.to_string(), "0.3333333333333333333333333333335");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec(Rational::from_signeds(-1, 3), 10);
assert_eq!(x.to_string(), "-0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) = Float::from_rational_prec(Rational::from_signeds(-1, 3), 100);
assert_eq!(x.to_string(), "-0.3333333333333333333333333333335");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Less);

pub fn from_rational_prec_round_ref( x: &Rational, prec: u64, rm: RoundingMode, ) -> (Self, Ordering)

Converts a Rational to a Float, taking the Rational by reference. If the Float is nonzero, it has the specified precision. If rounding is needed, the specified rounding mode is used. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If you’re only using Nearest, try using Float::from_rational_prec_ref instead.

• If the Rational rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$ if rm is Ceiling, Up, or Nearest, and rounds down to $(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.
• If the Rational rounds to a value less than or equal to $-2^{2^{30}-1}$), this function overflows to $-\infty$ if rm is Floor, Up, or Nearest, and rounds up to $-(1-(1/2)^p)2^{2^{30}-1}$ otherwise, where $p$ is prec.
• If the Rational rounds to a positive value less than $2^{-2^{30}}$), this function underflows to positive zero if rm is Floor or Down, rounds up to $2^{-2^{30}}$ if rm is Ceiling or Up, underflows to positive zero if rm is Nearest and the Rational rounds to a value less than or equal to $2^{-2^{30}-1}$, and rounds up to $2^{-2^{30}}$ if rm is Nearest and the Rational rounds to a value greater than $2^{-2^{30}-1}$.
• If the Rational rounds to a negative value greater than $-2^{-2^{30}}$), this function underflows to negative zero if rm is Ceiling or Down, rounds down to $-2^{-2^{30}}$ if rm is Floor or Up, underflows to negative zero if rm is Nearest and the Rational rounds to a value greater than or equal to $-2^{-2^{30}-1}$, and rounds down to $-2^{-2^{30}}$ if rm is Nearest and the Rational rounds to a value less than $-2^{-2^{30}-1}$.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), prec).

Panics

Panics if prec is zero, or if rm is exact and the Rational cannot be exactly represented with the specified precision.

Examples

use malachite_base::rounding_modes::RoundingMode::*;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (x, o) = Float::from_rational_prec_round_ref(&Rational::from_signeds(1, 3), 10, Floor);
assert_eq!(x.to_string(), "0.333");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) =
    Float::from_rational_prec_round_ref(&Rational::from_signeds(1, 3), 10, Ceiling);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) =
    Float::from_rational_prec_round_ref(&Rational::from_signeds(1, 3), 10, Nearest);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec_round_ref(&Rational::from_signeds(-1, 3), 10, Floor);
assert_eq!(x.to_string(), "-0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) =
    Float::from_rational_prec_round_ref(&Rational::from_signeds(-1, 3), 10, Ceiling);
assert_eq!(x.to_string(), "-0.333");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) =
    Float::from_rational_prec_round_ref(&Rational::from_signeds(-1, 3), 10, Nearest);
assert_eq!(x.to_string(), "-0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

pub fn from_rational_prec_ref(x: &Rational, prec: u64) -> (Self, Ordering)

Converts a Rational to a Float, taking the Rational by reference. If the Float is nonzero, it has the specified precision. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Rational is dyadic (its denominator is a power of 2), then you can convert it to a Float using try_from instead. The precision of the resulting Float will be the number of significant bits of the Rational’s numerator.

Rounding may occur, in which case Nearest is used by default. To specify a rounding mode as well as a precision, try Float::from_rational_prec_round_ref.

• If the Rational rounds to a value greater than or equal to $2^{2^{30}-1}$), this function overflows to $\infty$.
• If the Rational rounds to a value less than or equal to $-2^{2^{30}-1}$), this function overflows to $-\infty$.
• If the Rational rounds to a positive value less than $2^{-2^{30}}$), this function underflows to positive zero if the Rational rounds to a value less than or equal to $2^{-2^{30}-1}$ and rounds up to $2^{-2^{30}}$ if the Rational rounds to a value greater than $2^{-2^{30}-1}$.
• If the Rational rounds to a negative value greater than $2^{-2^{30}}$), this function underflows to negative zero if the Rational rounds to a value greater than or equal to $-2^{-2^{30}-1}$ and rounds down to $-2^{-2^{30}}$ if the Rational rounds to a value less than $-2^{-2^{30}-1}$.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(x.significant_bits(), prec).

Panics

Panics if prec is zero.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_q::Rational;
use std::cmp::Ordering::*;

let (x, o) = Float::from_rational_prec_ref(&Rational::ZERO, 10);
assert_eq!(x.to_string(), "0.0");
assert_eq!(o, Equal);

let (x, o) = Float::from_rational_prec_ref(&Rational::from_signeds(1, 3), 10);
assert_eq!(x.to_string(), "0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec_ref(&Rational::from_signeds(1, 3), 100);
assert_eq!(x.to_string(), "0.3333333333333333333333333333335");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Greater);

let (x, o) = Float::from_rational_prec_ref(&Rational::from_signeds(-1, 3), 10);
assert_eq!(x.to_string(), "-0.3335");
assert_eq!(x.get_prec(), Some(10));
assert_eq!(o, Less);

let (x, o) = Float::from_rational_prec_ref(&Rational::from_signeds(-1, 3), 100);
assert_eq!(x.to_string(), "-0.3333333333333333333333333333335");
assert_eq!(x.get_prec(), Some(100));
assert_eq!(o, Less);

impl Float

pub fn sci_mantissa_and_exponent_round<T: PrimitiveFloat>( &self, rm: RoundingMode, ) -> Option<(T, i32, Ordering)>

Returns a Float’s scientific mantissa and exponent, rounding according to the specified rounding mode. An Ordering is also returned, indicating whether the mantissa and exponent represent a value that is less than, equal to, or greater than the original value.

When $x$ is positive, we can write $x = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. We represent the rational mantissa as a float. The conversion might not be exact, so we round to the nearest float using the provided rounding mode. If the rounding mode is Exact but the conversion is not exact, None is returned. $$ f(x, r) \approx \left (\frac{x}{2^{\lfloor \log_2 x \rfloor}}, \lfloor \log_2 x \rfloor\right ). $$

This function does not overflow or underflow. The returned exponent is always in the range $[-2^{30}, 2^{30}-1]$. Notice that although a Float’s maximum scientific exponent is $2^{30}-2$, this function may return an exponent one larger than this limit due to rounding.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_base::num::float::NiceFloat;
use malachite_base::rounding_modes::RoundingMode::{self, *};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use std::cmp::Ordering::{self, *};

let test = |x: Float, rm: RoundingMode, out: Option<(f32, i32, Ordering)>| {
    assert_eq!(
        x.sci_mantissa_and_exponent_round(rm)
            .map(|(m, e, o)| (NiceFloat(m), e, o)),
        out.map(|(m, e, o)| (NiceFloat(m), e, o))
    );
};
test(Float::from(3u32), Floor, Some((1.5, 1, Equal)));
test(Float::from(3u32), Down, Some((1.5, 1, Equal)));
test(Float::from(3u32), Ceiling, Some((1.5, 1, Equal)));
test(Float::from(3u32), Up, Some((1.5, 1, Equal)));
test(Float::from(3u32), Nearest, Some((1.5, 1, Equal)));
test(Float::from(3u32), Exact, Some((1.5, 1, Equal)));

let x = Float::from(std::f64::consts::PI);
test(x.clone(), Floor, Some((1.5707963, 1, Less)));
test(x.clone(), Down, Some((1.5707963, 1, Less)));
test(x.clone(), Ceiling, Some((1.5707964, 1, Greater)));
test(x.clone(), Up, Some((1.5707964, 1, Greater)));
test(x.clone(), Nearest, Some((1.5707964, 1, Greater)));
test(x.clone(), Exact, None);

test(
    Float::from(1000000000u32),
    Nearest,
    Some((1.8626451, 29, Equal)),
);
test(
    Float::exact_from(Natural::from(10u32).pow(52)),
    Nearest,
    Some((1.670478, 172, Greater)),
);

test(Float::exact_from(Natural::from(10u32).pow(52)), Exact, None);

impl Float

pub const MAX_EXPONENT: i32 = 1_073_741_823i32

The maximum raw exponent of any Float, equal to $2^{30}-1$, or $1,073,741,823$. This is one more than the maximum scientific exponent. If we write a Float as $\pm m2^e$, with $1\leq m<2$ and $e$ an integer, we must have $e\leq 2^{30}-2$. If the result of a calculation would produce a Float with an exponent larger than this, $\pm\infty$ is returned instead.

pub const MIN_EXPONENT: i32 = -1_073_741_823i32

The minimum raw exponent of any Float, equal to $-(2^{30}-1)$, or $-1,073,741,823$. This is one more than the minimum scientific exponent. If we write a Float as $\pm m2^e$, with $1\leq m<2$ and $e$ an integer, we must have $e\geq -2^{30}$. If the result of a calculation would produce a Float with an exponent smaller than this, $\pm0.0$ is returned instead.

Trait Implementations

impl Abs for &Float

fn abs(self) -> Float

Takes the absolute value of a Float, taking the Float by reference.

$$ f(x) = |x|. $$

Special cases:

• $f(\text{NaN}) = \text{NaN}$
• $f(\infty) = f(-\infty) = \infty$
• $f(0.0) = f(-0.0) = 0.0$

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Abs;
use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

assert_eq!(
    ComparableFloat((&Float::NAN).abs()),
    ComparableFloat(Float::NAN)
);
assert_eq!((&Float::INFINITY).abs(), Float::INFINITY);
assert_eq!((&Float::NEGATIVE_INFINITY).abs(), Float::INFINITY);
assert_eq!(
    ComparableFloat((&Float::ZERO).abs()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(
    ComparableFloat((&Float::NEGATIVE_ZERO).abs()),
    ComparableFloat(Float::ZERO)
);
assert_eq!((&Float::ONE).abs(), Float::ONE);
assert_eq!((&Float::NEGATIVE_ONE).abs(), Float::ONE);

type Output = Float

impl Abs for Float

fn abs(self) -> Self

Takes the absolute value of a Float, taking the Float by value.

$$ f(x) = |x|. $$

Special cases:

• $f(\text{NaN}) = \text{NaN}$
• $f(\infty) = f(-\infty) = \infty$
• $f(0.0) = f(-0.0) = 0.0$

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::Abs;
use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

assert_eq!(
    ComparableFloat(Float::NAN.abs()),
    ComparableFloat(Float::NAN)
);
assert_eq!(Float::INFINITY.abs(), Float::INFINITY);
assert_eq!(Float::NEGATIVE_INFINITY.abs(), Float::INFINITY);
assert_eq!(
    ComparableFloat(Float::ZERO.abs()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(
    ComparableFloat(Float::NEGATIVE_ZERO.abs()),
    ComparableFloat(Float::ZERO)
);
assert_eq!(Float::ONE.abs(), Float::ONE);
assert_eq!(Float::NEGATIVE_ONE.abs(), Float::ONE);

type Output = Float

impl AbsAssign for Float

fn abs_assign(&mut self)

Replaces a Float with its absolute value.

$$ x \gets |x|. $$

Special cases:

• $\text{NaN} \gets \text{NaN}$
• $\infty \gets \infty$
• $-\infty \gets \infty$
• $0.0 \gets 0.0$
• $-0.0 \gets 0.0$

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::AbsAssign;
use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

let mut x = Float::NAN;
x.abs_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::NAN));

let mut x = Float::INFINITY;
x.abs_assign();
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x.abs_assign();
assert_eq!(x, Float::INFINITY);

let mut x = Float::ZERO;
x.abs_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::ZERO));

let mut x = Float::NEGATIVE_ZERO;
x.abs_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::ZERO));

let mut x = Float::ONE;
x.abs_assign();
assert_eq!(x, Float::ONE);

let mut x = Float::NEGATIVE_ONE;
x.abs_assign();
assert_eq!(x, Float::ONE);

impl Add<&Float> for &Float

fn add(self, other: &Float) -> Float

Adds two Floats, taking both by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,-\infty)=f(-\infty,\infty)=\text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0)=0.0$
• $f(-0.0,-0.0)=-0.0$
• $f(0.0,-0.0)=f(-0.0,0.0)=0.0$
• $f(0.0,x)=f(x,0.0)=f(-0.0,x)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_ref_ref instead. If you want to specify the output precision, consider using Float::add_round_ref_ref. If you want both of these things, consider using Float::add_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((&Float::from(1.5) + &Float::NAN).is_nan());
assert_eq!(&Float::from(1.5) + &Float::INFINITY, Float::INFINITY);
assert_eq!(
    &Float::from(1.5) + &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert!((&Float::INFINITY + &Float::NEGATIVE_INFINITY).is_nan());

assert_eq!(&Float::from(1.5) + &Float::from(2.5), 4.0);
assert_eq!(&Float::from(1.5) + &Float::from(-2.5), -1.0);
assert_eq!(&Float::from(-1.5) + &Float::from(2.5), 1.0);
assert_eq!(&Float::from(-1.5) + &Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<&Float> for &Rational

fn add(self, other: &Float) -> Float

Adds a Rational and a Float, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=\infty$
• $f(x,-\infty)=-\infty$
• $f(0,0.0)=0.0$
• $f(0,-0.0)=-0.0$
• $f(x,0.0)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) + &Float::NAN).is_nan());
assert_eq!(
    &Rational::exact_from(1.5) + &Float::INFINITY,
    Float::INFINITY
);
assert_eq!(
    &Rational::exact_from(1.5) + &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);

assert_eq!(&Rational::exact_from(1.5) + &Float::from(2.5), 4.0);
assert_eq!(&Rational::exact_from(1.5) + &Float::from(-2.5), -1.0);
assert_eq!(&Rational::exact_from(-1.5) + &Float::from(2.5), 1.0);
assert_eq!(&Rational::exact_from(-1.5) + &Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<&Float> for Float

fn add(self, other: &Self) -> Self

Adds two Floats, taking the first by value and the second by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,-\infty)=f(-\infty,\infty)=\text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0)=0.0$
• $f(-0.0,-0.0)=-0.0$
• $f(0.0,-0.0)=f(-0.0,0.0)=0.0$
• $f(0.0,x)=f(x,0.0)=f(-0.0,x)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_val_ref instead. If you want to specify the output precision, consider using Float::add_round_val_ref. If you want both of these things, consider using Float::add_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((Float::from(1.5) + &Float::NAN).is_nan());
assert_eq!(Float::from(1.5) + &Float::INFINITY, Float::INFINITY);
assert_eq!(
    Float::from(1.5) + &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert!((Float::INFINITY + &Float::NEGATIVE_INFINITY).is_nan());

assert_eq!(Float::from(1.5) + &Float::from(2.5), 4.0);
assert_eq!(Float::from(1.5) + &Float::from(-2.5), -1.0);
assert_eq!(Float::from(-1.5) + &Float::from(2.5), 1.0);
assert_eq!(Float::from(-1.5) + &Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<&Float> for Rational

fn add(self, other: &Float) -> Float

Adds a Rational and a Float, taking the Rational by value and the Float by reference.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=\infty$
• $f(x,-\infty)=-\infty$
• $f(0,0.0)=0.0$
• $f(0,-0.0)=-0.0$
• $f(x,0.0)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) + &Float::NAN).is_nan());
assert_eq!(
    Rational::exact_from(1.5) + &Float::INFINITY,
    Float::INFINITY
);
assert_eq!(
    Rational::exact_from(1.5) + &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);

assert_eq!(Rational::exact_from(1.5) + &Float::from(2.5), 4.0);
assert_eq!(Rational::exact_from(1.5) + &Float::from(-2.5), -1.0);
assert_eq!(Rational::exact_from(-1.5) + &Float::from(2.5), 1.0);
assert_eq!(Rational::exact_from(-1.5) + &Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<&Rational> for &Float

fn add(self, other: &Rational) -> Float

Adds a Float and a Rational, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(0.0,x)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_ref_ref instead. If you want to specify the output precision, consider using Float::add_rational_round_ref_ref. If you want both of these things, consider using Float::add_rational_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN + &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY + &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY + &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(&Float::from(2.5) + &Rational::exact_from(1.5), 4.0);
assert_eq!(&Float::from(2.5) + &Rational::exact_from(-1.5), 1.0);
assert_eq!(&Float::from(-2.5) + &Rational::exact_from(1.5), -1.0);
assert_eq!(&Float::from(-2.5) + &Rational::exact_from(-1.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<&Rational> for Float

fn add(self, other: &Rational) -> Self

Adds a Float and a Rational, taking the Float by value and the Rational by reference.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(0.0,x)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_val_ref instead. If you want to specify the output precision, consider using Float::add_rational_round_val_ref. If you want both of these things, consider using Float::add_rational_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN + &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    Float::INFINITY + &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY + &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(Float::from(2.5) + &Rational::exact_from(1.5), 4.0);
assert_eq!(Float::from(2.5) + &Rational::exact_from(-1.5), 1.0);
assert_eq!(Float::from(-2.5) + &Rational::exact_from(1.5), -1.0);
assert_eq!(Float::from(-2.5) + &Rational::exact_from(-1.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<Float> for &Float

fn add(self, other: Float) -> Float

Adds two Floats, taking the first by reference and the second by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,-\infty)=f(-\infty,\infty)=\text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0)=0.0$
• $f(-0.0,-0.0)=-0.0$
• $f(0.0,-0.0)=f(-0.0,0.0)=0.0$
• $f(0.0,x)=f(x,0.0)=f(-0.0,x)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_ref_val instead. If you want to specify the output precision, consider using Float::add_round_ref_val. If you want both of these things, consider using Float::add_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is self.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((&Float::from(1.5) + Float::NAN).is_nan());
assert_eq!(&Float::from(1.5) + Float::INFINITY, Float::INFINITY);
assert_eq!(
    &Float::from(1.5) + Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert!((&Float::INFINITY + Float::NEGATIVE_INFINITY).is_nan());

assert_eq!(&Float::from(1.5) + Float::from(2.5), 4.0);
assert_eq!(&Float::from(1.5) + Float::from(-2.5), -1.0);
assert_eq!(&Float::from(-1.5) + Float::from(2.5), 1.0);
assert_eq!(&Float::from(-1.5) + Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<Float> for &Rational

fn add(self, other: Float) -> Float

Adds a Rational and a Float, taking the Rational by reference and the Float by value.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=\infty$
• $f(x,-\infty)=-\infty$
• $f(0,0.0)=0.0$
• $f(0,-0.0)=-0.0$
• $f(x,0.0)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) + Float::NAN).is_nan());
assert_eq!(
    &Rational::exact_from(1.5) + Float::INFINITY,
    Float::INFINITY
);
assert_eq!(
    &Rational::exact_from(1.5) + Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);

assert_eq!(&Rational::exact_from(1.5) + Float::from(2.5), 4.0);
assert_eq!(&Rational::exact_from(1.5) + Float::from(-2.5), -1.0);
assert_eq!(&Rational::exact_from(-1.5) + Float::from(2.5), 1.0);
assert_eq!(&Rational::exact_from(-1.5) + Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<Float> for Rational

fn add(self, other: Float) -> Float

Adds a Rational and a Float, taking both by value.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=\infty$
• $f(x,-\infty)=-\infty$
• $f(0,0.0)=0.0$
• $f(0,-0.0)=-0.0$
• $f(x,0.0)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) + Float::NAN).is_nan());
assert_eq!(Rational::exact_from(1.5) + Float::INFINITY, Float::INFINITY);
assert_eq!(
    Rational::exact_from(1.5) + Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);

assert_eq!(Rational::exact_from(1.5) + Float::from(2.5), 4.0);
assert_eq!(Rational::exact_from(1.5) + Float::from(-2.5), -1.0);
assert_eq!(Rational::exact_from(-1.5) + Float::from(2.5), 1.0);
assert_eq!(Rational::exact_from(-1.5) + Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<Rational> for &Float

fn add(self, other: Rational) -> Float

Adds a Float and a Rational, taking the Float by reference and the Rational by value.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(0.0,x)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_ref_val instead. If you want to specify the output precision, consider using Float::add_rational_round_ref_val. If you want both of these things, consider using Float::add_rational_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN + Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY + Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY + Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(&Float::from(2.5) + Rational::exact_from(1.5), 4.0);
assert_eq!(&Float::from(2.5) + Rational::exact_from(-1.5), 1.0);
assert_eq!(&Float::from(-2.5) + Rational::exact_from(1.5), -1.0);
assert_eq!(&Float::from(-2.5) + Rational::exact_from(-1.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add<Rational> for Float

fn add(self, other: Rational) -> Self

Adds a Float and a Rational, taking both by value.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(0.0,x)=f(x,0)=f(-0.0,x)=x$
• $f(x,-x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec instead. If you want to specify the output precision, consider using Float::add_rational_round. If you want both of these things, consider using Float::add_rational_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN + Rational::exact_from(1.5)).is_nan());
assert_eq!(Float::INFINITY + Rational::exact_from(1.5), Float::INFINITY);
assert_eq!(
    Float::NEGATIVE_INFINITY + Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(Float::from(2.5) + Rational::exact_from(1.5), 4.0);
assert_eq!(Float::from(2.5) + Rational::exact_from(-1.5), 1.0);
assert_eq!(Float::from(-2.5) + Rational::exact_from(1.5), -1.0);
assert_eq!(Float::from(-2.5) + Rational::exact_from(-1.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl Add for Float

fn add(self, other: Self) -> Self

Adds two Floats, taking both by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,-\infty)=f(-\infty,\infty)=\text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,0.0)=0.0$
• $f(-0.0,-0.0)=-0.0$
• $f(0.0,-0.0)=f(-0.0,0.0)=0.0$
• $f(0.0,x)=f(x,0.0)=f(-0.0,x)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,-x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec instead. If you want to specify the output precision, consider using Float::add_round. If you want both of these things, consider using Float::add_prec_round.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((Float::from(1.5) + Float::NAN).is_nan());
assert_eq!(Float::from(1.5) + Float::INFINITY, Float::INFINITY);
assert_eq!(
    Float::from(1.5) + Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert!((Float::INFINITY + Float::NEGATIVE_INFINITY).is_nan());

assert_eq!(Float::from(1.5) + Float::from(2.5), 4.0);
assert_eq!(Float::from(1.5) + Float::from(-2.5), -1.0);
assert_eq!(Float::from(-1.5) + Float::from(2.5), 1.0);
assert_eq!(Float::from(-1.5) + Float::from(-2.5), -4.0);

type Output = Float

The resulting type after applying the + operator.

impl AddAssign<&Float> for Float

fn add_assign(&mut self, other: &Self)

Adds a Float to a Float in place, taking the Float on the right-hand side by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the + documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_assign_ref instead. If you want to specify the output precision, consider using Float::add_round_assign_ref. If you want both of these things, consider using Float::add_prec_round_assign_ref.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

let mut x = Float::from(1.5);
x += &Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x += &Float::INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x += &Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::INFINITY;
x += &Float::NEGATIVE_INFINITY;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x += &Float::from(2.5);
assert_eq!(x, 4.0);

let mut x = Float::from(1.5);
x += &Float::from(-2.5);
assert_eq!(x, -1.0);

let mut x = Float::from(-1.5);
x += &Float::from(2.5);
assert_eq!(x, 1.0);

let mut x = Float::from(-1.5);
x += &Float::from(-2.5);
assert_eq!(x, -4.0);

impl AddAssign<&Rational> for Float

fn add_assign(&mut self, other: &Rational)

Adds a Rational to a Float in place, taking the Rational by reference.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

See the + documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_assign instead. If you want to specify the output precision, consider using Float::add_rational_round_assign. If you want both of these things, consider using Float::add_rational_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x += &Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x += &Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x += &Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(2.5);
x += &Rational::exact_from(1.5);
assert_eq!(x, 4.0);

let mut x = Float::from(2.5);
x += &Rational::exact_from(-1.5);
assert_eq!(x, 1.0);

let mut x = Float::from(-2.5);
x += &Rational::exact_from(1.5);
assert_eq!(x, -1.0);

let mut x = Float::from(-2.5);
x += &Rational::exact_from(-1.5);
assert_eq!(x, -4.0);

impl AddAssign<Rational> for Float

fn add_assign(&mut self, other: Rational)

Adds a Rational to a Float in place, taking the Rational by value.

If the output has a precision, it is the precision of the input Float. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the precision of the input Float.

See the + documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_rational_prec_assign instead. If you want to specify the output precision, consider using Float::add_rational_round_assign. If you want both of these things, consider using Float::add_rational_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x += Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x += Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x += Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(2.5);
x += Rational::exact_from(1.5);
assert_eq!(x, 4.0);

let mut x = Float::from(2.5);
x += Rational::exact_from(-1.5);
assert_eq!(x, 1.0);

let mut x = Float::from(-2.5);
x += Rational::exact_from(1.5);
assert_eq!(x, -1.0);

let mut x = Float::from(-2.5);
x += Rational::exact_from(-1.5);
assert_eq!(x, -4.0);

impl AddAssign for Float

fn add_assign(&mut self, other: Self)

Adds a Float to a Float in place, taking the Float on the right-hand side by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the sum is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x+y+\varepsilon. $$

• If $x+y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x+y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x+y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the + documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::add_prec_assign instead. If you want to specify the output precision, consider using Float::add_round_assign. If you want both of these things, consider using Float::add_prec_round_assign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

let mut x = Float::from(1.5);
x += Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x += Float::INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x += Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::INFINITY;
x += Float::NEGATIVE_INFINITY;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x += Float::from(2.5);
assert_eq!(x, 4.0);

let mut x = Float::from(1.5);
x += Float::from(-2.5);
assert_eq!(x, -1.0);

let mut x = Float::from(-1.5);
x += Float::from(2.5);
assert_eq!(x, 1.0);

let mut x = Float::from(-1.5);
x += Float::from(-2.5);
assert_eq!(x, -4.0);

impl Clone for Float

fn clone(&self) -> Float

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl ConvertibleFrom<&Float> for Integer

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an Integer, taking the Float by reference.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::traits::ConvertibleFrom;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(Integer::convertible_from(&Float::ZERO), true);
assert_eq!(Integer::convertible_from(&Float::from(123.0)), true);
assert_eq!(Integer::convertible_from(&Float::from(-123.0)), true);

assert_eq!(Integer::convertible_from(&Float::from(1.5)), false);
assert_eq!(Integer::convertible_from(&Float::INFINITY), false);
assert_eq!(Integer::convertible_from(&Float::NAN), false);

impl ConvertibleFrom<&Float> for Natural

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a Natural (when the Float is non-negative and an integer), taking the Float by reference.

Both positive and negative zero are convertible to a Natural. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::traits::ConvertibleFrom;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(Natural::convertible_from(&Float::ZERO), true);
assert_eq!(Natural::convertible_from(&Float::from(123.0)), true);

assert_eq!(Natural::convertible_from(&Float::from(-123.0)), false);
assert_eq!(Natural::convertible_from(&Float::from(1.5)), false);
assert_eq!(Natural::convertible_from(&Float::INFINITY), false);
assert_eq!(Natural::convertible_from(&Float::NAN), false);

impl ConvertibleFrom<&Float> for Rational

fn convertible_from(x: &Float) -> bool

Determines whether a Float can be converted to a Rational (which is when the Float is finite), taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::traits::ConvertibleFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert_eq!(Rational::convertible_from(&Float::ZERO), true);
assert_eq!(Rational::convertible_from(&Float::from(123.0)), true);
assert_eq!(Rational::convertible_from(&Float::from(-123.0)), true);
assert_eq!(Rational::convertible_from(&Float::from(1.5)), true);

assert_eq!(Rational::convertible_from(&Float::INFINITY), false);
assert_eq!(Rational::convertible_from(&Float::NAN), false);

impl ConvertibleFrom<&Float> for f32

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a primitive float, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for f64

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a primitive float, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for i128

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a signed primitive integer, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for i16

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a signed primitive integer, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for i32

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a signed primitive integer, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for i64

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a signed primitive integer, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for i8

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a signed primitive integer, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for isize

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to a signed primitive integer, taking the Float by reference.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for u128

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an unsigned primitive integer, taking the Float by reference.

Both positive and negative zero are convertible to any unsigned primitive integer. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for u16

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an unsigned primitive integer, taking the Float by reference.

Both positive and negative zero are convertible to any unsigned primitive integer. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for u32

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an unsigned primitive integer, taking the Float by reference.

Both positive and negative zero are convertible to any unsigned primitive integer. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for u64

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an unsigned primitive integer, taking the Float by reference.

Both positive and negative zero are convertible to any unsigned primitive integer. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for u8

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an unsigned primitive integer, taking the Float by reference.

Both positive and negative zero are convertible to any unsigned primitive integer. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Float> for usize

fn convertible_from(f: &Float) -> bool

Determines whether a Float can be converted to an unsigned primitive integer, taking the Float by reference.

Both positive and negative zero are convertible to any unsigned primitive integer. (Although negative zero is nominally negative, the real number it represents is zero, which is not negative.)

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl ConvertibleFrom<&Integer> for Float

fn convertible_from(x: &Integer) -> bool

Determines whether an Integer can be converted to an Float, taking the Integer by reference.

The Integers that are convertible to Floats are those whose that would not overflow: that is, those whose absolute values are less than $2^{2^{30}-1}$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is x.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::num::conversion::traits::ConvertibleFrom;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(Float::convertible_from(&Integer::ZERO), true);
assert_eq!(Float::convertible_from(&Integer::from(3u8)), true);

impl ConvertibleFrom<&Natural> for Float

fn convertible_from(x: &Natural) -> bool

Determines whether a Natural can be converted to an Float, taking the Natural by reference.

The Naturals that are convertible to Floats are those whose that would not overflow: that is, those that are less than $2^{2^{30}-1}$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is x.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::num::conversion::traits::ConvertibleFrom;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(Float::convertible_from(&Natural::ZERO), true);
assert_eq!(Float::convertible_from(&Natural::from(3u8)), true);

impl ConvertibleFrom<&Rational> for Float

fn convertible_from(x: &Rational) -> bool

Determines whether a Rational can be converted to an Float, taking the Rational by reference.

The Rationals that are convertible to Floats are precisely those whose denominators are powers of two, and would not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is x.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::num::conversion::traits::ConvertibleFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert_eq!(Float::convertible_from(&Rational::ZERO), true);
assert_eq!(Float::convertible_from(&Rational::from_signeds(3, 8)), true);
assert_eq!(
    Float::convertible_from(&Rational::from_signeds(-3, 8)),
    true
);

assert_eq!(
    Float::convertible_from(&Rational::from_signeds(1, 3)),
    false
);
assert_eq!(
    Float::convertible_from(&Rational::from_signeds(-1, 3)),
    false
);

impl Debug for Float

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Default for Float

fn default() -> Self

The default value of a Float, NaN.

impl Display for Float

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Div<&Float> for &Float

fn div(self, other: &Float) -> Float

Divides two Floats, taking both by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm\infty)=f(\pm0.0,\pm0.0) = \text{NaN}$
• $f(\infty,x)=\infty$ if $0.0<x<\infty$
• $f(\infty,x)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0)=\infty$ if $x>0.0$
• $f(x,0.0)=-\infty$ if $x<0.0$
• $f(-\infty,x)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0)=-\infty$ if $x>0.0$
• $f(x,-0.0)=\infty$ if $x<0.0$
• $f(0.0,x)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_ref_ref instead. If you want to specify the output precision, consider using Float::div_round_ref_ref. If you want both of these things, consider using Float::div_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_float::Float;

assert!((&Float::from(1.5) / &Float::NAN).is_nan());
assert_eq!(&Float::from(1.5) / &Float::ZERO, Float::INFINITY);
assert_eq!(
    &Float::from(1.5) / &Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(&Float::from(-1.5) / &Float::ZERO, Float::NEGATIVE_INFINITY);
assert_eq!(&Float::from(-1.5) / &Float::NEGATIVE_ZERO, Float::INFINITY);
assert!((&Float::ZERO / &Float::ZERO).is_nan());

assert_eq!((&Float::from(1.5) / &Float::from(2.5)).to_string(), "0.6");
assert_eq!((&Float::from(1.5) / &Float::from(-2.5)).to_string(), "-0.6");
assert_eq!((&Float::from(-1.5) / &Float::from(2.5)).to_string(), "-0.6");
assert_eq!((&Float::from(-1.5) / &Float::from(-2.5)).to_string(), "0.6");

type Output = Float

The resulting type after applying the / operator.

impl Div<&Float> for &Rational

fn div(self, other: &Float) -> Float

Divides a Rational by a Float, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) / &Float::NAN).is_nan());
assert_eq!(&Rational::exact_from(1.5) / &Float::ZERO, Float::INFINITY);
assert_eq!(
    &Rational::exact_from(1.5) / &Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) / &Float::ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) / &Float::NEGATIVE_ZERO,
    Float::INFINITY
);

assert_eq!(
    (&Rational::exact_from(1.5) / &Float::from(2.5)).to_string(),
    "0.6"
);
assert_eq!(
    (&Rational::exact_from(-1.5) / &Float::from(2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (&Rational::exact_from(1.5) / &Float::from(-2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (&Rational::exact_from(-1.5) / &Float::from(-2.5)).to_string(),
    "0.6"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<&Float> for Float

fn div(self, other: &Self) -> Self

Divides two Floats, taking the first by value and the second by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm\infty)=f(\pm0.0,\pm0.0) = \text{NaN}$
• $f(\infty,x)=\infty$ if $0.0<x<\infty$
• $f(\infty,x)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0)=\infty$ if $x>0.0$
• $f(x,0.0)=-\infty$ if $x<0.0$
• $f(-\infty,x)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0)=-\infty$ if $x>0.0$
• $f(x,-0.0)=\infty$ if $x<0.0$
• $f(0.0,x)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_val_ref instead. If you want to specify the output precision, consider using Float::div_round_val_ref. If you want both of these things, consider using Float::div_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_float::Float;

assert!((Float::from(1.5) / &Float::NAN).is_nan());
assert_eq!(Float::from(1.5) / &Float::ZERO, Float::INFINITY);
assert_eq!(
    Float::from(1.5) / &Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(Float::from(-1.5) / &Float::ZERO, Float::NEGATIVE_INFINITY);
assert_eq!(Float::from(-1.5) / &Float::NEGATIVE_ZERO, Float::INFINITY);
assert!((Float::ZERO / &Float::ZERO).is_nan());

assert_eq!((Float::from(1.5) / &Float::from(2.5)).to_string(), "0.6");
assert_eq!((Float::from(1.5) / &Float::from(-2.5)).to_string(), "-0.6");
assert_eq!((Float::from(-1.5) / &Float::from(2.5)).to_string(), "-0.6");
assert_eq!((Float::from(-1.5) / &Float::from(-2.5)).to_string(), "0.6");

type Output = Float

The resulting type after applying the / operator.

impl Div<&Float> for Rational

fn div(self, other: &Float) -> Float

Divides a Rational by a Float, taking the Rational by value and the Float by reference.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) / &Float::NAN).is_nan());
assert_eq!(Rational::exact_from(1.5) / &Float::ZERO, Float::INFINITY);
assert_eq!(
    Rational::exact_from(1.5) / &Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) / &Float::ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) / &Float::NEGATIVE_ZERO,
    Float::INFINITY
);

assert_eq!(
    (Rational::exact_from(1.5) / &Float::from(2.5)).to_string(),
    "0.6"
);
assert_eq!(
    (Rational::exact_from(-1.5) / &Float::from(2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (Rational::exact_from(1.5) / &Float::from(-2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (Rational::exact_from(-1.5) / &Float::from(-2.5)).to_string(),
    "0.6"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<&Rational> for &Float

fn div(self, other: &Rational) -> Float

Divides a Float by a Rational, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=f(\pm0.0,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x\geq 0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x\geq 0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x>0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x>0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_ref_ref instead. If you want to specify the output precision, consider using Float::div_rational_round_ref_ref. If you want both of these things, consider using Float::div_rational_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN / &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY / &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY / &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::INFINITY / &Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY / &Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(
    (&Float::from(2.5) / &Rational::exact_from(1.5)).to_string(),
    "1.8"
);
assert_eq!(
    (&Float::from(2.5) / &Rational::exact_from(-1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (&Float::from(-2.5) / &Rational::exact_from(1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (&Float::from(-2.5) / &Rational::exact_from(-1.5)).to_string(),
    "1.8"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<&Rational> for Float

fn div(self, other: &Rational) -> Self

Divides a Float by a Rational, taking the first by value and the second by reference.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=f(\pm0.0,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x\geq 0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x\geq 0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x>0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x>0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_val_ref instead. If you want to specify the output precision, consider using Float::div_rational_round_val_ref. If you want both of these things, consider using Float::div_rational_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN / &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    Float::INFINITY / &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY / &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::INFINITY / &Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY / &Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(
    (Float::from(2.5) / &Rational::exact_from(1.5)).to_string(),
    "1.8"
);
assert_eq!(
    (Float::from(2.5) / &Rational::exact_from(-1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (Float::from(-2.5) / &Rational::exact_from(1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (Float::from(-2.5) / &Rational::exact_from(-1.5)).to_string(),
    "1.8"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<Float> for &Float

fn div(self, other: Float) -> Float

Divides two Floats, taking the first by reference and the second by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm\infty)=f(\pm0.0,\pm0.0) = \text{NaN}$
• $f(\infty,x)=\infty$ if $0.0<x<\infty$
• $f(\infty,x)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0)=\infty$ if $x>0.0$
• $f(x,0.0)=-\infty$ if $x<0.0$
• $f(-\infty,x)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0)=-\infty$ if $x>0.0$
• $f(x,-0.0)=\infty$ if $x<0.0$
• $f(0.0,x)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_ref_val instead. If you want to specify the output precision, consider using Float::div_round_ref_val. If you want both of these things, consider using Float::div_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_float::Float;

assert!((&Float::from(1.5) / Float::NAN).is_nan());
assert_eq!(&Float::from(1.5) / Float::ZERO, Float::INFINITY);
assert_eq!(
    &Float::from(1.5) / Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(&Float::from(-1.5) / Float::ZERO, Float::NEGATIVE_INFINITY);
assert_eq!(&Float::from(-1.5) / Float::NEGATIVE_ZERO, Float::INFINITY);
assert!((&Float::ZERO / Float::ZERO).is_nan());

assert_eq!((&Float::from(1.5) / Float::from(2.5)).to_string(), "0.6");
assert_eq!((&Float::from(1.5) / Float::from(-2.5)).to_string(), "-0.6");
assert_eq!((&Float::from(-1.5) / Float::from(2.5)).to_string(), "-0.6");
assert_eq!((&Float::from(-1.5) / Float::from(-2.5)).to_string(), "0.6");

type Output = Float

The resulting type after applying the / operator.

impl Div<Float> for &Rational

fn div(self, other: Float) -> Float

Divides a Rational by a Float, taking the Rational by reference and the Float by value.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) / Float::NAN).is_nan());
assert_eq!(&Rational::exact_from(1.5) / Float::ZERO, Float::INFINITY);
assert_eq!(
    &Rational::exact_from(1.5) / Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) / Float::ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) / Float::NEGATIVE_ZERO,
    Float::INFINITY
);

assert_eq!(
    (&Rational::exact_from(1.5) / Float::from(2.5)).to_string(),
    "0.6"
);
assert_eq!(
    (&Rational::exact_from(-1.5) / Float::from(2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (&Rational::exact_from(1.5) / Float::from(-2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (&Rational::exact_from(-1.5) / Float::from(-2.5)).to_string(),
    "0.6"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<Float> for Rational

fn div(self, other: Float) -> Float

Divides a Rational by a Float, taking both by value.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN},p,m)=f(0,\pm0.0,p,m)=\text{NaN}$
• $f(x,\infty,x,p,m)=0.0$ if $x>0.0$ or $x=0.0$
• $f(x,\infty,x,p,m)=-0.0$ if $x<0.0$ or #x=-0.0$
• $f(x,-\infty,x,p,m)=-0.0$ if $x>0.0$ or $x=0.0$
• $f(x,-\infty,x,p,m)=0.0$ if $x<0.0$ or #x=-0.0$
• $f(0,x,p,m)=0.0$ if $x>0$
• $f(0,x,p,m)=-0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) / Float::NAN).is_nan());
assert_eq!(Rational::exact_from(1.5) / Float::ZERO, Float::INFINITY);
assert_eq!(
    Rational::exact_from(1.5) / Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) / Float::ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) / Float::NEGATIVE_ZERO,
    Float::INFINITY
);

assert_eq!(
    (Rational::exact_from(1.5) / Float::from(2.5)).to_string(),
    "0.6"
);
assert_eq!(
    (Rational::exact_from(-1.5) / Float::from(2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (Rational::exact_from(1.5) / Float::from(-2.5)).to_string(),
    "-0.6"
);
assert_eq!(
    (Rational::exact_from(-1.5) / Float::from(-2.5)).to_string(),
    "0.6"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<Rational> for &Float

fn div(self, other: Rational) -> Float

Divides a Float by a Rational, taking the first by reference and the second by value.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=f(\pm0.0,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x\geq 0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x\geq 0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x>0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x>0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_ref_val instead. If you want to specify the output precision, consider using Float::div_rational_round_ref_val. If you want both of these things, consider using Float::div_rational_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN / Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY / Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY / Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::INFINITY / Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY / Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(
    (&Float::from(2.5) / Rational::exact_from(1.5)).to_string(),
    "1.8"
);
assert_eq!(
    (&Float::from(2.5) / Rational::exact_from(-1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (&Float::from(-2.5) / Rational::exact_from(1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (&Float::from(-2.5) / Rational::exact_from(-1.5)).to_string(),
    "1.8"
);

type Output = Float

The resulting type after applying the / operator.

impl Div<Rational> for Float

fn div(self, other: Rational) -> Self

Divides a Float by a Rational, taking both by value.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=f(\pm0.0,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x\geq 0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x\geq 0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x>0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x>0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec instead. If you want to specify the output precision, consider using Float::div_rational_round. If you want both of these things, consider using Float::div_rational_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN / Rational::exact_from(1.5)).is_nan());
assert_eq!(Float::INFINITY / Rational::exact_from(1.5), Float::INFINITY);
assert_eq!(
    Float::NEGATIVE_INFINITY / Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::INFINITY / Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY / Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(
    (Float::from(2.5) / Rational::exact_from(1.5)).to_string(),
    "1.8"
);
assert_eq!(
    (Float::from(2.5) / Rational::exact_from(-1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (Float::from(-2.5) / Rational::exact_from(1.5)).to_string(),
    "-1.8"
);
assert_eq!(
    (Float::from(-2.5) / Rational::exact_from(-1.5)).to_string(),
    "1.8"
);

type Output = Float

The resulting type after applying the / operator.

impl Div for Float

fn div(self, other: Self) -> Self

Divides two Floats, taking both by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm\infty)=f(\pm0.0,\pm0.0) = \text{NaN}$
• $f(\infty,x)=\infty$ if $0.0<x<\infty$
• $f(\infty,x)=-\infty$ if $-\infty<x<0.0$
• $f(x,0.0)=\infty$ if $x>0.0$
• $f(x,0.0)=-\infty$ if $x<0.0$
• $f(-\infty,x)=-\infty$ if $0.0<x<\infty$
• $f(-\infty,x)=\infty$ if $-\infty<x<0.0$
• $f(x,-0.0)=-\infty$ if $x>0.0$
• $f(x,-0.0)=\infty$ if $x<0.0$
• $f(0.0,x)=0.0$ if $x$ is not NaN and $x>0.0$
• $f(0.0,x)=-0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=-0.0$ if $x$ is not NaN and $x>0.0$
• $f(-0.0,x)=0.0$ if $x$ is not NaN and $x<0.0$
• $f(x,-\infty)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(x,-\infty)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec instead. If you want to specify the output precision, consider using Float::div_round. If you want both of these things, consider using Float::div_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_float::Float;

assert!((Float::from(1.5) / Float::NAN).is_nan());
assert_eq!(Float::from(1.5) / Float::ZERO, Float::INFINITY);
assert_eq!(
    Float::from(1.5) / Float::NEGATIVE_ZERO,
    Float::NEGATIVE_INFINITY
);
assert_eq!(Float::from(-1.5) / Float::ZERO, Float::NEGATIVE_INFINITY);
assert_eq!(Float::from(-1.5) / Float::NEGATIVE_ZERO, Float::INFINITY);
assert!((Float::ZERO / Float::ZERO).is_nan());

assert_eq!((Float::from(1.5) / Float::from(2.5)).to_string(), "0.6");
assert_eq!((Float::from(1.5) / Float::from(-2.5)).to_string(), "-0.6");
assert_eq!((Float::from(-1.5) / Float::from(2.5)).to_string(), "-0.6");
assert_eq!((Float::from(-1.5) / Float::from(-2.5)).to_string(), "0.6");

type Output = Float

The resulting type after applying the / operator.

impl DivAssign<&Float> for Float

fn div_assign(&mut self, other: &Self)

Divides a Float by a Float in place, taking the Float on the right-hand side by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the / documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_assign instead. If you want to specify the output precision, consider using Float::div_round_assign. If you want both of these things, consider using Float::div_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_float::Float;

let mut x = Float::from(1.5);
x /= &Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x /= &Float::ZERO;
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x /= &Float::NEGATIVE_ZERO;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x /= &Float::ZERO;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x /= &Float::NEGATIVE_ZERO;
assert_eq!(x, Float::INFINITY);

let mut x = Float::INFINITY;
x /= &Float::INFINITY;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x /= &Float::from(2.5);
assert_eq!(x.to_string(), "0.6");

let mut x = Float::from(1.5);
x /= &Float::from(-2.5);
assert_eq!(x.to_string(), "-0.6");

let mut x = Float::from(-1.5);
x /= &Float::from(2.5);
assert_eq!(x.to_string(), "-0.6");

let mut x = Float::from(-1.5);
x /= &Float::from(-2.5);
assert_eq!(x.to_string(), "0.6");

impl DivAssign<&Rational> for Float

fn div_assign(&mut self, other: &Rational)

Divides a Float by a Rational in place, taking the Rational by reference.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

See the / documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_assign_ref instead. If you want to specify the output precision, consider using Float::div_rational_round_assign_ref. If you want both of these things, consider using Float::div_rational_prec_round_assign_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x /= &Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x /= &Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x /= &Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::INFINITY;
x /= &Rational::exact_from(-1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x /= &Rational::exact_from(-1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(2.5);
x /= &Rational::exact_from(1.5);
assert_eq!(x.to_string(), "1.8");

impl DivAssign<Rational> for Float

fn div_assign(&mut self, other: Rational)

Divides a Float by a Rational in place, taking the Rational by value.

If the output has a precision, it is the precision of the input Float. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the precision of the input Float.

See the / documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_rational_prec_assign instead. If you want to specify the output precision, consider using Float::div_rational_round_assign. If you want both of these things, consider using Float::div_rational_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x /= Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x /= Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x /= Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::INFINITY;
x /= Rational::exact_from(-1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x /= Rational::exact_from(-1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(2.5);
x /= Rational::exact_from(1.5);
assert_eq!(x.to_string(), "1.8");

impl DivAssign for Float

fn div_assign(&mut self, other: Self)

Divides a Float by a Float in place, taking the Float on the right-hand side by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the quotient is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x/y+\varepsilon. $$

• If $x/y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x/y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x/y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the / documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::div_prec_assign instead. If you want to specify the output precision, consider using Float::div_round_assign. If you want both of these things, consider using Float::div_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeZero, Zero,
};
use malachite_float::Float;

let mut x = Float::from(1.5);
x /= Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x /= Float::ZERO;
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x /= Float::NEGATIVE_ZERO;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x /= Float::ZERO;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x /= Float::NEGATIVE_ZERO;
assert_eq!(x, Float::INFINITY);

let mut x = Float::INFINITY;
x /= Float::INFINITY;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x /= Float::from(2.5);
assert_eq!(x.to_string(), "0.6");

let mut x = Float::from(1.5);
x /= Float::from(-2.5);
assert_eq!(x.to_string(), "-0.6");

let mut x = Float::from(-1.5);
x /= Float::from(2.5);
assert_eq!(x.to_string(), "-0.6");

let mut x = Float::from(-1.5);
x /= Float::from(-2.5);
assert_eq!(x.to_string(), "0.6");

impl EqAbs<Float> for Integer

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of an Integer is equal to the absolute value of a Float.

No Integer is equal to $\infty$, $-\infty$, or NaN. The Integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_base::num::comparison::traits::EqAbs;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Integer::from(123).eq_abs(&Float::from(123)));
assert!(Integer::from(-123).eq_abs(&Float::from(-123)));
assert!(Integer::ONE.ne_abs(&Float::ONE_HALF));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for Natural

fn eq_abs(&self, other: &Float) -> bool

Determines whether a Natural is equal to the absolute value of a Float.

No Natural is equal to $\infty$, $-\infty$, or NaN. The Natural zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_base::num::comparison::traits::EqAbs;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Natural::from(123u32).eq_abs(&Float::from(123)));
assert!(Natural::ONE.ne_abs(&Float::ONE_HALF));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for Rational

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a Rational is equal to the absolute value of a Float.

No Rational is equal to $\infty$, $-\infty$, or NaN. The Rational zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::OneHalf;
use malachite_base::num::comparison::traits::EqAbs;
use malachite_float::Float;
use malachite_q::Rational;

assert!(Rational::from(123).eq_abs(&Float::from(123)));
assert!(Rational::from(-123).eq_abs(&Float::from(123)));
assert!(Rational::ONE_HALF.eq_abs(&Float::ONE_HALF));
assert!(Rational::from_unsigneds(1u8, 3).ne_abs(&Float::from(1.0f64 / 3.0)));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for f32

fn eq_abs(&self, other: &Float) -> bool

Determines whether a primitive float is equal to a Float.

The primitive float $\infty$ is equal to the Float $\infty$, and the primitive float $-\infty$ is equal to the Float $-\infty$. The primitive float NaN is not equal to anything, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for f64

fn eq_abs(&self, other: &Float) -> bool

Determines whether a primitive float is equal to a Float.

The primitive float $\infty$ is equal to the Float $\infty$, and the primitive float $-\infty$ is equal to the Float $-\infty$. The primitive float NaN is not equal to anything, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for i128

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a signed primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for i16

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a signed primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for i32

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a signed primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for i64

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a signed primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for i8

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a signed primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for isize

fn eq_abs(&self, other: &Float) -> bool

Determines whether the absolute value of a signed primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for u128

fn eq_abs(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for u16

fn eq_abs(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for u32

fn eq_abs(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for u64

fn eq_abs(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for u8

fn eq_abs(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Float> for usize

fn eq_abs(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to the absolute value of a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Integer> for Float

fn eq_abs(&self, other: &Integer) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of an Integer.

$\infty$, $-\infty$, and NaN are not equal to any Integer. Both the Float zero and the Float negative zero are equal to the Integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_base::num::comparison::traits::EqAbs;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Float::from(123).eq_abs(&Integer::from(123)));
assert!(Float::from(-123).eq_abs(&Integer::from(-123)));
assert!(Float::ONE_HALF.ne_abs(&Integer::ONE));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Natural> for Float

fn eq_abs(&self, other: &Natural) -> bool

Determines whether the absolute value of a Float is equal to a Natural.

$\infty$, $-\infty$, and NaN are not equal to any Natural. Both the Float zero and the Float negative zero are equal to the Natural zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_base::num::comparison::traits::EqAbs;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Float::from(123).eq_abs(&Natural::from(123u32)));
assert!(Float::ONE_HALF.ne_abs(&Natural::ONE));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<Rational> for Float

fn eq_abs(&self, other: &Rational) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a Rational.

$\infty$, $-\infty$, and NaN are not equal to any Rational. Both the Float zero and the Float negative zero are equal to the Rational zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::OneHalf;
use malachite_base::num::comparison::traits::EqAbs;
use malachite_float::Float;
use malachite_q::Rational;

assert!(Float::from(123).eq_abs(&Rational::from(123)));
assert!(Float::from(-123).eq_abs(&Rational::from(123)));
assert!(Float::ONE_HALF.eq_abs(&Rational::ONE_HALF));
assert!(Float::from(1.0f64 / 3.0).ne_abs(&Rational::from_unsigneds(1u8, 3)));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<f32> for Float

fn eq_abs(&self, other: &f32) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a primitive float.

The Float $\infty$ is equal to the primitive float $\infty$, and the Float $-\infty$ is equal to the primitive float $-\infty$. The Float NaN is not equal to anything, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<f64> for Float

fn eq_abs(&self, other: &f64) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a primitive float.

The Float $\infty$ is equal to the primitive float $\infty$, and the Float $-\infty$ is equal to the primitive float $-\infty$. The Float NaN is not equal to anything, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<i128> for Float

fn eq_abs(&self, other: &i128) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<i16> for Float

fn eq_abs(&self, other: &i16) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<i32> for Float

fn eq_abs(&self, other: &i32) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<i64> for Float

fn eq_abs(&self, other: &i64) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<i8> for Float

fn eq_abs(&self, other: &i8) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<isize> for Float

fn eq_abs(&self, other: &isize) -> bool

Determines whether the absolute value of a Float is equal to the absolute value of a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<u128> for Float

fn eq_abs(&self, other: &u128) -> bool

Determines whether the absolute value of a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<u16> for Float

fn eq_abs(&self, other: &u16) -> bool

Determines whether the absolute value of a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<u32> for Float

fn eq_abs(&self, other: &u32) -> bool

Determines whether the absolute value of a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<u64> for Float

fn eq_abs(&self, other: &u64) -> bool

Determines whether the absolute value of a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<u8> for Float

fn eq_abs(&self, other: &u8) -> bool

Determines whether the absolute value of a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs<usize> for Float

fn eq_abs(&self, other: &usize) -> bool

Determines whether the absolute value of a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl EqAbs for Float

fn eq_abs(&self, other: &Self) -> bool

Compares the absolute values of two Floats for equality.

This implementation follows the IEEE 754 standard. NaN is not equal to anything, not even itself. Positive zero is equal to negative zero. Floats with different precisions are equal if they represent the same numeric value.

For different equality behavior, consider using ComparableFloat or ComparableFloatRef.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{NaN, NegativeZero, One, Two, Zero};
use malachite_float::Float;

assert_ne!(Float::NAN, Float::NAN);
assert_eq!(Float::ZERO, Float::ZERO);
assert_eq!(Float::NEGATIVE_ZERO, Float::NEGATIVE_ZERO);
assert_eq!(Float::ZERO, Float::NEGATIVE_ZERO);

assert_eq!(Float::ONE, Float::ONE);
assert_ne!(Float::ONE, Float::TWO);
assert_eq!(Float::ONE, Float::one_prec(100));

fn ne_abs(&self, other: &Rhs) -> bool

Compares the absolute values of two numbers for inequality, taking both by reference.

impl From<f32> for Float

fn from(x: f32) -> Float

Converts a primitive float to a Float.

If the primitive float is finite and nonzero, the precision of the Float is the minimum possible precision to represent the primitive float exactly. If you want to specify a different precision, try Float::from_primitive_float_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_primitive_float_prec_round.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is x.sci_exponent().abs().

Examples

See here.

impl From<f64> for Float

fn from(x: f64) -> Float

Converts a primitive float to a Float.

If the primitive float is finite and nonzero, the precision of the Float is the minimum possible precision to represent the primitive float exactly. If you want to specify a different precision, try Float::from_primitive_float_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_primitive_float_prec_round.

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is x.sci_exponent().abs().

Examples

See here.

impl From<i128> for Float

fn from(i: i128) -> Float

Converts a signed primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_signed_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

If you want to create a Float from an signed primitive integer in a const context, try Float::const_from_signed instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<i16> for Float

fn from(i: i16) -> Float

Converts a signed primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_signed_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

If you want to create a Float from an signed primitive integer in a const context, try Float::const_from_signed instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<i32> for Float

fn from(i: i32) -> Float

Converts a signed primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_signed_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

If you want to create a Float from an signed primitive integer in a const context, try Float::const_from_signed instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<i64> for Float

fn from(i: i64) -> Float

Converts a signed primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_signed_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

If you want to create a Float from an signed primitive integer in a const context, try Float::const_from_signed instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<i8> for Float

fn from(i: i8) -> Float

Converts a signed primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_signed_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

If you want to create a Float from an signed primitive integer in a const context, try Float::const_from_signed instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<isize> for Float

fn from(i: isize) -> Float

Converts a signed primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_signed_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_signed_prec_round.

If you want to create a Float from an signed primitive integer in a const context, try Float::const_from_signed instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<u128> for Float

fn from(u: u128) -> Float

Converts an unsigned primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_unsigned_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

If you want to create a Float from an unsigned primitive integer in a const context, try Float::const_from_unsigned instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<u16> for Float

fn from(u: u16) -> Float

Converts an unsigned primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_unsigned_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

If you want to create a Float from an unsigned primitive integer in a const context, try Float::const_from_unsigned instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<u32> for Float

fn from(u: u32) -> Float

Converts an unsigned primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_unsigned_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

If you want to create a Float from an unsigned primitive integer in a const context, try Float::const_from_unsigned instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<u64> for Float

fn from(u: u64) -> Float

Converts an unsigned primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_unsigned_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

If you want to create a Float from an unsigned primitive integer in a const context, try Float::const_from_unsigned instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<u8> for Float

fn from(u: u8) -> Float

Converts an unsigned primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_unsigned_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

If you want to create a Float from an unsigned primitive integer in a const context, try Float::const_from_unsigned instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl From<usize> for Float

fn from(u: usize) -> Float

Converts an unsigned primitive integer to a Float.

If the integer is nonzero, the precision of the Float is equal to the integer’s number of significant bits. If you want to specify a different precision, try Float::from_unsigned_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_unsigned_prec_round.

If you want to create a Float from an unsigned primitive integer in a const context, try Float::const_from_unsigned instead.

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

impl FromStringBase for Float

fn from_string_base(base: u8, s: &str) -> Option<Self>

impl Infinity for Float

The constant $\infty$.

const INFINITY: Self

impl IntegerMantissaAndExponent<Natural, i64> for Float

fn integer_mantissa_and_exponent(self) -> (Natural, i64)

Returns a Float’s integer mantissa and exponent, taking the Float by value.

When $x$ is finite and nonzero, we can write $x = 2^{e_i}m_i$, where $e_i$ is an integer and $m_i$ is an odd integer. $$ f(x) = (\frac{|x|}{2^{e_i}}, e_i), $$ where $e_i$ is the unique integer such that $x/2^{e_i}$ is an odd integer.

The inverse operation is from_integer_mantissa_and_exponent.

The integer exponent is less than or equal to $2^{30}-2$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, IntegerMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;
use std::str::FromStr;

assert_eq!(
    Float::ONE.integer_mantissa_and_exponent(),
    (Natural::ONE, 0)
);
assert_eq!(
    Float::from(std::f64::consts::PI).integer_mantissa_and_exponent(),
    (Natural::from(884279719003555u64), -48)
);
assert_eq!(
    Float::exact_from(Natural::from(3u32).pow(50u64)).integer_mantissa_and_exponent(),
    (Natural::from_str("717897987691852588770249").unwrap(), 0)
);
assert_eq!(
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100)
        .0
        .integer_mantissa_and_exponent(),
    (
        Natural::from_str("1067349099133908271875104088939").unwrap(),
        -179
    )
);

fn integer_exponent(self) -> i64

Returns a Float’s integer exponent, taking the Float by value.

When $x$ is finite and nonzero, we can write $x = 2^{e_i}m_i$, where $e_i$ is an integer and $m_i$ is an odd integer. $$ f(x) = e_i, $$ where $e_i$ is the unique integer such that $x/2^{e_i}$ is an odd integer.

The integer exponent is less than or equal to $2^{30}-2$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, IntegerMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(Float::ONE.integer_exponent(), 0);
assert_eq!(Float::from(std::f64::consts::PI).integer_exponent(), -48);
assert_eq!(
    Float::exact_from(Natural::from(3u32).pow(50u64)).integer_exponent(),
    0
);
assert_eq!(
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100)
        .0
        .integer_exponent(),
    -179
);

fn from_integer_mantissa_and_exponent( integer_mantissa: Natural, integer_exponent: i64, ) -> Option<Self>

Constructs a Float from its integer mantissa and exponent.

When $x$ is finite and nonzero, we can write $x = 2^{e_i}m_i$, where $e_i$ is an integer and $m_i$ is an odd integer.

$$ f(x) = 2^{e_i}m_i. $$

The input does not have to be reduced; that is, the mantissa does not have to be odd. If the inputs correspond to a number too large in absolute value or too close to zero to be represented by a Float, None is returned.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is integer_mantissa.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::IntegerMantissaAndExponent;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;
use std::str::FromStr;

assert_eq!(
    Float::from_integer_mantissa_and_exponent(Natural::ONE, 0).unwrap(),
    1
);
assert_eq!(
    Float::from_integer_mantissa_and_exponent(Natural::from(884279719003555u64), -48)
        .unwrap(),
    std::f64::consts::PI
);
assert_eq!(
    Float::from_integer_mantissa_and_exponent(
        Natural::from_str("717897987691852588770249").unwrap(),
        0
    )
    .unwrap(),
    Natural::from(3u32).pow(50u64)
);
assert_eq!(
    Float::from_integer_mantissa_and_exponent(
        Natural::from_str("1067349099133908271875104088939").unwrap(),
        -179
    )
    .unwrap(),
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
);

fn integer_mantissa(self) -> M

Extracts the integer mantissa from a number.

impl IntegerMantissaAndExponent<Natural, i64, Float> for &Float

fn integer_mantissa_and_exponent(self) -> (Natural, i64)

Returns a Float’s integer mantissa and exponent, taking the Float by reference.

When $x$ is finite and nonzero, we can write $x = 2^{e_i}m_i$, where $e_i$ is an integer and $m_i$ is an odd integer. $$ f(x) = (\frac{|x|}{2^{e_i}}, e_i), $$ where $e_i$ is the unique integer such that $x/2^{e_i}$ is an odd integer.

The inverse operation is from_integer_mantissa_and_exponent.

The integer exponent is less than or equal to $2^{30}-2$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, IntegerMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;
use std::str::FromStr;

assert_eq!(
    (&Float::ONE).integer_mantissa_and_exponent(),
    (Natural::ONE, 0)
);
assert_eq!(
    (&Float::from(std::f64::consts::PI)).integer_mantissa_and_exponent(),
    (Natural::from(884279719003555u64), -48)
);
assert_eq!(
    (&Float::exact_from(Natural::from(3u32).pow(50u64))).integer_mantissa_and_exponent(),
    (Natural::from_str("717897987691852588770249").unwrap(), 0)
);
assert_eq!(
    (&Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0)
        .integer_mantissa_and_exponent(),
    (
        Natural::from_str("1067349099133908271875104088939").unwrap(),
        -179
    )
);

fn integer_exponent(self) -> i64

Returns a Float’s integer exponent, taking the Float by reference.

When $x$ is finite and nonzero, we can write $x = 2^{e_i}m_i$, where $e_i$ is an integer and $m_i$ is an odd integer. $$ f(x) = e_i, $$ where $e_i$ is the unique integer such that $x/2^{e_i}$ is an odd integer.

The integer exponent is less than or equal to $2^{30}-2$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, IntegerMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!((&Float::ONE).integer_exponent(), 0);
assert_eq!((&Float::from(std::f64::consts::PI)).integer_exponent(), -48);
assert_eq!(
    (&Float::exact_from(Natural::from(3u32).pow(50u64))).integer_exponent(),
    0
);
assert_eq!(
    (&Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0)
        .integer_exponent(),
    -179
);

fn from_integer_mantissa_and_exponent( integer_mantissa: Natural, integer_exponent: i64, ) -> Option<Float>

Constructs a Float from its integer mantissa and exponent.

When $x$ is finite and nonzero, we can write $x = 2^{e_i}m_i$, where $e_i$ is an integer and $m_i$ is an odd integer.

$$ f(x) = 2^{e_i}m_i. $$

The input does not have to be reduced; that is, the mantissa does not have to be odd. If the inputs correspond to a number too large in absolute value or too close to zero to be represented by a Float, None is returned.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is integer_mantissa.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::IntegerMantissaAndExponent;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;
use std::str::FromStr;

assert_eq!(
    <&Float as IntegerMantissaAndExponent<_, _, _>>::from_integer_mantissa_and_exponent(
        Natural::ONE,
        0
    )
    .unwrap(),
    1
);
assert_eq!(
    <&Float as IntegerMantissaAndExponent<_, _, _>>::from_integer_mantissa_and_exponent(
        Natural::from(884279719003555u64),
        -48
    )
    .unwrap(),
    std::f64::consts::PI
);
assert_eq!(
    <&Float as IntegerMantissaAndExponent<_, _, _>>::from_integer_mantissa_and_exponent(
        Natural::from_str("717897987691852588770249").unwrap(),
        0
    )
    .unwrap(),
    Natural::from(3u32).pow(50u64)
);
assert_eq!(
    <&Float as IntegerMantissaAndExponent<_, _, _>>::from_integer_mantissa_and_exponent(
        Natural::from_str("1067349099133908271875104088939").unwrap(),
        -179
    )
    .unwrap(),
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
);

fn integer_mantissa(self) -> M

Extracts the integer mantissa from a number.

impl IsInteger for &Float

fn is_integer(self) -> bool

Determines whether a Float is an integer.

$f(x) = x \in \Z$.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{One, OneHalf, Zero};
use malachite_base::num::conversion::traits::IsInteger;
use malachite_float::Float;

assert_eq!(Float::ZERO.is_integer(), true);
assert_eq!(Float::ONE.is_integer(), true);
assert_eq!(Float::from(100).is_integer(), true);
assert_eq!(Float::from(-100).is_integer(), true);
assert_eq!(Float::ONE_HALF.is_integer(), false);
assert_eq!((-Float::ONE_HALF).is_integer(), false);

impl IsPowerOf2 for Float

fn is_power_of_2(&self) -> bool

Determines whether a Float is an integer power of 2.

$f(x) = (\exists n \in \Z : 2^n = x)$.

Floats that are NaN, infinite, or zero are not powers of 2.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::IsPowerOf2;
use malachite_base::num::basic::traits::{NaN, One, OneHalf, Two};
use malachite_float::Float;

assert_eq!(Float::NAN.is_power_of_2(), false);

assert_eq!(Float::ONE.is_power_of_2(), true);
assert_eq!(Float::TWO.is_power_of_2(), true);
assert_eq!(Float::ONE_HALF.is_power_of_2(), true);
assert_eq!(Float::from(1024).is_power_of_2(), true);

assert_eq!(Float::from(3).is_power_of_2(), false);
assert_eq!(Float::from(1025).is_power_of_2(), false);
assert_eq!(Float::from(0.1f64).is_power_of_2(), false);

impl LowerHex for Float

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Max for Float

The highest value representable by this type, $\infty$.

const MAX: Self = Self::INFINITY

The maximum value of Self.

impl Min for Float

The lowest value representable by this type, $-\infty$.

const MIN: Self = Self::NEGATIVE_INFINITY

The minimum value of Self.

impl Mul<&Float> for &Float

fn mul(self, other: &Float) -> Float

Multiplies two Floats, taking both by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm0.0)=f(\pm0.0,\pm\infty) = \text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x>0.0$
• $f(\infty,x)=f(x,\infty)=-\infty$ if $x<0.0$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x>0.0$
• $f(-\infty,x)=f(x,-\infty)=\infty$ if $x<0.0$
• $f(0.0,x)=f(x,0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x)=f(x,0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=f(x,-0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x)=f(x,-0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_ref_ref instead. If you want to specify the output precision, consider using Float::mul_round_ref_ref. If you want both of these things, consider using Float::mul_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity, Zero};
use malachite_float::Float;

assert!((&Float::from(1.5) * &Float::NAN).is_nan());
assert_eq!(&Float::from(1.5) * &Float::INFINITY, Float::INFINITY);
assert_eq!(
    &Float::from(1.5) * &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::from(-1.5) * &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::from(-1.5) * &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((&Float::INFINITY * &Float::ZERO).is_nan());

assert_eq!(&Float::from(1.5) * &Float::from(2.5), 4.0);
assert_eq!(&Float::from(1.5) * &Float::from(-2.5), -4.0);
assert_eq!(&Float::from(-1.5) * &Float::from(2.5), -4.0);
assert_eq!(&Float::from(-1.5) * &Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<&Float> for &Rational

fn mul(self, other: &Float) -> Float

Multiplies a Rational by a Float, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=f(0,\pm\infty)=\text{NaN}$
• $f(x,\infty)=\infty$ if $x>0$
• $f(x,\infty)=-\infty$ if $x<0$
• $f(x,-\infty)=-\infty$ if $x>0$
• $f(x,-\infty)=\infty$ if $x<0$
• $f(x,0.0)=0.0$ if $x\geq0$
• $f(x,0.0)=-0.0$ if $x<0$
• $f(x,-0.0)=-0.0$ if $x\geq0$
• $f(x,-0.0)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) * &Float::NAN).is_nan());
assert_eq!(
    &Rational::exact_from(1.5) * &Float::INFINITY,
    Float::INFINITY
);
assert_eq!(
    &Rational::exact_from(1.5) * &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) * &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) * &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(&Rational::exact_from(1.5) * &Float::from(2.5), 4.0);
assert_eq!(&Rational::exact_from(-1.5) * &Float::from(2.5), -4.0);
assert_eq!(&Rational::exact_from(1.5) * &Float::from(-2.5), -4.0);
assert_eq!(&Rational::exact_from(-1.5) * &Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<&Float> for Float

fn mul(self, other: &Self) -> Self

Multiplies two Floats, taking the first by value and the second by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm0.0)=f(\pm0.0,\pm\infty) = \text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x>0.0$
• $f(\infty,x)=f(x,\infty)=-\infty$ if $x<0.0$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x>0.0$
• $f(-\infty,x)=f(x,-\infty)=\infty$ if $x<0.0$
• $f(0.0,x)=f(x,0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x)=f(x,0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=f(x,-0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x)=f(x,-0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_val_ref instead. If you want to specify the output precision, consider using Float::mul_round_val_ref. If you want both of these things, consider using Float::mul_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity, Zero};
use malachite_float::Float;

assert!((Float::from(1.5) * &Float::NAN).is_nan());
assert_eq!(Float::from(1.5) * &Float::INFINITY, Float::INFINITY);
assert_eq!(
    Float::from(1.5) * &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::from(-1.5) * &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::from(-1.5) * &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((Float::INFINITY * &Float::ZERO).is_nan());

assert_eq!(Float::from(1.5) * &Float::from(2.5), 4.0);
assert_eq!(Float::from(1.5) * &Float::from(-2.5), -4.0);
assert_eq!(Float::from(-1.5) * &Float::from(2.5), -4.0);
assert_eq!(Float::from(-1.5) * &Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<&Float> for Rational

fn mul(self, other: &Float) -> Float

Multiplies a Rational by a Float, taking the Rational by value and the Float by reference.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=f(0,\pm\infty)=\text{NaN}$
• $f(x,\infty)=\infty$ if $x>0$
• $f(x,\infty)=-\infty$ if $x<0$
• $f(x,-\infty)=-\infty$ if $x>0$
• $f(x,-\infty)=\infty$ if $x<0$
• $f(x,0.0)=0.0$ if $x\geq0$
• $f(x,0.0)=-0.0$ if $x<0$
• $f(x,-0.0)=-0.0$ if $x\geq0$
• $f(x,-0.0)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) * &Float::NAN).is_nan());
assert_eq!(
    Rational::exact_from(1.5) * &Float::INFINITY,
    Float::INFINITY
);
assert_eq!(
    Rational::exact_from(1.5) * &Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) * &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) * &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(Rational::exact_from(1.5) * &Float::from(2.5), 4.0);
assert_eq!(Rational::exact_from(-1.5) * &Float::from(2.5), -4.0);
assert_eq!(Rational::exact_from(1.5) * &Float::from(-2.5), -4.0);
assert_eq!(Rational::exact_from(-1.5) * &Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<&Rational> for &Float

fn mul(self, other: &Rational) -> Float

Multiplies a Float by a Rational, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x>0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x>0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x\geq0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x\geq0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_ref_ref instead. If you want to specify the output precision, consider using Float::mul_rational_round_ref_ref. If you want both of these things, consider using Float::mul_rational_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN * &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY * &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY * &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::INFINITY * &Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY * &Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(&Float::from(2.5) * &Rational::exact_from(1.5), 4.0);
assert_eq!(&Float::from(2.5) * &Rational::exact_from(-1.5), -4.0);
assert_eq!(&Float::from(-2.5) * &Rational::exact_from(1.5), -4.0);
assert_eq!(&Float::from(-2.5) * &Rational::exact_from(-1.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<&Rational> for Float

fn mul(self, other: &Rational) -> Self

Multiplies a Float by a Rational, taking the first by value and the second by reference.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x>0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x>0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x\geq0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x\geq0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_val_ref instead. If you want to specify the output precision, consider using Float::mul_rational_round_val_ref. If you want both of these things, consider using Float::mul_rational_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN * &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    Float::INFINITY * &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY * &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::INFINITY * &Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY * &Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(Float::from(2.5) * &Rational::exact_from(1.5), 4.0);
assert_eq!(Float::from(2.5) * &Rational::exact_from(-1.5), -4.0);
assert_eq!(Float::from(-2.5) * &Rational::exact_from(1.5), -4.0);
assert_eq!(Float::from(-2.5) * &Rational::exact_from(-1.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<Float> for &Float

fn mul(self, other: Float) -> Float

Multiplies two Floats, taking the first by reference and the second by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm0.0)=f(\pm0.0,\pm\infty) = \text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x>0.0$
• $f(\infty,x)=f(x,\infty)=-\infty$ if $x<0.0$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x>0.0$
• $f(-\infty,x)=f(x,-\infty)=\infty$ if $x<0.0$
• $f(0.0,x)=f(x,0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x)=f(x,0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=f(x,-0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x)=f(x,-0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_ref_val instead. If you want to specify the output precision, consider using Float::mul_round_ref_val. If you want both of these things, consider using Float::mul_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity, Zero};
use malachite_float::Float;

assert!((&Float::from(1.5) * Float::NAN).is_nan());
assert_eq!(&Float::from(1.5) * Float::INFINITY, Float::INFINITY);
assert_eq!(
    &Float::from(1.5) * Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::from(-1.5) * Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::from(-1.5) * Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((&Float::INFINITY * Float::ZERO).is_nan());

assert_eq!(&Float::from(1.5) * Float::from(2.5), 4.0);
assert_eq!(&Float::from(1.5) * Float::from(-2.5), -4.0);
assert_eq!(&Float::from(-1.5) * Float::from(2.5), -4.0);
assert_eq!(&Float::from(-1.5) * Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<Float> for &Rational

fn mul(self, other: Float) -> Float

Multiplies a Rational by a Float, taking the Rational by reference and the Float by value.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=f(0,\pm\infty)=\text{NaN}$
• $f(x,\infty)=\infty$ if $x>0$
• $f(x,\infty)=-\infty$ if $x<0$
• $f(x,-\infty)=-\infty$ if $x>0$
• $f(x,-\infty)=\infty$ if $x<0$
• $f(x,0.0)=0.0$ if $x\geq0$
• $f(x,0.0)=-0.0$ if $x<0$
• $f(x,-0.0)=-0.0$ if $x\geq0$
• $f(x,-0.0)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) * Float::NAN).is_nan());
assert_eq!(
    &Rational::exact_from(1.5) * Float::INFINITY,
    Float::INFINITY
);
assert_eq!(
    &Rational::exact_from(1.5) * Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) * Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(-1.5) * Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(&Rational::exact_from(1.5) * Float::from(2.5), 4.0);
assert_eq!(&Rational::exact_from(-1.5) * Float::from(2.5), -4.0);
assert_eq!(&Rational::exact_from(1.5) * Float::from(-2.5), -4.0);
assert_eq!(&Rational::exact_from(-1.5) * Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<Float> for Rational

fn mul(self, other: Float) -> Float

Multiplies a Rational by a Float, taking both by value.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=f(0,\pm\infty)=\text{NaN}$
• $f(x,\infty)=\infty$ if $x>0$
• $f(x,\infty)=-\infty$ if $x<0$
• $f(x,-\infty)=-\infty$ if $x>0$
• $f(x,-\infty)=\infty$ if $x<0$
• $f(x,0.0)=0.0$ if $x\geq0$
• $f(x,0.0)=-0.0$ if $x<0$
• $f(x,-0.0)=-0.0$ if $x\geq0$
• $f(x,-0.0)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) * Float::NAN).is_nan());
assert_eq!(Rational::exact_from(1.5) * Float::INFINITY, Float::INFINITY);
assert_eq!(
    Rational::exact_from(1.5) * Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) * Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(-1.5) * Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(Rational::exact_from(1.5) * Float::from(2.5), 4.0);
assert_eq!(Rational::exact_from(-1.5) * Float::from(2.5), -4.0);
assert_eq!(Rational::exact_from(1.5) * Float::from(-2.5), -4.0);
assert_eq!(Rational::exact_from(-1.5) * Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<Rational> for &Float

fn mul(self, other: Rational) -> Float

Multiplies a Float by a Rational, taking the first by reference and the second by value.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x>0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x>0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x\geq0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x\geq0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_ref_val instead. If you want to specify the output precision, consider using Float::mul_rational_round_ref_val. If you want both of these things, consider using Float::mul_rational_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN * Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY * Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY * Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::INFINITY * Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY * Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(&Float::from(2.5) * Rational::exact_from(1.5), 4.0);
assert_eq!(&Float::from(2.5) * Rational::exact_from(-1.5), -4.0);
assert_eq!(&Float::from(-2.5) * Rational::exact_from(1.5), -4.0);
assert_eq!(&Float::from(-2.5) * Rational::exact_from(-1.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul<Rational> for Float

fn mul(self, other: Rational) -> Self

Multiplies a Float by a Rational, taking both by value.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=f(\pm\infty,0)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x>0$
• $f(\infty,x)=-\infty$ if $x<0$
• $f(-\infty,x)=-\infty$ if $x>0$
• $f(-\infty,x)=\infty$ if $x<0$
• $f(0.0,x)=0.0$ if $x\geq0$
• $f(0.0,x)=-0.0$ if $x<0$
• $f(-0.0,x)=-0.0$ if $x\geq0$
• $f(-0.0,x)=0.0$ if $x<0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec instead. If you want to specify the output precision, consider using Float::mul_rational_round. If you want both of these things, consider using Float::mul_rational_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN * Rational::exact_from(1.5)).is_nan());
assert_eq!(Float::INFINITY * Rational::exact_from(1.5), Float::INFINITY);
assert_eq!(
    Float::NEGATIVE_INFINITY * Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::INFINITY * Rational::exact_from(-1.5),
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY * Rational::exact_from(-1.5),
    Float::INFINITY
);

assert_eq!(Float::from(2.5) * Rational::exact_from(1.5), 4.0);
assert_eq!(Float::from(2.5) * Rational::exact_from(-1.5), -4.0);
assert_eq!(Float::from(-2.5) * Rational::exact_from(1.5), -4.0);
assert_eq!(Float::from(-2.5) * Rational::exact_from(-1.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl Mul for Float

fn mul(self, other: Self) -> Self

Multiplies two Floats, taking both by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\pm\infty,\pm0.0)=f(\pm0.0,\pm\infty) = \text{NaN}$
• $f(\infty,x)=f(x,\infty)=\infty$ if $x>0.0$
• $f(\infty,x)=f(x,\infty)=-\infty$ if $x<0.0$
• $f(-\infty,x)=f(x,-\infty)=-\infty$ if $x>0.0$
• $f(-\infty,x)=f(x,-\infty)=\infty$ if $x<0.0$
• $f(0.0,x)=f(x,0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(0.0,x)=f(x,0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$
• $f(-0.0,x)=f(x,-0.0)=-0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=0.0$ or $x>0.0$
• $f(-0.0,x)=f(x,-0.0)=0.0$ if $x$ is not NaN or $\pm\infty$, and if $x=-0.0$ or $x<0.0$

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec instead. If you want to specify the output precision, consider using Float::mul_round. If you want both of these things, consider using Float::mul_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity, Zero};
use malachite_float::Float;

assert!((Float::from(1.5) * Float::NAN).is_nan());
assert_eq!(Float::from(1.5) * Float::INFINITY, Float::INFINITY);
assert_eq!(
    Float::from(1.5) * Float::NEGATIVE_INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::from(-1.5) * Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::from(-1.5) * Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((Float::INFINITY * Float::ZERO).is_nan());

assert_eq!(Float::from(1.5) * Float::from(2.5), 4.0);
assert_eq!(Float::from(1.5) * Float::from(-2.5), -4.0);
assert_eq!(Float::from(-1.5) * Float::from(2.5), -4.0);
assert_eq!(Float::from(-1.5) * Float::from(-2.5), 4.0);

type Output = Float

The resulting type after applying the * operator.

impl MulAssign<&Float> for Float

fn mul_assign(&mut self, other: &Self)

Multiplies a Float by a Float in place, taking the Float on the right-hand side by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the * documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_assign instead. If you want to specify the output precision, consider using Float::mul_round_assign. If you want both of these things, consider using Float::mul_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity, Zero};
use malachite_float::Float;

let mut x = Float::from(1.5);
x *= &Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x *= &Float::INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x *= &Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x *= &Float::INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x *= &Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::INFINITY;
x *= &Float::ZERO;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x *= &Float::from(2.5);
assert_eq!(x, 4.0);

let mut x = Float::from(1.5);
x *= &Float::from(-2.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-1.5);
x *= &Float::from(2.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-1.5);
x *= &Float::from(-2.5);
assert_eq!(x, 4.0);

impl MulAssign<&Rational> for Float

fn mul_assign(&mut self, other: &Rational)

Multiplies a Float by a Rational in place, taking the Rational by reference.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

See the * documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_assign_ref instead. If you want to specify the output precision, consider using Float::mul_rational_round_assign_ref. If you want both of these things, consider using Float::mul_rational_prec_round_assign_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x *= &Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x *= &Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x *= &Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::INFINITY;
x *= &Rational::exact_from(-1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x *= &Rational::exact_from(-1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(2.5);
x *= &Rational::exact_from(1.5);
assert_eq!(x, 4.0);

impl MulAssign<Rational> for Float

fn mul_assign(&mut self, other: Rational)

Multiplies a Float by a Rational in place, taking the Rational by value.

If the output has a precision, it is the precision of the input Float. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the precision of the input Float.

See the * documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_rational_prec_assign instead. If you want to specify the output precision, consider using Float::mul_rational_round_assign. If you want both of these things, consider using Float::mul_rational_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x *= Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x *= Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x *= Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::INFINITY;
x *= Rational::exact_from(-1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x *= Rational::exact_from(-1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(2.5);
x *= Rational::exact_from(1.5);
assert_eq!(x, 4.0);

impl MulAssign for Float

fn mul_assign(&mut self, other: Self)

Multiplies a Float by a Float in place, taking the Float on the right-hand side by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the product is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = xy+\varepsilon. $$

• If $xy$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $xy$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |xy|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the * documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::mul_prec_assign instead. If you want to specify the output precision, consider using Float::mul_round_assign. If you want both of these things, consider using Float::mul_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity, Zero};
use malachite_float::Float;

let mut x = Float::from(1.5);
x *= Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x *= Float::INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x *= Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x *= Float::INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(-1.5);
x *= Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::INFINITY;
x *= Float::ZERO;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x *= Float::from(2.5);
assert_eq!(x, 4.0);

let mut x = Float::from(1.5);
x *= Float::from(-2.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-1.5);
x *= Float::from(2.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-1.5);
x *= Float::from(-2.5);
assert_eq!(x, 4.0);

impl NaN for Float

The constant NaN.

const NAN: Self

impl Named for Float

const NAME: &'static str = "Float"

The name of this type, as given by the stringify macro.

See the documentation for impl_named for more details.

impl Neg for &Float

fn neg(self) -> Float

Negates a Float, taking it by reference.

$$ f(x) = -x. $$

Special cases:

• $f(\text{NaN}) = \text{NaN}$
• $f(\infty) = -\infty$
• $f(-\infty) = \infty$
• $f(0.0) = -0.0$
• $f(-0.0) = 0.0$

This function does not overflow or underflow.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

assert_eq!(ComparableFloat(-&Float::NAN), ComparableFloat(Float::NAN));
assert_eq!(-&Float::INFINITY, Float::NEGATIVE_INFINITY);
assert_eq!(-&Float::NEGATIVE_INFINITY, Float::INFINITY);
assert_eq!(
    ComparableFloat(-&Float::ZERO),
    ComparableFloat(Float::NEGATIVE_ZERO)
);
assert_eq!(
    ComparableFloat(-&Float::NEGATIVE_ZERO),
    ComparableFloat(Float::ZERO)
);
assert_eq!(-&Float::ONE, Float::NEGATIVE_ONE);
assert_eq!(-&Float::NEGATIVE_ONE, Float::ONE);

type Output = Float

The resulting type after applying the - operator.

impl Neg for Float

fn neg(self) -> Self

Negates a Float, taking it by value.

$$ f(x) = -x. $$

Special cases:

• $f(\text{NaN}) = \text{NaN}$
• $f(\infty) = -\infty$
• $f(-\infty) = \infty$
• $f(0.0) = -0.0$
• $f(-0.0) = 0.0$

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

assert_eq!(ComparableFloat(-Float::NAN), ComparableFloat(Float::NAN));
assert_eq!(-Float::INFINITY, Float::NEGATIVE_INFINITY);
assert_eq!(-Float::NEGATIVE_INFINITY, Float::INFINITY);
assert_eq!(
    ComparableFloat(-Float::ZERO),
    ComparableFloat(Float::NEGATIVE_ZERO)
);
assert_eq!(
    ComparableFloat(-Float::NEGATIVE_ZERO),
    ComparableFloat(Float::ZERO)
);
assert_eq!(-Float::ONE, Float::NEGATIVE_ONE);
assert_eq!(-Float::NEGATIVE_ONE, Float::ONE);

type Output = Float

The resulting type after applying the - operator.

impl NegAssign for Float

fn neg_assign(&mut self)

Negates a Float in place.

$$ x \gets -x. $$

Special cases:

• $\text{NaN} \gets \text{NaN}$
• $\infty \gets -\infty$
• $-\infty \gets \infty$
• $0.0 \gets -0.0$
• $-0.0 \gets 0.0$

This function does not overflow or underflow.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::NegAssign;
use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::{ComparableFloat, Float};

let mut x = Float::NAN;
x.neg_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::NAN));

let mut x = Float::INFINITY;
x.neg_assign();
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x.neg_assign();
assert_eq!(x, Float::INFINITY);

let mut x = Float::ZERO;
x.neg_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::NEGATIVE_ZERO));

let mut x = Float::NEGATIVE_ZERO;
x.neg_assign();
assert_eq!(ComparableFloat(x), ComparableFloat(Float::ZERO));

let mut x = Float::ONE;
x.neg_assign();
assert_eq!(x, Float::NEGATIVE_ONE);

let mut x = Float::NEGATIVE_ONE;
x.neg_assign();
assert_eq!(x, Float::ONE);

impl NegativeInfinity for Float

The constant $-\infty$.

const NEGATIVE_INFINITY: Self

impl NegativeOne for Float

The constant -1.0, with precision 1.

const NEGATIVE_ONE: Self

impl NegativeZero for Float

The constant -0.0, with precision 1.

const NEGATIVE_ZERO: Self

impl One for Float

The constant 1.0, with precision 1.

const ONE: Self

impl OneHalf for Float

The constant 0.5, with precision 1.

const ONE_HALF: Self

impl PartialEq<Float> for Integer

fn eq(&self, other: &Float) -> bool

Determines whether an Integer is equal to a Float.

No Integer is equal to $\infty$, $-\infty$, or NaN. The Integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Integer::from(123) == Float::from(123));
assert!(Integer::from(-123) == Float::from(-123));
assert!(Integer::ONE != Float::ONE_HALF);

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for Natural

fn eq(&self, other: &Float) -> bool

Determines whether a Natural is equal to a Float.

No Natural is equal to $\infty$, $-\infty$, or NaN. The Natural zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Natural::from(123u32) == Float::from(123));
assert!(Natural::ONE != Float::ONE_HALF);

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for Rational

fn eq(&self, other: &Float) -> bool

Determines whether a Rational is equal to a Float.

No Rational is equal to $\infty$, $-\infty$, or NaN. The Rational zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::OneHalf;
use malachite_float::Float;
use malachite_q::Rational;

assert!(Rational::from(123) == Float::from(123));
assert!(Rational::from(-123) == Float::from(-123));
assert!(Rational::ONE_HALF == Float::ONE_HALF);
assert!(Rational::from_unsigneds(1u8, 3) != Float::from(1.0f64 / 3.0));

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for f32

fn eq(&self, other: &Float) -> bool

Determines whether a primitive float is equal to a Float.

The primitive float $\infty$ is equal to the Float $\infty$, and the primitive float $-\infty$ is equal to the Float $-\infty$. The primitive float NaN is not equal to anything, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for f64

fn eq(&self, other: &Float) -> bool

Determines whether a primitive float is equal to a Float.

The primitive float $\infty$ is equal to the Float $\infty$, and the primitive float $-\infty$ is equal to the Float $-\infty$. The primitive float NaN is not equal to anything, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for i128

fn eq(&self, other: &Float) -> bool

Determines whether a signed primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for i16

fn eq(&self, other: &Float) -> bool

Determines whether a signed primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for i32

fn eq(&self, other: &Float) -> bool

Determines whether a signed primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for i64

fn eq(&self, other: &Float) -> bool

Determines whether a signed primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for i8

fn eq(&self, other: &Float) -> bool

Determines whether a signed primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for isize

fn eq(&self, other: &Float) -> bool

Determines whether a signed primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for u128

fn eq(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for u16

fn eq(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for u32

fn eq(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for u64

fn eq(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for u8

fn eq(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Float> for usize

fn eq(&self, other: &Float) -> bool

Determines whether an unsigned primitive integer is equal to a Float.

No primitive integer is equal to $\infty$, $-\infty$, or NaN. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Integer> for Float

fn eq(&self, other: &Integer) -> bool

Determines whether a Float is equal to an Integer.

$\infty$, $-\infty$, and NaN are not equal to any Integer. Both the Float zero and the Float negative zero are equal to the Integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Float::from(123) == Integer::from(123));
assert!(Float::from(-123) == Integer::from(-123));
assert!(Float::ONE_HALF != Integer::ONE);

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Natural> for Float

fn eq(&self, other: &Natural) -> bool

Determines whether a Float is equal to a Natural.

$\infty$, $-\infty$, and NaN are not equal to any Natural. Both the Float zero and the Float negative zero are equal to the Natural zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{One, OneHalf};
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Float::from(123) == Natural::from(123u32));
assert!(Float::ONE_HALF != Natural::ONE);

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<Rational> for Float

fn eq(&self, other: &Rational) -> bool

Determines whether a Float is equal to a Rational.

$\infty$, $-\infty$, and NaN are not equal to any Rational. Both the Float zero and the Float negative zero are equal to the Rational zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::OneHalf;
use malachite_float::Float;
use malachite_q::Rational;

assert!(Float::from(123) == Rational::from(123));
assert!(Float::from(-123) == Rational::from(-123));
assert!(Float::ONE_HALF == Rational::ONE_HALF);
assert!(Float::from(1.0f64 / 3.0) != Rational::from_unsigneds(1u8, 3));

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<f32> for Float

fn eq(&self, other: &f32) -> bool

Determines whether a Float is equal to a primitive float.

The Float $\infty$ is equal to the primitive float $\infty$, and the Float $-\infty$ is equal to the primitive float $-\infty$. The Float NaN is not equal to anything, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<f64> for Float

fn eq(&self, other: &f64) -> bool

Determines whether a Float is equal to a primitive float.

The Float $\infty$ is equal to the primitive float $\infty$, and the Float $-\infty$ is equal to the primitive float $-\infty$. The Float NaN is not equal to anything, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<i128> for Float

fn eq(&self, other: &i128) -> bool

Determines whether a Float is equal to a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<i16> for Float

fn eq(&self, other: &i16) -> bool

Determines whether a Float is equal to a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<i32> for Float

fn eq(&self, other: &i32) -> bool

Determines whether a Float is equal to a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<i64> for Float

fn eq(&self, other: &i64) -> bool

Determines whether a Float is equal to a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<i8> for Float

fn eq(&self, other: &i8) -> bool

Determines whether a Float is equal to a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<isize> for Float

fn eq(&self, other: &isize) -> bool

Determines whether a Float is equal to a signed primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<u128> for Float

fn eq(&self, other: &u128) -> bool

Determines whether a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<u16> for Float

fn eq(&self, other: &u16) -> bool

Determines whether a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<u32> for Float

fn eq(&self, other: &u32) -> bool

Determines whether a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<u64> for Float

fn eq(&self, other: &u64) -> bool

Determines whether a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<u8> for Float

fn eq(&self, other: &u8) -> bool

Determines whether a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq<usize> for Float

fn eq(&self, other: &usize) -> bool

Determines whether a Float is equal to an unsigned primitive integer.

$\infty$, $-\infty$, and NaN are not equal to any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialEq for Float

fn eq(&self, other: &Self) -> bool

Compares two Floats for equality.

This implementation follows the IEEE 754 standard. NaN is not equal to anything, not even itself. Positive zero is equal to negative zero. Floats with different precisions are equal if they represent the same numeric value.

For different equality behavior, consider using ComparableFloat or ComparableFloatRef.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{NaN, NegativeZero, One, Two, Zero};
use malachite_float::Float;

assert_ne!(Float::NAN, Float::NAN);
assert_eq!(Float::ZERO, Float::ZERO);
assert_eq!(Float::NEGATIVE_ZERO, Float::NEGATIVE_ZERO);
assert_eq!(Float::ZERO, Float::NEGATIVE_ZERO);

assert_eq!(Float::ONE, Float::ONE);
assert_ne!(Float::ONE, Float::TWO);
assert_eq!(Float::ONE, Float::one_prec(100));

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PartialOrd<Float> for Integer

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an Integer to a Float.

No Integer is comparable to NaN. Every Integer is smaller than $\infty$ and greater than $-\infty$. The Integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Integer::from(100) > Float::from(80));
assert!(Integer::from(-100) < Float::from(-80));
assert!(Integer::from(100) < Float::INFINITY);
assert!(Integer::from(-100) > Float::NEGATIVE_INFINITY);

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for Natural

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a Natural to a Float.

No Natural is comparable to NaN. Every Natural is smaller than $\infty$ and greater than $-\infty$. The Natural zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Natural::from(100u32) > Float::from(80));
assert!(Natural::from(100u32) < Float::INFINITY);
assert!(Natural::from(100u32) > Float::NEGATIVE_INFINITY);

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for Rational

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an Rational to a Float.

No Rational is comparable to NaN. Every Rational is smaller than $\infty$ and greater than $-\infty$. The Rational zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_float::Float;
use malachite_q::Rational;

assert!(Rational::from(100) > Float::from(80));
assert!(Rational::from(-100) < Float::from(-80));
assert!(Rational::from(100) < Float::INFINITY);
assert!(Rational::from(-100) > Float::NEGATIVE_INFINITY);
assert!(Rational::from_unsigneds(1u8, 3) > Float::from(1.0f64 / 3.0));

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for f32

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a primitive float to a Float.

The primitive float NaN is not comparable to any primitive float, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for f64

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a primitive float to a Float.

The primitive float NaN is not comparable to any primitive float, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for i128

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a signed primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for i16

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a signed primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for i32

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a signed primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for i64

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a signed primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for i8

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a signed primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for isize

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares a signed primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for u128

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an unsigned primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for u16

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an unsigned primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for u32

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an unsigned primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for u64

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an unsigned primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for u8

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an unsigned primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Float> for usize

fn partial_cmp(&self, other: &Float) -> Option<Ordering>

Compares an unsigned primitive integer to a Float.

No integer is comparable to NaN. Every integer is smaller than $\infty$ and greater than $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Integer> for Float

fn partial_cmp(&self, other: &Integer) -> Option<Ordering>

Compares a Float to an Integer.

NaN is not comparable to any Integer. $\infty$ is greater than any Integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the Integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Float::from(80) < Integer::from(100));
assert!(Float::from(-80) > Integer::from(-100));
assert!(Float::INFINITY > Integer::from(100));
assert!(Float::NEGATIVE_INFINITY < Integer::from(-100));

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Natural> for Float

fn partial_cmp(&self, other: &Natural) -> Option<Ordering>

Compares a Float to a Natural.

NaN is not comparable to any Natural. $\infty$ is greater than any Natural, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the Natural zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Float::from(80) < Natural::from(100u32));
assert!(Float::INFINITY > Natural::from(100u32));
assert!(Float::NEGATIVE_INFINITY < Natural::from(100u32));

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<Rational> for Float

fn partial_cmp(&self, other: &Rational) -> Option<Ordering>

Compares a Float to a Rational.

NaN is not comparable to any Rational. $\infty$ is greater than any Rational, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the Rational zero.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_float::Float;
use malachite_q::Rational;

assert!(Float::from(80) < Rational::from(100));
assert!(Float::from(-80) > Rational::from(-100));
assert!(Float::INFINITY > Rational::from(100));
assert!(Float::NEGATIVE_INFINITY < Rational::from(-100));
assert!(Float::from(1.0f64 / 3.0) < Rational::from_unsigneds(1u8, 3));

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<f32> for Float

fn partial_cmp(&self, other: &f32) -> Option<Ordering>

Compares a Float to a primitive float.

The Float NaN is not comparable to any primitive float, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<f64> for Float

fn partial_cmp(&self, other: &f64) -> Option<Ordering>

Compares a Float to a primitive float.

The Float NaN is not comparable to any primitive float, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<i128> for Float

fn partial_cmp(&self, other: &i128) -> Option<Ordering>

Compares a Float to a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<i16> for Float

fn partial_cmp(&self, other: &i16) -> Option<Ordering>

Compares a Float to a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<i32> for Float

fn partial_cmp(&self, other: &i32) -> Option<Ordering>

Compares a Float to a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<i64> for Float

fn partial_cmp(&self, other: &i64) -> Option<Ordering>

Compares a Float to a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<i8> for Float

fn partial_cmp(&self, other: &i8) -> Option<Ordering>

Compares a Float to a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<isize> for Float

fn partial_cmp(&self, other: &isize) -> Option<Ordering>

Compares a Float to a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<u128> for Float

fn partial_cmp(&self, other: &u128) -> Option<Ordering>

Compares a Float to an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<u16> for Float

fn partial_cmp(&self, other: &u16) -> Option<Ordering>

Compares a Float to an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<u32> for Float

fn partial_cmp(&self, other: &u32) -> Option<Ordering>

Compares a Float to an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<u64> for Float

fn partial_cmp(&self, other: &u64) -> Option<Ordering>

Compares a Float to an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<u8> for Float

fn partial_cmp(&self, other: &u8) -> Option<Ordering>

Compares a Float to an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd<usize> for Float

fn partial_cmp(&self, other: &usize) -> Option<Ordering>

Compares a Float to an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ is greater than any primitive integer, and $-\infty$ is less. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrd for Float

fn partial_cmp(&self, other: &Self) -> Option<Ordering>

Compares two Floats.

This implementation follows the IEEE 754 standard. NaN is not comparable to anything, not even itself. Positive zero is equal to negative zero. Floats with different precisions are equal if they represent the same numeric value.

For different comparison behavior that provides a total order, consider using ComparableFloat or ComparableFloatRef.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, OneHalf, Zero,
};
use malachite_float::Float;
use std::cmp::Ordering::*;

assert_eq!(Float::NAN.partial_cmp(&Float::NAN), None);
assert_eq!(Float::ZERO.partial_cmp(&Float::NEGATIVE_ZERO), Some(Equal));
assert_eq!(Float::ONE.partial_cmp(&Float::one_prec(100)), Some(Equal));
assert!(Float::INFINITY > Float::ONE);
assert!(Float::NEGATIVE_INFINITY < Float::ONE);
assert!(Float::ONE_HALF < Float::ONE);
assert!(Float::ONE_HALF > Float::NEGATIVE_ONE);

fn lt(&self, other: &Rhs) -> bool

Tests less than (for self and other) and is used by the < operator.

fn le(&self, other: &Rhs) -> bool

Tests less than or equal to (for self and other) and is used by the <= operator.

fn gt(&self, other: &Rhs) -> bool

Tests greater than (for self and other) and is used by the > operator.

fn ge(&self, other: &Rhs) -> bool

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PartialOrdAbs<Float> for Integer

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an Integer and a Float.

No Integer is comparable to NaN. Every Integer is smaller in absolute value than $\infty$ and $-\infty$. The Integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Integer::from(100).gt_abs(&Float::from(80)));
assert!(Integer::from(100).lt_abs(&Float::INFINITY));
assert!(Integer::from(-100).lt_abs(&Float::INFINITY));
assert!(Integer::from(-100).lt_abs(&Float::NEGATIVE_INFINITY));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for Natural

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares a Natural to the absolute value of a Float.

No Natural is comparable to NaN. Every Natural is smaller in absolute value than $\infty$ and $-\infty$. The Natural zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Natural::from(100u32).gt_abs(&Float::from(80)));
assert!(Natural::from(100u32).lt_abs(&Float::INFINITY));
assert!(Natural::from(100u32).lt_abs(&Float::NEGATIVE_INFINITY));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for Rational

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a Rational and a Float.

No Rational is comparable to NaN. Every Rational is smaller in absolute value than $\infty$ and $-\infty$. The Rational zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use malachite_q::Rational;

assert!(Rational::from(100).gt_abs(&Float::from(80)));
assert!(Rational::from(-100).gt_abs(&Float::from(-80)));
assert!(Rational::from(100).lt_abs(&Float::INFINITY));
assert!(Rational::from(-100).lt_abs(&Float::NEGATIVE_INFINITY));
assert!(Rational::from_unsigneds(1u8, 3).gt_abs(&Float::from(1.0f64 / 3.0)));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for f32

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a primitive float and a Float.

The primitive float NaN is not comparable to any primitive float, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for f64

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a primitive float and a Float.

The primitive float NaN is not comparable to any primitive float, not even the Float NaN. Every primitive float zero is equal to every Float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for i128

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a signed primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for i16

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a signed primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for i32

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a signed primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for i64

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a signed primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for i8

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a signed primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for isize

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of a signed primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for u128

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an unsigned primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for u16

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an unsigned primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for u32

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an unsigned primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for u64

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an unsigned primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for u8

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an unsigned primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Float> for usize

fn partial_cmp_abs(&self, other: &Float) -> Option<Ordering>

Compares the absolute values of an unsigned primitive integer and a Float.

No primitive integer is comparable to NaN. Every primitive integer is smaller in absolute value than $\infty$ and $-\infty$. The integer zero is equal to both the Float zero and the Float negative zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is other.significant_bits().

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Integer> for Float

fn partial_cmp_abs(&self, other: &Integer) -> Option<Ordering>

Compares the absolute values of a Float and an Integer.

NaN is not comparable to any Integer. $\infty$ and $-\infty$ are greater in absolute value than any Integer. Both the Float zero and the Float negative zero are equal to the Integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert!(Float::from(80).lt_abs(&Integer::from(100)));
assert!(Float::from(-80).lt_abs(&Integer::from(-100)));
assert!(Float::INFINITY.gt_abs(&Integer::from(100)));
assert!(Float::NEGATIVE_INFINITY.gt_abs(&Integer::from(-100)));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Natural> for Float

fn partial_cmp_abs(&self, other: &Natural) -> Option<Ordering>

Compares the absolute value of a Float to a Natural.

NaN is not comparable to any Natural. $\infty$ and $-\infty$ are greater in absolute value than any Natural. Both the Float zero and the Float negative zero are equal to the Natural zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is min(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert!(Float::from(80).lt_abs(&Natural::from(100u32)));
assert!(Float::INFINITY.gt_abs(&Natural::from(100u32)));
assert!(Float::NEGATIVE_INFINITY.gt_abs(&Natural::from(100u32)));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<Rational> for Float

fn partial_cmp_abs(&self, other: &Rational) -> Option<Ordering>

Compares the absolute values of a Float and a Rational.

NaN is not comparable to any Rational. $\infty$ and $-\infty$ are greater in absolute value than any Rational. Both the Float zero and the Float negative zero are equal to the Rational zero.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NegativeInfinity};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use malachite_q::Rational;

assert!(Float::from(80).lt_abs(&Rational::from(100)));
assert!(Float::from(-80).lt_abs(&Rational::from(-100)));
assert!(Float::INFINITY.gt_abs(&Rational::from(100)));
assert!(Float::NEGATIVE_INFINITY.gt_abs(&Rational::from(-100)));
assert!(Float::from(1.0f64 / 3.0).lt_abs(&Rational::from_unsigneds(1u8, 3)));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<f32> for Float

fn partial_cmp_abs(&self, other: &f32) -> Option<Ordering>

Compares the absolute values of a Float and a primitive float.

The Float NaN is not comparable to any primitive float, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<f64> for Float

fn partial_cmp_abs(&self, other: &f64) -> Option<Ordering>

Compares the absolute values of a Float and a primitive float.

The Float NaN is not comparable to any primitive float, not even the primitive float NaN. Every Float zero is equal to every primitive float zero, regardless of sign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.sci_exponent().abs()).

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<i128> for Float

fn partial_cmp_abs(&self, other: &i128) -> Option<Ordering>

Compares the absolute values of a Float and a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<i16> for Float

fn partial_cmp_abs(&self, other: &i16) -> Option<Ordering>

Compares the absolute values of a Float and a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<i32> for Float

fn partial_cmp_abs(&self, other: &i32) -> Option<Ordering>

Compares the absolute values of a Float and a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<i64> for Float

fn partial_cmp_abs(&self, other: &i64) -> Option<Ordering>

Compares the absolute values of a Float and a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<i8> for Float

fn partial_cmp_abs(&self, other: &i8) -> Option<Ordering>

Compares the absolute values of a Float and a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<isize> for Float

fn partial_cmp_abs(&self, other: &isize) -> Option<Ordering>

Compares the absolute values of a Float and a signed primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<u128> for Float

fn partial_cmp_abs(&self, other: &u128) -> Option<Ordering>

Compares the absolute values of a Float and an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<u16> for Float

fn partial_cmp_abs(&self, other: &u16) -> Option<Ordering>

Compares the absolute values of a Float and an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<u32> for Float

fn partial_cmp_abs(&self, other: &u32) -> Option<Ordering>

Compares the absolute values of a Float and an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<u64> for Float

fn partial_cmp_abs(&self, other: &u64) -> Option<Ordering>

Compares the absolute values of a Float and an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<u8> for Float

fn partial_cmp_abs(&self, other: &u8) -> Option<Ordering>

Compares the absolute values of a Float and an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs<usize> for Float

fn partial_cmp_abs(&self, other: &usize) -> Option<Ordering>

Compares the absolute values of a Float and an unsigned primitive integer.

NaN is not comparable to any primitive integer. $\infty$ and $-\infty$ are greater in absolute value than any primitive integer. Both the Float zero and the Float negative zero are equal to the integer zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

See here.

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PartialOrdAbs for Float

fn partial_cmp_abs(&self, other: &Self) -> Option<Ordering>

Compares the absolute values of two Floats.

This implementation follows the IEEE 754 standard. NaN is not comparable to anything, not even itself. Floats with different precisions are equal if they represent the same numeric value.

For different comparison behavior that provides a total order, consider using ComparableFloat or ComparableFloatRef.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{
    Infinity, NaN, NegativeInfinity, NegativeOne, NegativeZero, One, OneHalf, Zero,
};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_float::Float;
use std::cmp::Ordering::*;

assert_eq!(Float::NAN.partial_cmp_abs(&Float::NAN), None);
assert_eq!(
    Float::ZERO.partial_cmp_abs(&Float::NEGATIVE_ZERO),
    Some(Equal)
);
assert_eq!(
    Float::ONE.partial_cmp_abs(&Float::one_prec(100)),
    Some(Equal)
);
assert!(Float::INFINITY.gt_abs(&Float::ONE));
assert!(Float::NEGATIVE_INFINITY.gt_abs(&Float::ONE));
assert!(Float::ONE_HALF.lt_abs(&Float::ONE));
assert!(Float::ONE_HALF.lt_abs(&Float::NEGATIVE_ONE));

fn lt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than the absolute value of another.

fn le_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is less than or equal to the absolute value of another.

fn gt_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than the absolute value of another.

fn ge_abs(&self, other: &Rhs) -> bool

Determines whether the absolute value of one number is greater than or equal to the absolute value of another.

impl PowerOf2<i64> for Float

fn power_of_2(pow: i64) -> Self

Raises 2 to an integer power, returning a Float with precision 1.

To get a Float with a higher precision, try Float::power_of_2_prec.

$f(k) = 2^k$.

If pow is greater than $2^{30}-2$, $\infty$ is returned. If pow is less than $-2^{30}$, positive zero is returned.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_float::Float;

assert_eq!(Float::power_of_2(0i64).to_string(), "1.0");
assert_eq!(Float::power_of_2(3i64).to_string(), "8.0");
assert_eq!(Float::power_of_2(100i64).to_string(), "1.0e30");
assert_eq!(Float::power_of_2(-3i64).to_string(), "0.1");
assert_eq!(Float::power_of_2(-100i64).to_string(), "8.0e-31");
assert_eq!(
    Float::power_of_2(i64::power_of_2(30) - 1).to_string(),
    "Infinity"
);
assert_eq!(
    Float::power_of_2(-i64::power_of_2(30) - 1).to_string(),
    "0.0"
);

impl PowerOf2<u64> for Float

fn power_of_2(pow: u64) -> Self

Raises 2 to an integer power, returning a Float with precision 1.

To get a Float with a higher precision, try Float::power_of_2_prec.

$f(k) = 2^k$.

If pow is greater than $2^{30}-2$, $\infty$ is returned.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_float::Float;

assert_eq!(Float::power_of_2(0u64).to_string(), "1.0");
assert_eq!(Float::power_of_2(3u64).to_string(), "8.0");
assert_eq!(Float::power_of_2(100u64).to_string(), "1.0e30");
assert_eq!(
    Float::power_of_2(u64::power_of_2(30) - 1).to_string(),
    "Infinity"
);

impl RawMantissaAndExponent<Natural, i32> for Float

fn raw_mantissa_and_exponent(self) -> (Natural, i32)

Returns the raw mantissa and exponent of a Float, taking the Float by value.

The raw exponent and raw mantissa are the actual bit patterns used to represent the components of self. When self is finite and nonzero, the raw mantissa is an integer whose number of significant bits is a multiple of the limb width, and which is equal to the absolute value of self multiplied by some integer power of 2. The raw exponent is one more than the floor of the base-2 logarithm of the absolute value of self.

The inverse operation is Self::from_raw_mantissa_and_exponent.

The raw exponent is in the range $[-(2^{30}-1), 2^{30}-1]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not finite or not zero.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, RawMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_nz::platform::Limb;
use malachite_q::Rational;

if Limb::WIDTH == u64::WIDTH {
    let (m, e) = Float::ONE.raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "9223372036854775808");
    assert_eq!(e, 1);

    let (m, e) = Float::from(std::f64::consts::PI).raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "14488038916154245120");
    assert_eq!(e, 2);

    let (m, e) =
        Float::exact_from(Natural::from(3u32).pow(50u64)).raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "202070319366191015160784900114134073344");
    assert_eq!(e, 80);

    let (m, e) = Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100)
        .0
        .raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "286514342137199872022965541161805021184");
    assert_eq!(e, -79);
}

fn raw_exponent(self) -> i32

Returns the raw exponent of a Float, taking the Float by value.

The raw exponent is one more than the floor of the base-2 logarithm of the absolute value of self.

The raw exponent is in the range $[-(2^{30}-1), 2^{30}-1]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not finite or not zero.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, RawMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(Float::ONE.raw_exponent(), 1);
assert_eq!(Float::from(std::f64::consts::PI).raw_exponent(), 2);
assert_eq!(
    Float::exact_from(Natural::from(3u32).pow(50u64)).raw_exponent(),
    80
);
assert_eq!(
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100)
        .0
        .raw_exponent(),
    -79
);

fn from_raw_mantissa_and_exponent( raw_mantissa: Natural, raw_exponent: i32, ) -> Self

Constructs a Float from its raw mantissa and exponent. The resulting Float is positive and has the smallest precision possible.

The number of significant bits of the raw mantissa must be divisible by the limb width. The raw exponent must be in the range $[-(2^{30}-1), 2^{30}-1]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if raw_mantissa is zero, if its number of significant bits is not divisible by the limb width, or if raw_exponent is out of range.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::conversion::traits::RawMantissaAndExponent;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_nz::platform::Limb;
use malachite_q::Rational;
use std::str::FromStr;

if Limb::WIDTH == u64::WIDTH {
    assert_eq!(
        Float::from_raw_mantissa_and_exponent(Natural::from(9223372036854775808u64), 1),
        1
    );
    assert_eq!(
        Float::from_raw_mantissa_and_exponent(Natural::from(14488038916154245120u64), 2),
        std::f64::consts::PI
    );
    assert_eq!(
        Float::from_raw_mantissa_and_exponent(
            Natural::from_str("202070319366191015160784900114134073344").unwrap(),
            80
        ),
        Natural::from(3u32).pow(50u64)
    );
    assert_eq!(
        Float::from_raw_mantissa_and_exponent(
            Natural::from_str("286514342137199872022965541161805021184").unwrap(),
            -79
        ),
        Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
    );
}

fn raw_mantissa(self) -> M

Extracts the raw mantissa from a number.

impl RawMantissaAndExponent<Natural, i32, Float> for &Float

fn raw_mantissa_and_exponent(self) -> (Natural, i32)

Returns the raw mantissa and exponent of a Float, taking the Float by reference.

The raw exponent and raw mantissa are the actual bit patterns used to represent the components of self. When self is finite and nonzero, the raw mantissa is an integer whose number of significant bits is a multiple of the limb width, and which is equal to the absolute value of self multiplied by some integer power of 2. The raw exponent is one more than the floor of the base-2 logarithm of the absolute value of self.

The raw exponent is in the range $[-(2^{30}-1), 2^{30}-1]$.

The inverse operation is Float::from_raw_mantissa_and_exponent.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.significant_bits().

Panics

Panics if the Float is not finite or not zero.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, RawMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_nz::platform::Limb;
use malachite_q::Rational;

if Limb::WIDTH == u64::WIDTH {
    let (m, e) = (&Float::ONE).raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "9223372036854775808");
    assert_eq!(e, 1);

    let (m, e) = (&Float::from(std::f64::consts::PI)).raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "14488038916154245120");
    assert_eq!(e, 2);

    let (m, e) =
        (&Float::exact_from(Natural::from(3u32).pow(50u64))).raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "202070319366191015160784900114134073344");
    assert_eq!(e, 80);

    let (m, e) = (&Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0)
        .raw_mantissa_and_exponent();
    assert_eq!(m.to_string(), "286514342137199872022965541161805021184");
    assert_eq!(e, -79);
}

fn raw_exponent(self) -> i32

Returns the raw exponent of a Float, taking the Float by reference.

The raw exponent is one more than the floor of the base-2 logarithm of the absolute value of self.

The raw exponent is in the range $[-(2^{30}-1), 2^{30}-1]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not finite or not zero.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, RawMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!((&Float::ONE).raw_exponent(), 1);
assert_eq!((&Float::from(std::f64::consts::PI)).raw_exponent(), 2);
assert_eq!(
    (&Float::exact_from(Natural::from(3u32).pow(50u64))).raw_exponent(),
    80
);
assert_eq!(
    (&Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0).raw_exponent(),
    -79
);

fn from_raw_mantissa_and_exponent( raw_mantissa: Natural, raw_exponent: i32, ) -> Float

Constructs a Float from its raw mantissa and exponent. The resulting Float is positive and has the smallest precision possible.

Worst-case complexity

Constant time and additional memory.

The number of significant bits of the raw mantissa must be divisible by the limb width. The raw exponent must be in the range $[-(2^{30}-1), 2^{30}-1]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if raw_mantissa is zero, if its number of significant bits is not divisible by the limb width, or if raw_exponent is out of range.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::conversion::traits::RawMantissaAndExponent;
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_nz::platform::Limb;
use malachite_q::Rational;
use std::str::FromStr;

if Limb::WIDTH == u64::WIDTH {
    assert_eq!(
        <&Float as RawMantissaAndExponent<_, _, _>>::from_raw_mantissa_and_exponent(
            Natural::from(9223372036854775808u64),
            1
        ),
        1
    );
    assert_eq!(
        <&Float as RawMantissaAndExponent<_, _, _>>::from_raw_mantissa_and_exponent(
            Natural::from(14488038916154245120u64),
            2
        ),
        std::f64::consts::PI
    );
    assert_eq!(
        <&Float as RawMantissaAndExponent<_, _, _>>::from_raw_mantissa_and_exponent(
            Natural::from_str("202070319366191015160784900114134073344").unwrap(),
            80
        ),
        Natural::from(3u32).pow(50u64)
    );
    assert_eq!(
        <&Float as RawMantissaAndExponent<_, _, _>>::from_raw_mantissa_and_exponent(
            Natural::from_str("286514342137199872022965541161805021184").unwrap(),
            -79
        ),
        Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
    );
}

fn raw_mantissa(self) -> M

Extracts the raw mantissa from a number.

impl Reciprocal for &Float

fn reciprocal(self) -> Float

Takes the reciprocal of a Float, taking it by reference.

If the output has a precision, it is the precision of the input. If the reciprocal is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN})=\text{NaN}$
• $f(\infty)=0.0$
• $f(-\infty)=-0.0$
• $f(0.0)=\infty$
• $f(-0.0)=-\infty$

Overflow:

• If $f(x)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.

This function cannot underflow.

If you want to use a rounding mode other than Nearest, consider using Float::reciprocal_prec_ref instead. If you want to specify the output precision, consider using Float::reciprocal_round_ref. If you want both of these things, consider using Float::reciprocal_prec_round_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Reciprocal;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((&Float::NAN).reciprocal().is_nan());
assert_eq!((&Float::INFINITY).reciprocal().to_string(), "0.0");
assert_eq!((&Float::NEGATIVE_INFINITY).reciprocal().to_string(), "-0.0");
assert_eq!((&Float::from(1.5)).reciprocal().to_string(), "0.8");
assert_eq!((&Float::from(-1.5)).reciprocal().to_string(), "-0.8");

type Output = Float

impl Reciprocal for Float

fn reciprocal(self) -> Self

Takes the reciprocal of a Float, taking it by value.

If the output has a precision, it is the precision of the input. If the reciprocal is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN})=\text{NaN}$
• $f(\infty)=0.0$
• $f(-\infty)=-0.0$
• $f(0.0)=\infty$
• $f(-0.0)=-\infty$

Overflow:

• If $f(x)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.

This function cannot underflow.

If you want to use a rounding mode other than Nearest, consider using Float::reciprocal_prec instead. If you want to specify the output precision, consider using Float::reciprocal_round. If you want both of these things, consider using Float::reciprocal_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Reciprocal;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!(Float::NAN.reciprocal().is_nan());
assert_eq!(Float::INFINITY.reciprocal().to_string(), "0.0");
assert_eq!(Float::NEGATIVE_INFINITY.reciprocal().to_string(), "-0.0");
assert_eq!(Float::from(1.5).reciprocal().to_string(), "0.8");
assert_eq!(Float::from(-1.5).reciprocal().to_string(), "-0.8");

type Output = Float

impl ReciprocalAssign for Float

fn reciprocal_assign(&mut self)

Takes the reciprocal of a Float in place.

If the output has a precision, it is the precision of the input. If the reciprocal is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = 1/x+\varepsilon. $$

• If $1/x$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $1/x$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |1/x|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the Float::reciprocal documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::reciprocal_prec_assign instead. If you want to specify the output precision, consider using Float::reciprocal_round_assign. If you want both of these things, consider using Float::reciprocal_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::ReciprocalAssign;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

let mut x = Float::NAN;
x.reciprocal_assign();
assert!(x.is_nan());

let mut x = Float::INFINITY;
x.reciprocal_assign();
assert_eq!(x.to_string(), "0.0");

let mut x = Float::NEGATIVE_INFINITY;
x.reciprocal_assign();
assert_eq!(x.to_string(), "-0.0");

let mut x = Float::from(1.5);
x.reciprocal_assign();
assert_eq!(x.to_string(), "0.8");

let mut x = Float::from(-1.5);
x.reciprocal_assign();
assert_eq!(x.to_string(), "-0.8");

impl RoundingFrom<&Float> for Integer

fn rounding_from(f: &Float, rm: RoundingMode) -> (Self, Ordering)

Converts a Float to an Integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is NaN or infinite, the function will panic regardless of the rounding mode.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is NaN or infinite.

Examples

use malachite_base::num::conversion::traits::RoundingFrom;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_base::strings::ToDebugString;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(
    Integer::rounding_from(&Float::from(1.5), Floor).to_debug_string(),
    "(1, Less)"
);
assert_eq!(
    Integer::rounding_from(&Float::from(1.5), Ceiling).to_debug_string(),
    "(2, Greater)"
);
assert_eq!(
    Integer::rounding_from(&Float::from(1.5), Nearest).to_debug_string(),
    "(2, Greater)"
);

assert_eq!(
    Integer::rounding_from(&Float::from(-1.5), Floor).to_debug_string(),
    "(-2, Less)"
);
assert_eq!(
    Integer::rounding_from(&Float::from(-1.5), Ceiling).to_debug_string(),
    "(-1, Greater)"
);
assert_eq!(
    Integer::rounding_from(&Float::from(-1.5), Nearest).to_debug_string(),
    "(-2, Less)"
);

impl RoundingFrom<&Float> for Natural

fn rounding_from(f: &Float, rm: RoundingMode) -> (Self, Ordering)

Converts a Float to a Natural, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN or $\infty$, the function will panic regardless of the rounding mode.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, or if the Float is NaN or $\infty$.

Examples

use malachite_base::num::basic::traits::NegativeInfinity;
use malachite_base::num::conversion::traits::RoundingFrom;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_base::strings::ToDebugString;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(
    Natural::rounding_from(&Float::from(1.5), Floor).to_debug_string(),
    "(1, Less)"
);
assert_eq!(
    Natural::rounding_from(&Float::from(1.5), Ceiling).to_debug_string(),
    "(2, Greater)"
);
assert_eq!(
    Natural::rounding_from(&Float::from(1.5), Nearest).to_debug_string(),
    "(2, Greater)"
);

assert_eq!(
    Natural::rounding_from(&Float::NEGATIVE_INFINITY, Down).to_debug_string(),
    "(0, Greater)"
);
assert_eq!(
    Natural::rounding_from(&Float::NEGATIVE_INFINITY, Ceiling).to_debug_string(),
    "(0, Greater)"
);
assert_eq!(
    Natural::rounding_from(&Float::NEGATIVE_INFINITY, Nearest).to_debug_string(),
    "(0, Greater)"
);

impl RoundingFrom<&Float> for f32

fn rounding_from(f: &Float, rm: RoundingMode) -> (f32, Ordering)

Converts a Float to a primitive float, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value. (Although a NaN is not comparable to any Float, converting a NaN to a NaN will also return Equal, indicating an exact conversion.)

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not exactly equal to any float of the target type, and rm is Exact.

Examples

See here.

impl RoundingFrom<&Float> for f64

fn rounding_from(f: &Float, rm: RoundingMode) -> (f64, Ordering)

Converts a Float to a primitive float, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value. (Although a NaN is not comparable to any Float, converting a NaN to a NaN will also return Equal, indicating an exact conversion.)

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not exactly equal to any float of the target type, and rm is Exact.

Examples

See here.

impl RoundingFrom<&Float> for i128

fn rounding_from(f: &Float, rm: RoundingMode) -> (i128, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for i16

fn rounding_from(f: &Float, rm: RoundingMode) -> (i16, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for i32

fn rounding_from(f: &Float, rm: RoundingMode) -> (i32, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for i64

fn rounding_from(f: &Float, rm: RoundingMode) -> (i64, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for i8

fn rounding_from(f: &Float, rm: RoundingMode) -> (i8, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for isize

fn rounding_from(f: &Float, rm: RoundingMode) -> (isize, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for u128

fn rounding_from(f: &Float, rm: RoundingMode) -> (u128, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for u16

fn rounding_from(f: &Float, rm: RoundingMode) -> (u16, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for u32

fn rounding_from(f: &Float, rm: RoundingMode) -> (u32, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for u64

fn rounding_from(f: &Float, rm: RoundingMode) -> (u64, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for u8

fn rounding_from(f: &Float, rm: RoundingMode) -> (u8, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<&Float> for usize

fn rounding_from(f: &Float, rm: RoundingMode) -> (usize, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by reference. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for Integer

fn rounding_from(f: Float, rm: RoundingMode) -> (Self, Ordering)

Converts a Float to an Integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is NaN or infinite, the function will panic regardless of the rounding mode.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is NaN or infinite.

Examples

use malachite_base::num::conversion::traits::RoundingFrom;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_base::strings::ToDebugString;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(
    Integer::rounding_from(Float::from(1.5), Floor).to_debug_string(),
    "(1, Less)"
);
assert_eq!(
    Integer::rounding_from(Float::from(1.5), Ceiling).to_debug_string(),
    "(2, Greater)"
);
assert_eq!(
    Integer::rounding_from(Float::from(1.5), Nearest).to_debug_string(),
    "(2, Greater)"
);

assert_eq!(
    Integer::rounding_from(Float::from(-1.5), Floor).to_debug_string(),
    "(-2, Less)"
);
assert_eq!(
    Integer::rounding_from(Float::from(-1.5), Ceiling).to_debug_string(),
    "(-1, Greater)"
);
assert_eq!(
    Integer::rounding_from(Float::from(-1.5), Nearest).to_debug_string(),
    "(-2, Less)"
);

impl RoundingFrom<Float> for Natural

fn rounding_from(f: Float, rm: RoundingMode) -> (Self, Ordering)

Converts a Float to a Natural, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN or $\infty$, the function will panic regardless of the rounding mode.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, or if the Float is NaN or $\infty$.

Examples

use malachite_base::num::basic::traits::NegativeInfinity;
use malachite_base::num::conversion::traits::RoundingFrom;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_base::strings::ToDebugString;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(
    Natural::rounding_from(Float::from(1.5), Floor).to_debug_string(),
    "(1, Less)"
);
assert_eq!(
    Natural::rounding_from(Float::from(1.5), Ceiling).to_debug_string(),
    "(2, Greater)"
);
assert_eq!(
    Natural::rounding_from(Float::from(1.5), Nearest).to_debug_string(),
    "(2, Greater)"
);

assert_eq!(
    Natural::rounding_from(Float::NEGATIVE_INFINITY, Down).to_debug_string(),
    "(0, Greater)"
);
assert_eq!(
    Natural::rounding_from(Float::NEGATIVE_INFINITY, Ceiling).to_debug_string(),
    "(0, Greater)"
);
assert_eq!(
    Natural::rounding_from(Float::NEGATIVE_INFINITY, Nearest).to_debug_string(),
    "(0, Greater)"
);

impl RoundingFrom<Float> for f32

fn rounding_from(f: Float, rm: RoundingMode) -> (f32, Ordering)

Converts a Float to a primitive float, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value. (Although a NaN is not comparable to any Float, converting a NaN to a NaN will also return Equal, indicating an exact conversion.)

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not exactly equal to any float of the target type, and rm is Exact.

Examples

See here.

impl RoundingFrom<Float> for f64

fn rounding_from(f: Float, rm: RoundingMode) -> (f64, Ordering)

Converts a Float to a primitive float, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value. (Although a NaN is not comparable to any Float, converting a NaN to a NaN will also return Equal, indicating an exact conversion.)

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not exactly equal to any float of the target type, and rm is Exact.

Examples

See here.

impl RoundingFrom<Float> for i128

fn rounding_from(f: Float, rm: RoundingMode) -> (i128, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for i16

fn rounding_from(f: Float, rm: RoundingMode) -> (i16, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for i32

fn rounding_from(f: Float, rm: RoundingMode) -> (i32, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for i64

fn rounding_from(f: Float, rm: RoundingMode) -> (i64, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for i8

fn rounding_from(f: Float, rm: RoundingMode) -> (i8, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for isize

fn rounding_from(f: Float, rm: RoundingMode) -> (isize, Ordering)

Converts a Float to a signed primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is less than the minimum representable value of the signed type (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the signed type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is smaller than the minimum representable value of the signed type and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the signed type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for u128

fn rounding_from(f: Float, rm: RoundingMode) -> (u128, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for u16

fn rounding_from(f: Float, rm: RoundingMode) -> (u16, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for u32

fn rounding_from(f: Float, rm: RoundingMode) -> (u32, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for u64

fn rounding_from(f: Float, rm: RoundingMode) -> (u64, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for u8

fn rounding_from(f: Float, rm: RoundingMode) -> (u8, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl RoundingFrom<Float> for usize

fn rounding_from(f: Float, rm: RoundingMode) -> (usize, Ordering)

Converts a Float to an unsigned primitive integer, using a specified RoundingMode and taking the Float by value. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.

If the Float is negative (including $-\infty$), then it will be rounded to zero when the RoundingMode is Ceiling, Down, or Nearest. Otherwise, this function will panic.

If the Float is greater than the maximum representable value of the unsigned type (including $\infty$), then it will be rounded to the maximum value when the RoundingMode is Floor, Down, or Nearest. Otherwise, this function will panic.

If the Float is NaN, the function will panic regardless of the rounding mode.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if the Float is not an integer and rm is Exact, or if the Float is less than zero and rm is not Down, Ceiling, or Nearest, if the Float is greater than the maximum representable value of the unsigned type and rm is not Down, Floor, or Nearest, or if the Float is NaN.

Examples

See here.

impl SciMantissaAndExponent<Float, i32> for Float

fn sci_mantissa_and_exponent(self) -> (Self, i32)

Returns a Float’s scientific mantissa and exponent, taking the Float by value.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. We represent the rational mantissa as a Float. $$ f(x) = (\frac{|x|}{2^{\lfloor \log_2 |x| \rfloor}}, \lfloor \log_2 |x| \rfloor). $$

The returned exponent is always in the range $[-2^{30}, 2^{30}-2]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, SciMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(Float::ONE.sci_mantissa_and_exponent(), (Float::ONE, 0));

let (m, e) = Float::from(std::f64::consts::PI).sci_mantissa_and_exponent();
assert_eq!(m.to_string(), "1.570796326794897");
assert_eq!(e, 1);

let (m, e) = Float::exact_from(Natural::from(3u32).pow(50u64)).sci_mantissa_and_exponent();
assert_eq!(m.to_string(), "1.187662594419065093441695");
assert_eq!(e, 79);

let (m, e) = Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100)
    .0
    .sci_mantissa_and_exponent();
assert_eq!(m.to_string(), "1.683979953059212693885095551367");
assert_eq!(e, -80);

fn sci_exponent(self) -> i32

Returns a Float’s scientific exponent, taking the Float by value.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. $$ f(x) = \lfloor \log_2 |x| \rfloor. $$

The returned exponent is always in the range $[-2^{30}, 2^{30}-2]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, SciMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(Float::ONE.sci_exponent(), 0);
assert_eq!(Float::from(std::f64::consts::PI).sci_exponent(), 1);
assert_eq!(
    Float::exact_from(Natural::from(3u32).pow(50u64)).sci_exponent(),
    79
);
assert_eq!(
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100)
        .0
        .sci_exponent(),
    -80
);

fn from_sci_mantissa_and_exponent( sci_mantissa: Self, sci_exponent: i32, ) -> Option<Self>

Constructs a Float from its scientific mantissa and exponent.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$.

$$ f(x) = 2^{e_i}m_i. $$

If the mantissa is zero or not finite, this function panics. If it is finite but not in the interval $[1, 2)$, None is returned. If the inputs correspond to a number too large in absolute value or too close to zero to be represented by a Float, None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{FromStringBase, SciMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(
    Float::from_sci_mantissa_and_exponent(Float::ONE, 0).unwrap(),
    1
);
assert_eq!(
    Float::from_sci_mantissa_and_exponent(
        Float::from_string_base(16, "0x1.921fb54442d18#53").unwrap(),
        1
    )
    .unwrap(),
    std::f64::consts::PI
);
assert_eq!(
    Float::from_sci_mantissa_and_exponent(
        Float::from_string_base(16, "0x1.300aa7e1b65fa13bc792#80").unwrap(),
        79
    )
    .unwrap(),
    Natural::from(3u32).pow(50u64)
);
assert_eq!(
    Float::from_sci_mantissa_and_exponent(
        Float::from_string_base(16, "0x1.af194f6982497a23f9dc546d6#100").unwrap(),
        -80
    )
    .unwrap(),
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
);

fn sci_mantissa(self) -> M

Extracts the scientific mantissa from a number.

impl SciMantissaAndExponent<Float, i32, Float> for &Float

fn sci_mantissa_and_exponent(self) -> (Float, i32)

Returns a Float’s scientific mantissa and exponent, taking the Float by reference.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. We represent the rational mantissa as a Float. $$ f(x) = (\frac{|x|}{2^{\lfloor \log_2 |x| \rfloor}}, \lfloor \log_2 |x| \rfloor). $$

The returned exponent is always in the range $[-2^{30}, 2^{30}-2]$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, SciMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!((&Float::ONE).sci_mantissa_and_exponent(), (Float::ONE, 0));

let (m, e): (Float, i32) = (&Float::from(std::f64::consts::PI)).sci_mantissa_and_exponent();
assert_eq!(m.to_string(), "1.570796326794897");
assert_eq!(e, 1);

let (m, e): (Float, i32) =
    (&Float::exact_from(Natural::from(3u32).pow(50u64))).sci_mantissa_and_exponent();
assert_eq!(m.to_string(), "1.187662594419065093441695");
assert_eq!(e, 79);

let (m, e): (Float, i32) =
    (&Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0)
        .sci_mantissa_and_exponent();
assert_eq!(m.to_string(), "1.683979953059212693885095551367");
assert_eq!(e, -80);

fn sci_exponent(self) -> i32

Returns a Float’s scientific exponent, taking the Float by reference.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. $$ f(x) = \lfloor \log_2 |x| \rfloor. $$

The returned exponent is always in the range $[-2^{30}, 2^{30}-2]$.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if self is zero or not finite.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{ExactFrom, SciMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::sci_exponent(&Float::ONE),
    0
);
assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::sci_exponent(&Float::from(
        std::f64::consts::PI
    )),
    1
);
assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::sci_exponent(&Float::exact_from(
        Natural::from(3u32).pow(50u64)
    )),
    79
);
assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::sci_exponent(
        &Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
    ),
    -80
);

fn from_sci_mantissa_and_exponent( sci_mantissa: Float, sci_exponent: i32, ) -> Option<Float>

Constructs a Float from its scientific mantissa and exponent.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$.

$$ f(x) = 2^{e_i}m_i. $$

If the mantissa is zero or not finite, this function panics. If it is finite but not in the interval $[1, 2)$, this function returns None.

If the mantissa is zero or not finite, this function panics. If it is finite but not in the interval $[1, 2)$, None is returned. If the inputs correspond to a number too large in absolute value or too close to zero to be represented by a Float, None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::Pow;
use malachite_base::num::basic::traits::One;
use malachite_base::num::conversion::traits::{FromStringBase, SciMantissaAndExponent};
use malachite_float::Float;
use malachite_nz::natural::Natural;
use malachite_q::Rational;

assert_eq!(
    Float::from_sci_mantissa_and_exponent(Float::ONE, 0).unwrap(),
    1
);
assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::from_sci_mantissa_and_exponent(
        Float::from_string_base(16, "0x1.921fb54442d18#53").unwrap(),
        1
    )
    .unwrap(),
    std::f64::consts::PI
);
assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::from_sci_mantissa_and_exponent(
        Float::from_string_base(16, "0x1.300aa7e1b65fa13bc792#80").unwrap(),
        79
    )
    .unwrap(),
    Natural::from(3u32).pow(50u64)
);
assert_eq!(
    <&Float as SciMantissaAndExponent<Float, _, _>>::from_sci_mantissa_and_exponent(
        Float::from_string_base(16, "0x1.af194f6982497a23f9dc546d6#100").unwrap(),
        -80
    )
    .unwrap(),
    Float::from_rational_prec(Rational::from(3u32).pow(-50i64), 100).0
);

fn sci_mantissa(self) -> M

Extracts the scientific mantissa from a number.

impl SciMantissaAndExponent<f32, i32, Float> for &Float

fn sci_mantissa_and_exponent(self) -> (f32, i32)

Returns a Float’s scientific mantissa and exponent, taking the Float by value.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. We represent the rational mantissa as a primitive float. The conversion might not be exact, so we round to the nearest float using the Nearest rounding mode. To use other rounding modes, use sci_mantissa_and_exponent_round. $$ f(x) \approx (\frac{|x|}{2^{\lfloor \log_2 |x| \rfloor}}, \lfloor \log_2 |x| \rfloor). $$

The returned exponent is always in the range $[-2^{30}, 2^{30}-2]$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is zero or not finite.

Examples

See here.

fn from_sci_mantissa_and_exponent( sci_mantissa: f32, sci_exponent: i32, ) -> Option<Float>

Constructs a Float from its scientific mantissa and exponent.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$.

$$ f(x) = 2^{e_i}m_i. $$

If the mantissa is zero or not finite, this function panics. If it is finite but not in the interval $[1, 2)$, None is returned. If the inputs correspond to a number too large in absolute value or too close to zero to be represented by a Float, None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

fn sci_mantissa(self) -> M

Extracts the scientific mantissa from a number.

fn sci_exponent(self) -> E

Extracts the scientific exponent from a number.

impl SciMantissaAndExponent<f64, i32, Float> for &Float

fn sci_mantissa_and_exponent(self) -> (f64, i32)

Returns a Float’s scientific mantissa and exponent, taking the Float by value.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$. We represent the rational mantissa as a primitive float. The conversion might not be exact, so we round to the nearest float using the Nearest rounding mode. To use other rounding modes, use sci_mantissa_and_exponent_round. $$ f(x) \approx (\frac{|x|}{2^{\lfloor \log_2 |x| \rfloor}}, \lfloor \log_2 |x| \rfloor). $$

The returned exponent is always in the range $[-2^{30}, 2^{30}-2]$.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if self is zero or not finite.

Examples

See here.

fn from_sci_mantissa_and_exponent( sci_mantissa: f64, sci_exponent: i32, ) -> Option<Float>

Constructs a Float from its scientific mantissa and exponent.

When $x$ is finite and nonzero, we can write $|x| = 2^{e_s}m_s$, where $e_s$ is an integer and $m_s$ is a rational number with $1 \leq m_s < 2$.

$$ f(x) = 2^{e_i}m_i. $$

If the mantissa is zero or not finite, this function panics. If it is finite but not in the interval $[1, 2)$, None is returned. If the inputs correspond to a number too large in absolute value or too close to zero to be represented by a Float, None is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

fn sci_mantissa(self) -> M

Extracts the scientific mantissa from a number.

fn sci_exponent(self) -> E

Extracts the scientific exponent from a number.

impl Shl<i128> for &Float

fn shl(self, bits: i128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i128> for Float

fn shl(self, bits: i128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i16> for &Float

fn shl(self, bits: i16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i16> for Float

fn shl(self, bits: i16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i32> for &Float

fn shl(self, bits: i32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i32> for Float

fn shl(self, bits: i32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i64> for &Float

fn shl(self, bits: i64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i64> for Float

fn shl(self, bits: i64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i8> for &Float

fn shl(self, bits: i8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<i8> for Float

fn shl(self, bits: i8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<isize> for &Float

fn shl(self, bits: isize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<isize> for Float

fn shl(self, bits: isize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u128> for &Float

fn shl(self, bits: u128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u128> for Float

fn shl(self, bits: u128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u16> for &Float

fn shl(self, bits: u16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u16> for Float

fn shl(self, bits: u16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u32> for &Float

fn shl(self, bits: u32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u32> for Float

fn shl(self, bits: u32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u64> for &Float

fn shl(self, bits: u64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u64> for Float

fn shl(self, bits: u64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u8> for &Float

fn shl(self, bits: u8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<u8> for Float

fn shl(self, bits: u8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<usize> for &Float

fn shl(self, bits: usize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl Shl<usize> for Float

fn shl(self, bits: usize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the << operator.

impl ShlAssign<i128> for Float

fn shl_assign(&mut self, bits: i128)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<i16> for Float

fn shl_assign(&mut self, bits: i16)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<i32> for Float

fn shl_assign(&mut self, bits: i32)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<i64> for Float

fn shl_assign(&mut self, bits: i64)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<i8> for Float

fn shl_assign(&mut self, bits: i8)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<isize> for Float

fn shl_assign(&mut self, bits: isize)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<u128> for Float

fn shl_assign(&mut self, bits: u128)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<u16> for Float

fn shl_assign(&mut self, bits: u16)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<u32> for Float

fn shl_assign(&mut self, bits: u32)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<u64> for Float

fn shl_assign(&mut self, bits: u64)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<u8> for Float

fn shl_assign(&mut self, bits: u8)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlAssign<usize> for Float

fn shl_assign(&mut self, bits: usize)

Left-shifts a Float (multiplies it by a power of 2), in place. If the Float has a precision, the precision is unchanged.

NaN, infinities, and zeros are unchanged.

$$ x \gets x2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShlRound<i128> for &Float

fn shl_round(self, bits: i128, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i128> for Float

fn shl_round(self, bits: i128, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i16> for &Float

fn shl_round(self, bits: i16, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i16> for Float

fn shl_round(self, bits: i16, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i32> for &Float

fn shl_round(self, bits: i32, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i32> for Float

fn shl_round(self, bits: i32, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i64> for &Float

fn shl_round(self, bits: i64, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i64> for Float

fn shl_round(self, bits: i64, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i8> for &Float

fn shl_round(self, bits: i8, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<i8> for Float

fn shl_round(self, bits: i8, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<isize> for &Float

fn shl_round(self, bits: isize, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<isize> for Float

fn shl_round(self, bits: isize, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u128> for &Float

fn shl_round(self, bits: u128, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u128> for Float

fn shl_round(self, bits: u128, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u16> for &Float

fn shl_round(self, bits: u16, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u16> for Float

fn shl_round(self, bits: u16, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u32> for &Float

fn shl_round(self, bits: u32, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u32> for Float

fn shl_round(self, bits: u32, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u64> for &Float

fn shl_round(self, bits: u64, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u64> for Float

fn shl_round(self, bits: u64, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u8> for &Float

fn shl_round(self, bits: u8, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<u8> for Float

fn shl_round(self, bits: u8, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<usize> for &Float

fn shl_round(self, bits: usize, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRound<usize> for Float

fn shl_round(self, bits: usize, rm: RoundingMode) -> (Float, Ordering)

Left-shifts a Float (multiplies it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use << instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShlRoundAssign<i128> for Float

fn shl_round_assign(&mut self, bits: i128, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<i16> for Float

fn shl_round_assign(&mut self, bits: i16, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<i32> for Float

fn shl_round_assign(&mut self, bits: i32, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<i64> for Float

fn shl_round_assign(&mut self, bits: i64, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<i8> for Float

fn shl_round_assign(&mut self, bits: i8, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<isize> for Float

fn shl_round_assign(&mut self, bits: isize, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<u128> for Float

fn shl_round_assign(&mut self, bits: u128, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<u16> for Float

fn shl_round_assign(&mut self, bits: u16, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<u32> for Float

fn shl_round_assign(&mut self, bits: u32, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<u64> for Float

fn shl_round_assign(&mut self, bits: u64, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<u8> for Float

fn shl_round_assign(&mut self, bits: u8, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShlRoundAssign<usize> for Float

fn shl_round_assign(&mut self, bits: usize, rm: RoundingMode) -> Ordering

Left-shifts a Float (multiplies it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use <<= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl Shr<i128> for &Float

fn shr(self, bits: i128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i128> for Float

fn shr(self, bits: i128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i16> for &Float

fn shr(self, bits: i16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i16> for Float

fn shr(self, bits: i16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i32> for &Float

fn shr(self, bits: i32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i32> for Float

fn shr(self, bits: i32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i64> for &Float

fn shr(self, bits: i64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i64> for Float

fn shr(self, bits: i64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i8> for &Float

fn shr(self, bits: i8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<i8> for Float

fn shr(self, bits: i8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<isize> for &Float

fn shr(self, bits: isize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<isize> for Float

fn shr(self, bits: isize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u128> for &Float

fn shr(self, bits: u128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u128> for Float

fn shr(self, bits: u128) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u16> for &Float

fn shr(self, bits: u16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u16> for Float

fn shr(self, bits: u16) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u32> for &Float

fn shr(self, bits: u32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u32> for Float

fn shr(self, bits: u32) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u64> for &Float

fn shr(self, bits: u64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u64> for Float

fn shr(self, bits: u64) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u8> for &Float

fn shr(self, bits: u8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<u8> for Float

fn shr(self, bits: u8) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<usize> for &Float

fn shr(self, bits: usize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl Shr<usize> for Float

fn shr(self, bits: usize) -> Float

Left-shifts a Float (multiplies it by a power of 2), taking it by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x, k) = x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Constant time and additional memory.

Examples

See here.

type Output = Float

The resulting type after applying the >> operator.

impl ShrAssign<i128> for Float

fn shr_assign(&mut self, bits: i128)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<i16> for Float

fn shr_assign(&mut self, bits: i16)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<i32> for Float

fn shr_assign(&mut self, bits: i32)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<i64> for Float

fn shr_assign(&mut self, bits: i64)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<i8> for Float

fn shr_assign(&mut self, bits: i8)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<isize> for Float

fn shr_assign(&mut self, bits: isize)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<u128> for Float

fn shr_assign(&mut self, bits: u128)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<u16> for Float

fn shr_assign(&mut self, bits: u16)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<u32> for Float

fn shr_assign(&mut self, bits: u32)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<u64> for Float

fn shr_assign(&mut self, bits: u64)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<u8> for Float

fn shr_assign(&mut self, bits: u8)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrAssign<usize> for Float

fn shr_assign(&mut self, bits: usize)

Left-shifts a Float (multiplies it by a power of 2), in place.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x \gets x/2^k. $$

• If $f(x,k)\geq 2^{2^{30}-1}$, $\infty$ is assigned instead.
• If $f(x,k)\leq -2^{2^{30}-1}$, $-\infty$ is assigned instead.
• If $0<f(x,k)\leq2^{-2^{30}-1}$, $0.0$ is assigned instead.
• If $2^{-2^{30}-1}<f(x,k)<2^{-2^{30}}$, $2^{-2^{30}}$ is assigned instead.
• If $-2^{-2^{30}-1}\leq f(x,k)<0$, $-0.0$ is assigned instead.
• If $-2^{-2^{30}}<f(x,k)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is assigned instead.

Constant time and additional memory.

Examples

See here.

impl ShrRound<i128> for &Float

fn shr_round(self, bits: i128, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i128> for Float

fn shr_round(self, bits: i128, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i16> for &Float

fn shr_round(self, bits: i16, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i16> for Float

fn shr_round(self, bits: i16, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i32> for &Float

fn shr_round(self, bits: i32, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i32> for Float

fn shr_round(self, bits: i32, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i64> for &Float

fn shr_round(self, bits: i64, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i64> for Float

fn shr_round(self, bits: i64, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i8> for &Float

fn shr_round(self, bits: i8, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<i8> for Float

fn shr_round(self, bits: i8, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<isize> for &Float

fn shr_round(self, bits: isize, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<isize> for Float

fn shr_round(self, bits: isize, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u128> for &Float

fn shr_round(self, bits: u128, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u128> for Float

fn shr_round(self, bits: u128, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u16> for &Float

fn shr_round(self, bits: u16, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u16> for Float

fn shr_round(self, bits: u16, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u32> for &Float

fn shr_round(self, bits: u32, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u32> for Float

fn shr_round(self, bits: u32, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u64> for &Float

fn shr_round(self, bits: u64, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u64> for Float

fn shr_round(self, bits: u64, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u8> for &Float

fn shr_round(self, bits: u8, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<u8> for Float

fn shr_round(self, bits: u8, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<usize> for &Float

fn shr_round(self, bits: usize, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by reference.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRound<usize> for Float

fn shr_round(self, bits: usize, rm: RoundingMode) -> (Float, Ordering)

Right-shifts a Float (divides it by a power of 2), rounding the result with the specified rounding mode and taking the Float by value.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the output has the same precision.

$$ f(x,k,m) = x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >> instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

type Output = Float

impl ShrRoundAssign<i128> for Float

fn shr_round_assign(&mut self, bits: i128, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<i16> for Float

fn shr_round_assign(&mut self, bits: i16, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<i32> for Float

fn shr_round_assign(&mut self, bits: i32, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<i64> for Float

fn shr_round_assign(&mut self, bits: i64, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<i8> for Float

fn shr_round_assign(&mut self, bits: i8, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<isize> for Float

fn shr_round_assign(&mut self, bits: isize, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<u128> for Float

fn shr_round_assign(&mut self, bits: u128, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<u16> for Float

fn shr_round_assign(&mut self, bits: u16, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<u32> for Float

fn shr_round_assign(&mut self, bits: u32, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<u64> for Float

fn shr_round_assign(&mut self, bits: u64, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<u8> for Float

fn shr_round_assign(&mut self, bits: u8, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl ShrRoundAssign<usize> for Float

fn shr_round_assign(&mut self, bits: usize, rm: RoundingMode) -> Ordering

Right-shifts a Float (divides it by a power of 2), in place, rounding the result with the specified rounding mode.

NaN, infinities, and zeros are unchanged. If the Float has a precision, the precision is unchanged.

$$ x\gets x/2^k. $$

• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling, Up, or Nearest, $\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor or Down, $(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Floor, Up, or Nearest, $-\infty$ is returned instead.
• If $f(x,k,m)\geq 2^{2^{30}-1}$ and $m$ is Ceiling or Down, $-(1-(1/2)^p)2^{2^{30}-1}$ is returned instead, where p is the precision of the input.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Floor or Down, $0.0$ is returned instead.
• If $0<f(x,k,m)<2^{-2^{30}}$, and $m$ is Ceiling or Up, $2^{-2^{30}}$ is returned instead.
• If $0<f(x,k,m)\leq2^{-2^{30}-1}$, and $m$ is Nearest, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,k,m)<2^{-2^{30}}$, and $m$ is Nearest, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Ceiling or Down, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<0$, and $m$ is Floor or Up, $-2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,k,m)<0$, and $m$ is Nearest, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,k,m)<-2^{-2^{30}-1}$, and $m$ is Nearest, $-2^{-2^{30}}$ is returned instead.

If you don’t care about overflow or underflow behavior, or only want the behavior of the Nearest rounding mode, you can just use >>= instead.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Panics

Panics if the result overflows or underflows and rm is Exact.

Examples

See here.

impl Sign for Float

fn sign(&self) -> Ordering

Returns the sign of a Float.

Returns Greater if the sign is positive and Less if the sign is negative. Never returns Equal. $\infty$ and positive zero have a positive sign, and $-\infty$ and negative zero have a negative sign.

Worst-case complexity

Constant time and additional memory.

Panics

Panics if self is NaN.

Examples

use malachite_base::num::arithmetic::traits::Sign;
use malachite_base::num::basic::traits::{
    Infinity, NegativeInfinity, NegativeOne, NegativeZero, One, Zero,
};
use malachite_float::Float;
use std::cmp::Ordering::*;

assert_eq!(Float::INFINITY.sign(), Greater);
assert_eq!(Float::NEGATIVE_INFINITY.sign(), Less);
assert_eq!(Float::ZERO.sign(), Greater);
assert_eq!(Float::NEGATIVE_ZERO.sign(), Less);
assert_eq!(Float::ONE.sign(), Greater);
assert_eq!(Float::NEGATIVE_ONE.sign(), Less);

impl SignificantBits for &Float

fn significant_bits(self) -> u64

Returns the number of significant bits of a Float. This is defined as follows:

$$ f(\text{NaN}) = f(\pm\infty) = f(\pm 0.0) = 1, $$

and, if $x$ is finite and nonzero,

$$ f(x) = p, $$

where $p$ is the precision of $x$.

See also the complexity function.

Worst-case complexity

Constant time and additional memory.

Examples

use malachite_base::num::arithmetic::traits::PowerOf2;
use malachite_base::num::basic::traits::{NaN, One};
use malachite_base::num::logic::traits::SignificantBits;
use malachite_float::Float;

assert_eq!(Float::NAN.significant_bits(), 1);
assert_eq!(Float::ONE.significant_bits(), 1);
assert_eq!(Float::one_prec(100).significant_bits(), 100);
assert_eq!(Float::from(std::f64::consts::PI).significant_bits(), 50);
assert_eq!(Float::power_of_2(100u64).significant_bits(), 1);
assert_eq!(Float::power_of_2(-100i64).significant_bits(), 1);

impl Square for &Float

fn square(self) -> Float

Squares a Float, taking it by reference.

If the output has a precision, it is the precision of the input. If the square is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN})=\text{NaN}$
• $f(\pm\infty)=\infty$
• $f(\pm0.0)=0.0$

Overflow and underflow:

• If $f(x)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::square_prec_ref instead. If you want to specify the output precision, consider using Float::square_round_ref. If you want both of these things, consider using Float::square_prec_round_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Square;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((&Float::NAN).square().is_nan());
assert_eq!((&Float::INFINITY).square(), Float::INFINITY);
assert_eq!((&Float::NEGATIVE_INFINITY).square(), Float::INFINITY);
assert_eq!((&Float::from(1.5)).square(), 2.0);
assert_eq!((&Float::from(-1.5)).square(), 2.0);

type Output = Float

impl Square for Float

fn square(self) -> Self

Squares a Float, taking it by value.

If the output has a precision, it is the precision of the input. If the square is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN})=\text{NaN}$
• $f(\pm\infty)=\infty$
• $f(\pm0.0)=0.0$

Overflow and underflow:

• If $f(x)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::square_prec instead. If you want to specify the output precision, consider using Float::square_round. If you want both of these things, consider using Float::square_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::Square;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!(Float::NAN.square().is_nan());
assert_eq!(Float::INFINITY.square(), Float::INFINITY);
assert_eq!(Float::NEGATIVE_INFINITY.square(), Float::INFINITY);
assert_eq!(Float::from(1.5).square(), 2.0);
assert_eq!(Float::from(-1.5).square(), 2.0);

type Output = Float

impl SquareAssign for Float

fn square_assign(&mut self)

Squares a Float in place.

If the output has a precision, it is the precision of the input. If the square is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x^2+\varepsilon. $$

• If $x^2$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x^2$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x^2|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the Float::square documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::square_prec_assign instead. If you want to specify the output precision, consider using Float::square_round_assign. If you want both of these things, consider using Float::square_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is self.significant_bits().

Examples

use malachite_base::num::arithmetic::traits::SquareAssign;
use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

let mut x = Float::NAN;
x.square_assign();
assert!(x.is_nan());

let mut x = Float::INFINITY;
x.square_assign();
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x.square_assign();
assert_eq!(x, Float::INFINITY);

let mut x = Float::from(1.5);
x.square_assign();
assert_eq!(x, 2.0);

let mut x = Float::from(-1.5);
x.square_assign();
assert_eq!(x, 2.0);

impl Sub<&Float> for &Float

fn sub(self, other: &Float) -> Float

Subtracts two Floats, taking both by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,\infty)=f(-\infty,-\infty)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0)=0.0$
• $f(-0.0,0.0)=-0.0$
• $f(0.0,0.0)=f(-0.0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x)=f(-0.0,x)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec instead. If you want to specify the output precision, consider using Float::sub_round. If you want both of these things, consider using Float::sub_prec_round.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((&Float::from(1.5) - &Float::NAN).is_nan());
assert_eq!(
    &Float::from(1.5) - &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::from(1.5) - &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((&Float::INFINITY - &Float::INFINITY).is_nan());

assert_eq!(&Float::from(1.5) - &Float::from(2.5), -1.0);
assert_eq!(&Float::from(1.5) - &Float::from(-2.5), 4.0);
assert_eq!(&Float::from(-1.5) - &Float::from(2.5), -4.0);
assert_eq!(&Float::from(-1.5) - &Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<&Float> for &Rational

fn sub(self, other: &Float) -> Float

Subtracts a Rational by a Float, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=-\infty$
• $f(x,-\infty)=\infty$
• $f(0,0.0)=-0.0$
• $f(0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$
• $f(0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) - &Float::NAN).is_nan());
assert_eq!(
    &Rational::exact_from(1.5) - &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(1.5) - &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(&Rational::exact_from(1.5) - &Float::from(2.5), -1.0);
assert_eq!(&Rational::exact_from(1.5) - &Float::from(-2.5), 4.0);
assert_eq!(&Rational::exact_from(-1.5) - &Float::from(2.5), -4.0);
assert_eq!(&Rational::exact_from(-1.5) - &Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<&Float> for Float

fn sub(self, other: &Self) -> Self

Subtracts two Floats, taking the first by value and the second by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,\infty)=f(-\infty,-\infty)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0)=0.0$
• $f(-0.0,0.0)=-0.0$
• $f(0.0,0.0)=f(-0.0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x)=f(-0.0,x)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_val_ref instead. If you want to specify the output precision, consider using Float::sub_round_val_ref. If you want both of these things, consider using Float::sub_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is other.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((Float::from(1.5) - &Float::NAN).is_nan());
assert_eq!(
    Float::from(1.5) - &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Float::from(1.5) - &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((Float::INFINITY - &Float::INFINITY).is_nan());

assert_eq!(Float::from(1.5) - &Float::from(2.5), -1.0);
assert_eq!(Float::from(1.5) - &Float::from(-2.5), 4.0);
assert_eq!(Float::from(-1.5) - &Float::from(2.5), -4.0);
assert_eq!(Float::from(-1.5) - &Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<&Float> for Rational

fn sub(self, other: &Float) -> Float

Subtracts a Rational by a Float, taking the Rational by value and the Float by reference.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=-\infty$
• $f(x,-\infty)=\infty$
• $f(0,0.0)=-0.0$
• $f(0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$
• $f(0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) - &Float::NAN).is_nan());
assert_eq!(
    Rational::exact_from(1.5) - &Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(1.5) - &Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(Rational::exact_from(1.5) - &Float::from(2.5), -1.0);
assert_eq!(Rational::exact_from(1.5) - &Float::from(-2.5), 4.0);
assert_eq!(Rational::exact_from(-1.5) - &Float::from(2.5), -4.0);
assert_eq!(Rational::exact_from(-1.5) - &Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<&Rational> for &Float

fn sub(self, other: &Rational) -> Float

Subtracts a Float by a Rational, taking both by reference.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(x,0)=x$
• $f(0.0,x)=f(-0.0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_ref_ref instead. If you want to specify the output precision, consider using Float::sub_rational_round_ref_ref. If you want both of these things, consider using Float::sub_rational_prec_round_ref_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN - Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY - Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY - Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(&Float::from(2.5) - &Rational::exact_from(1.5), 1.0);
assert_eq!(&Float::from(2.5) - &Rational::exact_from(-1.5), 4.0);
assert_eq!(&Float::from(-2.5) - &Rational::exact_from(1.5), -4.0);
assert_eq!(&Float::from(-2.5) - &Rational::exact_from(-1.5), -1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<&Rational> for Float

fn sub(self, other: &Rational) -> Self

Subtracts a Float by a Rational, taking the first by value and the second by reference.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(x,0)=x$
• $f(0.0,x)=f(-0.0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_val_ref instead. If you want to specify the output precision, consider using Float::sub_rational_round_val_ref. If you want both of these things, consider using Float::sub_rational_prec_round_val_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN - &Rational::exact_from(1.5)).is_nan());
assert_eq!(
    Float::INFINITY - &Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    Float::NEGATIVE_INFINITY - &Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(Float::from(2.5) - &Rational::exact_from(1.5), 1.0);
assert_eq!(Float::from(2.5) - &Rational::exact_from(-1.5), 4.0);
assert_eq!(Float::from(-2.5) - &Rational::exact_from(1.5), -4.0);
assert_eq!(Float::from(-2.5) - &Rational::exact_from(-1.5), -1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<Float> for &Float

fn sub(self, other: Float) -> Float

Subtracts two Floats, taking the first by reference and the second by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,\infty)=f(-\infty,-\infty)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0)=0.0$
• $f(-0.0,0.0)=-0.0$
• $f(0.0,0.0)=f(-0.0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x)=f(-0.0,x)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_ref_val instead. If you want to specify the output precision, consider using Float::sub_round_ref_val. If you want both of these things, consider using Float::sub_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(m)$

where $T$ is time, $M$ is additional memory, $n$ is max(self.significant_bits(), other.significant_bits()), and $m$ is self.significant_bits().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((&Float::from(1.5) - Float::NAN).is_nan());
assert_eq!(
    &Float::from(1.5) - Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Float::from(1.5) - Float::NEGATIVE_INFINITY,
    Float::INFINITY
);
assert!((&Float::INFINITY - Float::INFINITY).is_nan());

assert_eq!(&Float::from(1.5) - Float::from(2.5), -1.0);
assert_eq!(&Float::from(1.5) - Float::from(-2.5), 4.0);
assert_eq!(&Float::from(-1.5) - Float::from(2.5), -4.0);
assert_eq!(&Float::from(-1.5) - Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<Float> for &Rational

fn sub(self, other: Float) -> Float

Subtracts a Rational by a Float, taking the Rational by value and the Float by reference.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=-\infty$
• $f(x,-\infty)=\infty$
• $f(0,0.0)=-0.0$
• $f(0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$
• $f(0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Rational::exact_from(1.5) - Float::NAN).is_nan());
assert_eq!(
    &Rational::exact_from(1.5) - Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    &Rational::exact_from(1.5) - Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(&Rational::exact_from(1.5) - Float::from(2.5), -1.0);
assert_eq!(&Rational::exact_from(1.5) - Float::from(-2.5), 4.0);
assert_eq!(&Rational::exact_from(-1.5) - Float::from(2.5), -4.0);
assert_eq!(&Rational::exact_from(-1.5) - Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<Float> for Rational

fn sub(self, other: Float) -> Float

Subtracts a Rational by a Float, taking both by value.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(x,\text{NaN})=\text{NaN}$
• $f(x,\infty)=-\infty$
• $f(x,-\infty)=\infty$
• $f(0,0.0)=-0.0$
• $f(0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$
• $f(0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Rational::exact_from(1.5) - Float::NAN).is_nan());
assert_eq!(
    Rational::exact_from(1.5) - Float::INFINITY,
    Float::NEGATIVE_INFINITY
);
assert_eq!(
    Rational::exact_from(1.5) - Float::NEGATIVE_INFINITY,
    Float::INFINITY
);

assert_eq!(Rational::exact_from(1.5) - Float::from(2.5), -1.0);
assert_eq!(Rational::exact_from(1.5) - Float::from(-2.5), 4.0);
assert_eq!(Rational::exact_from(-1.5) - Float::from(2.5), -4.0);
assert_eq!(Rational::exact_from(-1.5) - Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<Rational> for &Float

fn sub(self, other: Rational) -> Float

Subtracts a Float by a Rational, taking the first by reference and the second by value.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(x,0)=x$
• $f(0.0,x)=f(-0.0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_ref_val instead. If you want to specify the output precision, consider using Float::sub_rational_round_ref_val. If you want both of these things, consider using Float::sub_rational_prec_round_ref_val.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((&Float::NAN - Rational::exact_from(1.5)).is_nan());
assert_eq!(
    &Float::INFINITY - Rational::exact_from(1.5),
    Float::INFINITY
);
assert_eq!(
    &Float::NEGATIVE_INFINITY - Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(&Float::from(2.5) - Rational::exact_from(1.5), 1.0);
assert_eq!(&Float::from(2.5) - Rational::exact_from(-1.5), 4.0);
assert_eq!(&Float::from(-2.5) - Rational::exact_from(1.5), -4.0);
assert_eq!(&Float::from(-2.5) - Rational::exact_from(-1.5), -1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub<Rational> for Float

fn sub(self, other: Rational) -> Self

Subtracts a Float by a Rational, taking both by value.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

Special cases:

• $f(\text{NaN},x)=\text{NaN}$
• $f(\infty,x)=\infty$
• $f(-\infty,x)=-\infty$
• $f(0.0,0)=0.0$
• $f(-0.0,0)=-0.0$
• $f(x,0)=x$
• $f(0.0,x)=f(-0.0,x)=-x$
• $f(x,x)=0.0$ if $x$ is nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec instead. If you want to specify the output precision, consider using Float::sub_rational_round. If you want both of these things, consider using Float::sub_rational_prec_round.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

assert!((Float::NAN - Rational::exact_from(1.5)).is_nan());
assert_eq!(Float::INFINITY - Rational::exact_from(1.5), Float::INFINITY);
assert_eq!(
    Float::NEGATIVE_INFINITY - Rational::exact_from(1.5),
    Float::NEGATIVE_INFINITY
);

assert_eq!(Float::from(2.5) - Rational::exact_from(1.5), 1.0);
assert_eq!(Float::from(2.5) - Rational::exact_from(-1.5), 4.0);
assert_eq!(Float::from(-2.5) - Rational::exact_from(1.5), -4.0);
assert_eq!(Float::from(-2.5) - Rational::exact_from(-1.5), -1.0);

type Output = Float

The resulting type after applying the - operator.

impl Sub for Float

fn sub(self, other: Self) -> Self

Subtracts two Floats, taking both by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ f(x,y) = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

Special cases:

• $f(\text{NaN},x)=f(x,\text{NaN})=f(\infty,\infty)=f(-\infty,-\infty)=\text{NaN}$
• $f(\infty,x)=\infty$ if $x$ is not NaN or $\infty$
• $f(x,-\infty)=\infty$ if $x$ is not NaN or $-\infty$
• $f(-\infty,x)=-\infty$ if $x$ is not NaN or $-\infty$
• $f(x,\infty)=-\infty$ if $x$ is not NaN or $\infty$
• $f(0.0,-0.0)=0.0$
• $f(-0.0,0.0)=-0.0$
• $f(0.0,0.0)=f(-0.0,-0.0)=0.0$
• $f(x,0.0)=f(x,-0.0)=x$ if $x$ is not NaN and $x$ is nonzero
• $f(0.0,x)=f(-0.0,x)=-x$ if $x$ is not NaN and $x$ is nonzero
• $f(x,x)=0.0$ if $x$ is finite and nonzero

Overflow and underflow:

• If $f(x,y)\geq 2^{2^{30}-1}$, $\infty$ is returned instead.
• If $f(x,y)\geq 2^{2^{30}-1}$, $-\infty$ is returned instead.
• If $0<f(x,y)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
• If $2^{-2^{30}-1}<f(x,y)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
• If $-2^{-2^{30}-1}\leq f(x,y)<0$, $-0.0$ is returned instead.
• If $-2^{-2^{30}}<f(x,y)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec instead. If you want to specify the output precision, consider using Float::sub_round. If you want both of these things, consider using Float::sub_prec_round.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

assert!((Float::from(1.5) - Float::NAN).is_nan());
assert_eq!(Float::from(1.5) - Float::INFINITY, Float::NEGATIVE_INFINITY);
assert_eq!(Float::from(1.5) - Float::NEGATIVE_INFINITY, Float::INFINITY);
assert!((Float::INFINITY - Float::INFINITY).is_nan());

assert_eq!(Float::from(1.5) - Float::from(2.5), -1.0);
assert_eq!(Float::from(1.5) - Float::from(-2.5), 4.0);
assert_eq!(Float::from(-1.5) - Float::from(2.5), -4.0);
assert_eq!(Float::from(-1.5) - Float::from(-2.5), 1.0);

type Output = Float

The resulting type after applying the - operator.

impl SubAssign<&Float> for Float

fn sub_assign(&mut self, other: &Self)

Subtracts a Float by a Float in place, taking the Float on the right-hand side by reference.

If the output has a precision, it is the maximum of the precisions of the inputs. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the - documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_assign instead. If you want to specify the output precision, consider using Float::sub_round_assign. If you want both of these things, consider using Float::sub_prec_round_assign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

let mut x = Float::from(1.5);
x -= &Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x -= &Float::INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(1.5);
x -= &Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::INFINITY;
x -= &Float::INFINITY;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x -= &Float::from(2.5);
assert_eq!(x, -1.0);

let mut x = Float::from(1.5);
x -= &Float::from(-2.5);
assert_eq!(x, 4.0);

let mut x = Float::from(-1.5);
x -= &Float::from(2.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-1.5);
x -= &Float::from(-2.5);
assert_eq!(x, 1.0);

impl SubAssign<&Rational> for Float

fn sub_assign(&mut self, other: &Rational)

Subtracts a Rational by a Float in place, taking the Rational by reference.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

See the - documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_assign_ref instead. If you want to specify the output precision, consider using Float::sub_rational_round_assign_ref. If you want both of these things, consider using Float::sub_rational_prec_round_assign_ref.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x -= &Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x -= &Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x -= &Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(2.5);
x -= &Rational::exact_from(1.5);
assert_eq!(x, 1.0);

let mut x = Float::from(2.5);
x -= &Rational::exact_from(-1.5);
assert_eq!(x, 4.0);

let mut x = Float::from(-2.5);
x -= &Rational::exact_from(1.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-2.5);
x -= &Rational::exact_from(-1.5);
assert_eq!(x, -1.0);

impl SubAssign<Rational> for Float

fn sub_assign(&mut self, other: Rational)

Subtracts a Rational by a Float in place, taking the Rational by value.

If the output has a precision, it is the precision of the input Float. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the precision of the input Float.

See the - documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_rational_prec_assign instead. If you want to specify the output precision, consider using Float::sub_rational_round_assign. If you want both of these things, consider using Float::sub_rational_prec_round_assign.

Worst-case complexity

$T(n) = O(n \log n \log\log n)$

$M(n) = O(n \log n)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_base::num::conversion::traits::ExactFrom;
use malachite_float::Float;
use malachite_q::Rational;

let mut x = Float::NAN;
x -= Rational::exact_from(1.5);
assert!(x.is_nan());

let mut x = Float::INFINITY;
x -= Rational::exact_from(1.5);
assert_eq!(x, Float::INFINITY);

let mut x = Float::NEGATIVE_INFINITY;
x -= Rational::exact_from(1.5);
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(2.5);
x -= Rational::exact_from(1.5);
assert_eq!(x, 1.0);

let mut x = Float::from(2.5);
x -= Rational::exact_from(-1.5);
assert_eq!(x, 4.0);

let mut x = Float::from(-2.5);
x -= Rational::exact_from(1.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-2.5);
x -= Rational::exact_from(-1.5);
assert_eq!(x, -1.0);

impl SubAssign for Float

fn sub_assign(&mut self, other: Self)

Subtracts a Float by a Float in place, taking the Float on the right-hand side by value.

If the output has a precision, it is the maximum of the precisions of the inputs. If the difference is equidistant from two Floats with the specified precision, the Float with fewer 1s in its binary expansion is chosen. See RoundingMode for a description of the Nearest rounding mode.

$$ x\gets = x-y+\varepsilon. $$

• If $x-y$ is infinite, zero, or NaN, $\varepsilon$ may be ignored or assumed to be 0.
• If $x-y$ is finite and nonzero, then $|\varepsilon| < 2^{\lfloor\log_2 |x-y|\rfloor-p}$, where $p$ is the maximum precision of the inputs.

See the - documentation for information on special cases, overflow, and underflow.

If you want to use a rounding mode other than Nearest, consider using Float::sub_prec_assign instead. If you want to specify the output precision, consider using Float::sub_round_assign. If you want both of these things, consider using Float::sub_prec_round_assign.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is max(self.significant_bits(), other.significant_bits()).

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, NegativeInfinity};
use malachite_float::Float;

let mut x = Float::from(1.5);
x -= Float::NAN;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x -= Float::INFINITY;
assert_eq!(x, Float::NEGATIVE_INFINITY);

let mut x = Float::from(1.5);
x -= Float::NEGATIVE_INFINITY;
assert_eq!(x, Float::INFINITY);

let mut x = Float::INFINITY;
x -= Float::INFINITY;
assert!(x.is_nan());

let mut x = Float::from(1.5);
x -= Float::from(2.5);
assert_eq!(x, -1.0);

let mut x = Float::from(1.5);
x -= Float::from(-2.5);
assert_eq!(x, 4.0);

let mut x = Float::from(-1.5);
x -= Float::from(2.5);
assert_eq!(x, -4.0);

let mut x = Float::from(-1.5);
x -= Float::from(-2.5);
assert_eq!(x, 1.0);

impl TryFrom<&Float> for Integer

fn try_from(f: &Float) -> Result<Self, Self::Error>

Converts a Float to an Integer, taking the Float by reference. If the Float is not equal to an integer, an error is returned.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::from::SignedFromFloatError::*;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(Integer::try_from(&Float::ZERO).unwrap(), 0);
assert_eq!(Integer::try_from(&Float::from(123.0)).unwrap(), 123);
assert_eq!(Integer::try_from(&Float::from(-123.0)).unwrap(), -123);

assert_eq!(
    Integer::try_from(&Float::from(1.5)),
    Err(FloatNonIntegerOrOutOfRange)
);
assert_eq!(Integer::try_from(&Float::INFINITY), Err(FloatInfiniteOrNan));
assert_eq!(Integer::try_from(&Float::NAN), Err(FloatInfiniteOrNan));

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for Natural

fn try_from(f: &Float) -> Result<Self, Self::Error>

Converts a Float to a Natural, taking the Float by reference. If the Float is not equal to a non-negative integer, an error is returned.

Both positive and negative zero convert to a Natural zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::from::UnsignedFromFloatError::*;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(Natural::try_from(&Float::ZERO).unwrap(), 0);
assert_eq!(Natural::try_from(&Float::from(123.0)).unwrap(), 123);

assert_eq!(Natural::try_from(&Float::from(-123.0)), Err(FloatNegative));
assert_eq!(
    Natural::try_from(&Float::from(1.5)),
    Err(FloatNonIntegerOrOutOfRange)
);
assert_eq!(Natural::try_from(&Float::INFINITY), Err(FloatInfiniteOrNan));
assert_eq!(Natural::try_from(&Float::NAN), Err(FloatInfiniteOrNan));

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for Rational

fn try_from(x: &Float) -> Result<Self, Self::Error>

Converts a Float to a Rational, taking the Float by reference. If the Float is not finite, an error is returned.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is x.complexity().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_float::conversion::rational_from_float::RationalFromFloatError;
use malachite_float::Float;
use malachite_q::Rational;

assert_eq!(Rational::try_from(&Float::ZERO).unwrap(), 0);
assert_eq!(
    Rational::try_from(&Float::from(1.5)).unwrap().to_string(),
    "3/2"
);
assert_eq!(
    Rational::try_from(&Float::from(-1.5)).unwrap().to_string(),
    "-3/2"
);

assert_eq!(
    Rational::try_from(&Float::INFINITY),
    Err(RationalFromFloatError)
);
assert_eq!(Rational::try_from(&Float::NAN), Err(RationalFromFloatError));

type Error = RationalFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for f32

fn try_from(f: &Float) -> Result<f32, Self::Error>

Converts a Float to a primitive float, taking the Float by reference. If the Float is not equal to a primitive float of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = FloatFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for f64

fn try_from(f: &Float) -> Result<f64, Self::Error>

Converts a Float to a primitive float, taking the Float by reference. If the Float is not equal to a primitive float of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = FloatFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for i128

fn try_from(f: &Float) -> Result<i128, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by reference. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for i16

fn try_from(f: &Float) -> Result<i16, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by reference. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for i32

fn try_from(f: &Float) -> Result<i32, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by reference. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for i64

fn try_from(f: &Float) -> Result<i64, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by reference. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for i8

fn try_from(f: &Float) -> Result<i8, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by reference. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for isize

fn try_from(f: &Float) -> Result<isize, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by reference. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for u128

fn try_from(f: &Float) -> Result<u128, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by reference. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for u16

fn try_from(f: &Float) -> Result<u16, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by reference. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for u32

fn try_from(f: &Float) -> Result<u32, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by reference. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for u64

fn try_from(f: &Float) -> Result<u64, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by reference. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for u8

fn try_from(f: &Float) -> Result<u8, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by reference. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Float> for usize

fn try_from(f: &Float) -> Result<usize, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by reference. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<&Integer> for Float

fn try_from(n: &Integer) -> Result<Self, Self::Error>

Converts an Integer to a Float, taking the Integer by reference.

If the Integer is nonzero, the precision of the Float is the minimum possible precision to represent the Integer exactly. If you want to specify some other precision, try Float::from_integer_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_integer_prec_round.

If the absolue value of the Integer is greater than or equal to $2^{2^{30}-1}$, this function returns an overflow error.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is n.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(Float::try_from(&Integer::ZERO).unwrap().to_string(), "0.0");
assert_eq!(
    Float::try_from(&Integer::from(123)).unwrap().to_string(),
    "123.0"
);
assert_eq!(
    Float::try_from(&Integer::from(123)).unwrap().get_prec(),
    Some(7)
);
assert_eq!(
    Float::try_from(&Integer::from(10)).unwrap().to_string(),
    "10.0"
);
assert_eq!(
    Float::try_from(&Integer::from(10)).unwrap().get_prec(),
    Some(3)
);
assert_eq!(
    Float::try_from(&Integer::from(-123)).unwrap().to_string(),
    "-123.0"
);
assert_eq!(
    Float::try_from(&Integer::from(-123)).unwrap().get_prec(),
    Some(7)
);
assert_eq!(
    Float::try_from(&Integer::from(-10)).unwrap().to_string(),
    "-10.0"
);
assert_eq!(
    Float::try_from(&Integer::from(-10)).unwrap().get_prec(),
    Some(3)
);

type Error = FloatConversionError

The type returned in the event of a conversion error.

impl TryFrom<&Natural> for Float

fn try_from(x: &Natural) -> Result<Self, Self::Error>

Converts a Natural to a Float, taking the Natural by reference.

If the Natural is nonzero, the precision of the Float is the minimum possible precision to represent the Natural exactly. If you want to specify some other precision, try Float::from_natural_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_natural_prec_round.

If the Natural is greater than or equal to $2^{2^{30}-1}$, this function returns an overflow error.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is n.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(Float::try_from(&Natural::ZERO).unwrap().to_string(), "0.0");
assert_eq!(
    Float::try_from(&Natural::from(123u32)).unwrap().to_string(),
    "123.0"
);
assert_eq!(
    Float::try_from(&Natural::from(123u32)).unwrap().get_prec(),
    Some(7)
);
assert_eq!(
    Float::try_from(&Natural::from(10u32)).unwrap().to_string(),
    "10.0"
);
assert_eq!(
    Float::try_from(&Natural::from(10u32)).unwrap().get_prec(),
    Some(3)
);

type Error = FloatConversionError

The type returned in the event of a conversion error.

impl TryFrom<&Rational> for Float

fn try_from(x: &Rational) -> Result<Self, Self::Error>

Converts a Rational to an Float, taking the Rational by reference. If the Rational’s denominator is not a power of 2, or if the Rational is too far from zero or too close to zero to be represented as a Float, an error is returned.

The Float’s precision is the minimum number of bits needed to exactly represent the Rational.

• If the Rational is greater than or equal to $2^{2^{30}-1}$), this function returns an overflow error.
• If the Rational is less than or equal to $-2^{2^{30}-1}$), this function returns an overflow error.
• If the Rational is positive and less than $2^{-2^{30}}$), this function returns an underflow error.
• If the Rational is negative and greater than $-2^{-2^{30}}$), this function returns an underflow error.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is x.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_q::conversion::primitive_float_from_rational::FloatConversionError;
use malachite_q::Rational;

assert_eq!(Float::try_from(&Rational::ZERO).unwrap(), 0);
assert_eq!(
    Float::try_from(&Rational::from_signeds(1, 8)).unwrap(),
    0.125
);
assert_eq!(
    Float::try_from(&Rational::from_signeds(-1, 8)).unwrap(),
    -0.125
);

assert_eq!(
    Float::try_from(&Rational::from_signeds(1, 3)),
    Err(FloatConversionError::Inexact)
);
assert_eq!(
    Float::try_from(&Rational::from_signeds(-1, 3)),
    Err(FloatConversionError::Inexact)
);

type Error = FloatConversionError

The type returned in the event of a conversion error.

impl TryFrom<Float> for Integer

fn try_from(f: Float) -> Result<Self, Self::Error>

Converts a Float to an Integer, taking the Float by value. If the Float is not equal to an integer, an error is returned.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::from::SignedFromFloatError::*;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(Integer::try_from(Float::ZERO).unwrap(), 0);
assert_eq!(Integer::try_from(Float::from(123.0)).unwrap(), 123);
assert_eq!(Integer::try_from(Float::from(-123.0)).unwrap(), -123);

assert_eq!(
    Integer::try_from(Float::from(1.5)),
    Err(FloatNonIntegerOrOutOfRange)
);
assert_eq!(Integer::try_from(Float::INFINITY), Err(FloatInfiniteOrNan));
assert_eq!(Integer::try_from(Float::NAN), Err(FloatInfiniteOrNan));

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for Natural

fn try_from(f: Float) -> Result<Self, Self::Error>

Converts a Float to a Natural, taking the Float by value. If the Float is not equal to a non-negative integer, an error is returned.

Both positive and negative zero convert to a Natural zero.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is f.complexity().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_base::num::conversion::from::UnsignedFromFloatError::*;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(Natural::try_from(Float::ZERO).unwrap(), 0);
assert_eq!(Natural::try_from(Float::from(123.0)).unwrap(), 123);

assert_eq!(Natural::try_from(Float::from(-123.0)), Err(FloatNegative));
assert_eq!(
    Natural::try_from(Float::from(1.5)),
    Err(FloatNonIntegerOrOutOfRange)
);
assert_eq!(Natural::try_from(Float::INFINITY), Err(FloatInfiniteOrNan));
assert_eq!(Natural::try_from(Float::NAN), Err(FloatInfiniteOrNan));

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for Rational

fn try_from(x: Float) -> Result<Self, Self::Error>

Converts a Float to a Rational, taking the Float by value. If the Float is not finite, an error is returned.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(n)$

where $T$ is time, $M$ is additional memory, and $n$ is x.complexity().

Examples

use malachite_base::num::basic::traits::{Infinity, NaN, Zero};
use malachite_float::conversion::rational_from_float::RationalFromFloatError;
use malachite_float::Float;
use malachite_q::Rational;

assert_eq!(Rational::try_from(Float::ZERO).unwrap(), 0);
assert_eq!(
    Rational::try_from(Float::from(1.5)).unwrap().to_string(),
    "3/2"
);
assert_eq!(
    Rational::try_from(Float::from(-1.5)).unwrap().to_string(),
    "-3/2"
);

assert_eq!(
    Rational::try_from(Float::INFINITY),
    Err(RationalFromFloatError)
);
assert_eq!(Rational::try_from(Float::NAN), Err(RationalFromFloatError));

type Error = RationalFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for f32

fn try_from(f: Float) -> Result<f32, Self::Error>

Converts a Float to a primitive float, taking the Float by value. If the Float is not equal to a primitive float of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = FloatFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for f64

fn try_from(f: Float) -> Result<f64, Self::Error>

Converts a Float to a primitive float, taking the Float by value. If the Float is not equal to a primitive float of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = FloatFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for i128

fn try_from(f: Float) -> Result<i128, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by value. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for i16

fn try_from(f: Float) -> Result<i16, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by value. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for i32

fn try_from(f: Float) -> Result<i32, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by value. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for i64

fn try_from(f: Float) -> Result<i64, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by value. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for i8

fn try_from(f: Float) -> Result<i8, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by value. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for isize

fn try_from(f: Float) -> Result<isize, Self::Error>

Converts a Float to a primitive signed integer, taking the Float by value. If the Float is not equal to a signed primitive integer of the given type, an error is returned.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = SignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for u128

fn try_from(f: Float) -> Result<u128, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by value. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for u16

fn try_from(f: Float) -> Result<u16, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by value. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for u32

fn try_from(f: Float) -> Result<u32, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by value. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for u64

fn try_from(f: Float) -> Result<u64, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by value. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for u8

fn try_from(f: Float) -> Result<u8, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by value. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Float> for usize

fn try_from(f: Float) -> Result<usize, Self::Error>

Converts a Float to a primitive unsigned integer, taking the Float by value. If the Float is not equal to an unsigned primitive integer of the given type, an error is returned.

Both positive and negative zero convert to a primitive unsigned integer zero.

Worst-case complexity

Constant time and additional memory.

Examples

See here.

type Error = UnsignedFromFloatError

The type returned in the event of a conversion error.

impl TryFrom<Integer> for Float

fn try_from(n: Integer) -> Result<Self, Self::Error>

Converts an Integer to a Float, taking the Integer by value.

If the Integer is nonzero, the precision of the Float is the minimum possible precision to represent the Integer exactly. If you want to specify some other precision, try Float::from_integer_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_integer_prec_round.

If the absolue value of the Integer is greater than or equal to $2^{2^{30}-1}$, this function returns an overflow error.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is n.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::integer::Integer;

assert_eq!(Float::try_from(Integer::ZERO).unwrap().to_string(), "0.0");
assert_eq!(
    Float::try_from(Integer::from(123)).unwrap().to_string(),
    "123.0"
);
assert_eq!(
    Float::try_from(Integer::from(123)).unwrap().get_prec(),
    Some(7)
);
assert_eq!(
    Float::try_from(Integer::from(10)).unwrap().to_string(),
    "10.0"
);
assert_eq!(
    Float::try_from(Integer::from(10)).unwrap().get_prec(),
    Some(3)
);
assert_eq!(
    Float::try_from(Integer::from(-123)).unwrap().to_string(),
    "-123.0"
);
assert_eq!(
    Float::try_from(Integer::from(-123)).unwrap().get_prec(),
    Some(7)
);
assert_eq!(
    Float::try_from(Integer::from(-10)).unwrap().to_string(),
    "-10.0"
);
assert_eq!(
    Float::try_from(Integer::from(-10)).unwrap().get_prec(),
    Some(3)
);

type Error = FloatConversionError

The type returned in the event of a conversion error.

impl TryFrom<Natural> for Float

fn try_from(x: Natural) -> Result<Self, Self::Error>

Converts a Natural to a Float, taking the Natural by value.

If the Natural is nonzero, the precision of the Float is the minimum possible precision to represent the Natural exactly. If you want to specify some other precision, try Float::from_natural_prec. This may require rounding, which uses Nearest by default. To specify a rounding mode as well as a precision, try Float::from_natural_prec_round.

If the Natural is greater than or equal to $2^{2^{30}-1}$, this function returns an overflow error.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is n.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_nz::natural::Natural;

assert_eq!(Float::try_from(Natural::ZERO).unwrap().to_string(), "0.0");
assert_eq!(
    Float::try_from(Natural::from(123u32)).unwrap().to_string(),
    "123.0"
);
assert_eq!(
    Float::try_from(Natural::from(123u32)).unwrap().get_prec(),
    Some(7)
);
assert_eq!(
    Float::try_from(Natural::from(10u32)).unwrap().to_string(),
    "10.0"
);
assert_eq!(
    Float::try_from(Natural::from(10u32)).unwrap().get_prec(),
    Some(3)
);

type Error = FloatConversionError

The type returned in the event of a conversion error.

impl TryFrom<Rational> for Float

fn try_from(x: Rational) -> Result<Self, Self::Error>

Converts a Rational to an Float, taking the Rational by value. If the Rational’s denominator is not a power of 2, or if the Rational is too far from zero or too close to zero to be represented as a Float, an error is returned.

The Float’s precision is the minimum number of bits needed to exactly represent the Rational.

• If the Rational is greater than or equal to $2^{2^{30}-1}$), this function returns an overflow error.
• If the Rational is less than or equal to $-2^{2^{30}-1}$), this function returns an overflow error.
• If the Rational is positive and less than $2^{-2^{30}}$), this function returns an underflow error.
• If the Rational is negative and greater than $-2^{-2^{30}}$), this function returns an underflow error.

Worst-case complexity

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ is x.significant_bits().

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_float::Float;
use malachite_q::conversion::primitive_float_from_rational::FloatConversionError;
use malachite_q::Rational;

assert_eq!(Float::try_from(Rational::ZERO).unwrap(), 0);
assert_eq!(
    Float::try_from(Rational::from_signeds(1, 8)).unwrap(),
    0.125
);
assert_eq!(
    Float::try_from(Rational::from_signeds(-1, 8)).unwrap(),
    -0.125
);

assert_eq!(
    Float::try_from(Rational::from_signeds(1, 3)),
    Err(FloatConversionError::Inexact)
);
assert_eq!(
    Float::try_from(Rational::from_signeds(-1, 3)),
    Err(FloatConversionError::Inexact)
);

type Error = FloatConversionError

The type returned in the event of a conversion error.

impl Two for Float

The constant 2.0, with precision 1.

const TWO: Self

impl Zero for Float

The constant 0.0 (positive zero), with precision 1.

const ZERO: Self