Struct inari::Interval [−][src]
#[repr(C)]pub struct Interval { /* fields omitted */ }
Expand description
An interval with f64
bounds.
It is sometimes referred to as a bare interval in contrast to a decorated interval (DecInterval
).
Implementations
Returns $(\self Γ \rhs) + \addend$.
Domain | Range |
---|---|
$\R^3$ | $\R$ |
Returns the multiplicative inverse of self
.
Domain | Range |
---|---|
$\R β \set 0$ | $\R β \set 0$ |
Returns true
if rhs
is a member of self
: $\rhs β \self$.
The result is false
whenever rhs
is infinite or NaN.
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).contains(1.0));
assert!(!Interval::EMPTY.contains(1.0));
assert!(Interval::ENTIRE.contains(1.0));
$Β±β$ and NaN are not real numbers, thus do not belong to any interval:
use inari::*;
assert!(!Interval::ENTIRE.contains(f64::INFINITY));
assert!(!Interval::ENTIRE.contains(f64::NEG_INFINITY));
assert!(!Interval::ENTIRE.contains(f64::NAN));
Returns true
if self
and rhs
are disjoint:
$$ \self β© \rhs = β , $$
or equivalently,
$$ βx β \self, βy β \rhs : x β y, $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | true |
$\self = [a, b]$ | true | $b < c β¨ d < a$ |
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).disjoint(const_interval!(3.0, 4.0)));
assert!(!const_interval!(1.0, 3.0).disjoint(const_interval!(3.0, 4.0)));
assert!(!const_interval!(1.0, 5.0).disjoint(const_interval!(3.0, 4.0)));
assert!(Interval::EMPTY.disjoint(Interval::EMPTY));
assert!(Interval::EMPTY.disjoint(Interval::ENTIRE));
Returns true
if self
is interior to rhs
:
$$ (βx β \self, βy β \rhs : x < y) β§ (βx β \self, βy β \rhs : y < x), $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | true |
$\self = [a, b]$ | false | $c <β² a β§ b <β² d$ |
where $<β²$ is defined as:
$$ x <β² y :βΊ x < y β¨ x = y = -β β¨ x = y = +β. $$
Examples
use inari::*;
assert!(const_interval!(1.1, 1.9).interior(const_interval!(1.0, 2.0)));
assert!(!const_interval!(1.1, 2.0).interior(const_interval!(1.0, 2.0)));
assert!(Interval::EMPTY.interior(Interval::EMPTY));
assert!(Interval::ENTIRE.interior(Interval::ENTIRE));
Returns true
if self
is nonempty and bounded.
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).is_common_interval());
assert!(!const_interval!(1.0, f64::INFINITY).is_common_interval());
assert!(!Interval::EMPTY.is_common_interval());
assert!(!Interval::ENTIRE.is_common_interval());
Returns true
if self
is empty: $\self = β
$.
Examples
use inari::*;
assert!(!const_interval!(1.0, 1.0).is_empty());
assert!(Interval::EMPTY.is_empty());
assert!(!Interval::ENTIRE.is_empty());
Returns true
if $\self = [-β, +β]$.
Examples
use inari::*;
assert!(!const_interval!(1.0, f64::INFINITY).is_entire());
assert!(!Interval::EMPTY.is_entire());
assert!(Interval::ENTIRE.is_entire());
Returns true
if self
consists of a single real number:
$$ βx β β : \self = [x, x]. $$
The result is false
whenever self
is empty or unbounded.
Examples
use inari::*;
assert!(const_interval!(1.0, 1.0).is_singleton());
assert!(!const_interval!(1.0, 2.0).is_singleton());
assert!(!Interval::EMPTY.is_singleton());
assert!(!Interval::ENTIRE.is_singleton());
0.1 is not representable as a f64
number:
use inari::*;
// The singleton interval that consists of the closest [`f64`] number to 0.1.
assert!(const_interval!(0.1, 0.1).is_singleton());
// The tightest interval that encloses 0.1.
#[cfg(feature = "gmp")]
assert!(!interval!("[0.1, 0.1]").unwrap().is_singleton());
Returns true
if self
is weakly less than rhs
:
$$ (βx β \self, βy β \rhs : x β€ y) β§ (βy β \rhs, βx β \self : x β€ y), $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | false |
$\self = [a, b]$ | false | $a β€ c β§ b β€ d$ |
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).less(const_interval!(3.0, 4.0)));
assert!(const_interval!(1.0, 3.0).less(const_interval!(2.0, 4.0)));
assert!(const_interval!(1.0, 4.0).less(const_interval!(1.0, 4.0)));
assert!(Interval::EMPTY.less(Interval::EMPTY));
assert!(!Interval::EMPTY.less(Interval::ENTIRE));
assert!(!Interval::ENTIRE.less(Interval::EMPTY));
assert!(Interval::ENTIRE.less(Interval::ENTIRE));
Returns true
if self
is to the left of rhs
but may touch it:
$$ βx β \self, βy β \rhs : x β€ y, $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | true |
$\self = [a, b]$ | true | $b β€ c$ |
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).precedes(const_interval!(3.0, 4.0)));
assert!(const_interval!(1.0, 3.0).precedes(const_interval!(3.0, 4.0)));
assert!(!const_interval!(1.0, 3.0).precedes(const_interval!(2.0, 4.0)));
assert!(Interval::EMPTY.precedes(Interval::EMPTY));
assert!(Interval::EMPTY.precedes(Interval::ENTIRE));
assert!(Interval::ENTIRE.precedes(Interval::EMPTY));
assert!(!Interval::ENTIRE.precedes(Interval::ENTIRE));
Returns true
if self
is strictly less than rhs
:
$$ (βx β \self, βy β \rhs : x < y) β§ (βy β \self, βx β \rhs : x < y), $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | false |
$\self = [a, b]$ | false | $a <β² c β§ b <β² d$ |
where $<β²$ is defined as:
$$ x <β² y :βΊ x < y β¨ x = y = -β β¨ x = y = +β. $$
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).strict_less(const_interval!(3.0, 4.0)));
assert!(const_interval!(1.0, 3.0).strict_less(const_interval!(2.0, 4.0)));
assert!(!const_interval!(1.0, 4.0).strict_less(const_interval!(2.0, 4.0)));
assert!(const_interval!(1.0, f64::INFINITY).strict_less(const_interval!(2.0, f64::INFINITY)));
assert!(Interval::EMPTY.strict_less(Interval::EMPTY));
assert!(!Interval::EMPTY.strict_less(Interval::ENTIRE));
assert!(!Interval::ENTIRE.strict_less(Interval::EMPTY));
assert!(Interval::ENTIRE.strict_less(Interval::ENTIRE));
Returns true
if self
is strictly to the left of rhs
:
$$ βx β \self, βy β \rhs : x < y, $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | true |
$\self = [a, b]$ | true | $b < c$ |
Returns true
if self
is a subset of rhs
:
$$ \self β \rhs, $$
or equivalently,
$$ βx β \self, βy β \rhs : x = y, $$
or equivalently,
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | true | true |
$\self = [a, b]$ | false | $c β€ a β§ b β€ d$ |
Examples
use inari::*;
assert!(const_interval!(1.0, 2.0).subset(const_interval!(1.0, 2.0)));
assert!(Interval::EMPTY.subset(Interval::EMPTY));
assert!(Interval::EMPTY.subset(Interval::ENTIRE));
assert!(Interval::ENTIRE.subset(Interval::ENTIRE));
Returns the interchange representation of self
in the big-endian byte order.
Returns the interchange representation of self
in the little-endian byte order.
Returns the interchange representation of self
in the native byte order of the target platform.
Creates an Interval
from its interchange representation in the big-endian byte order.
Creates an Interval
from its interchange representation in the little-endian byte order.
The tightest interval enclosing $1 / \sqrt{2}$.
The tightest interval enclosing $2 / \sqrt{Ο}$.
Returns the inverse hyperbolic cosine of self
.
Domain | Range |
---|---|
$[1, β)$ | $[0, β)$ |
Returns the inverse sine of self
.
Domain | Range |
---|---|
$[-1, 1]$ | $[-Ο/2, Ο/2]$ |
Returns the angle of the point $(\rhs, \self)$ measured counterclockwise from the positive $x$-axis in the Euclidean plane.
Domain | Range |
---|---|
$\R^2 β \set{(0, 0)}$ | $(-Ο, Ο]$ |
Returns the inverse hyperbolic tangent of self
.
Domain | Range |
---|---|
$(-1, 1)$ | $\R$ |
Returns self
raised to the power of rhs
.
Domain | Range |
---|---|
$((0, β) Γ \R) βͺ (\set 0 Γ (0, β))$ | $[0, β)$ |
Returns self
raised to the power of rhs
.
For a fixed $n β \Z$, the domain and the range of the point function $\operatorname{pown}(x, n)$ are:
Domain | Range | |
---|---|---|
$n > 0$, odd | $\R$ | $\R$ |
$n > 0$, even | $\R$ | $[0, β)$ |
$n = 0$ | $\R$ | $\set 1$ |
$n < 0$, odd | $\R β \set 0$ | $\R β \set 0$ |
$n < 0$, even | $\R β \set 0$ | $(0, β)$ |
Returns the tangent of self
.
Domain | Range |
---|---|
$\R β \set{(n + 1/2) Ο β£ n β \Z}$ | $\R$ |
Rounds self
to the closest integer toward $+β$.
Domain | Range |
---|---|
$\R$ | $\Z$ |
Examples
use inari::*;
assert_eq!(const_interval!(0.2, 1.2).ceil(), const_interval!(1.0, 2.0));
assert_eq!(const_interval!(0.8, 1.8).ceil(), const_interval!(1.0, 2.0));
assert_eq!(const_interval!(-1.2, -0.2).ceil(), const_interval!(-1.0, 0.0));
assert_eq!(const_interval!(-1.8, -0.8).ceil(), const_interval!(-1.0, 0.0));
assert_eq!(Interval::EMPTY.ceil(), Interval::EMPTY);
assert_eq!(Interval::ENTIRE.ceil(), Interval::ENTIRE);
See also: Interval::floor
, Interval::trunc
.
Rounds self
to the closest integer toward $-β$.
Domain | Range |
---|---|
$\R$ | $\Z$ |
Examples
use inari::*;
assert_eq!(const_interval!(0.2, 1.2).floor(), const_interval!(0.0, 1.0));
assert_eq!(const_interval!(0.8, 1.8).floor(), const_interval!(0.0, 1.0));
assert_eq!(const_interval!(-1.2, -0.2).floor(), const_interval!(-2.0, -1.0));
assert_eq!(const_interval!(-1.8, -0.8).floor(), const_interval!(-2.0, -1.0));
assert_eq!(Interval::EMPTY.floor(), Interval::EMPTY);
assert_eq!(Interval::ENTIRE.floor(), Interval::ENTIRE);
See also: Interval::ceil
, Interval::trunc
.
Rounds self
to the closest integer, away from zero in case of ties.
Domain | Range |
---|---|
$\R$ | $\Z$ |
Examples
use inari::*;
assert_eq!(const_interval!(0.2, 1.2).round(), const_interval!(0.0, 1.0));
assert_eq!(const_interval!(0.5, 1.5).round(), const_interval!(1.0, 2.0));
assert_eq!(const_interval!(0.8, 1.8).round(), const_interval!(1.0, 2.0));
assert_eq!(const_interval!(-1.2, -0.2).round(), const_interval!(-1.0, 0.0));
assert_eq!(const_interval!(-1.5, -0.5).round(), const_interval!(-2.0, -1.0));
assert_eq!(const_interval!(-1.8, -0.8).round(), const_interval!(-2.0, -1.0));
assert_eq!(Interval::EMPTY.round(), Interval::EMPTY);
assert_eq!(Interval::ENTIRE.round(), Interval::ENTIRE);
See also: Interval::round_ties_to_even
.
Rounds self
to the closest integer, the even number in case of ties.
Domain | Range |
---|---|
$\R$ | $\Z$ |
Examples
use inari::*;
assert_eq!(const_interval!(0.2, 1.2).round_ties_to_even(), const_interval!(0.0, 1.0));
assert_eq!(const_interval!(0.5, 1.5).round_ties_to_even(), const_interval!(0.0, 2.0));
assert_eq!(const_interval!(0.8, 1.8).round_ties_to_even(), const_interval!(1.0, 2.0));
assert_eq!(const_interval!(-1.2, -0.2).round_ties_to_even(), const_interval!(-1.0, 0.0));
assert_eq!(const_interval!(-1.5, -0.5).round_ties_to_even(), const_interval!(-2.0, 0.0));
assert_eq!(const_interval!(-1.8, -0.8).round_ties_to_even(), const_interval!(-2.0, -1.0));
assert_eq!(Interval::EMPTY.round_ties_to_even(), Interval::EMPTY);
assert_eq!(Interval::ENTIRE.round_ties_to_even(), Interval::ENTIRE);
See also: Interval::round
.
Returns the sign of self
.
Domain | Range |
---|---|
$\R$ | $\set{-1, 0, 1}$ |
Note the difference in definition between f64::signum
and this function;
+0.0_f64.signum()
and -0.0_f64.signum()
return +1.0
and -1.0
, respectively,
while the sign of zero is just zero.
Examples
use inari::*;
assert_eq!(const_interval!(-10.0, -0.1).sign(), const_interval!(-1.0, -1.0));
assert_eq!(const_interval!(0.0, 0.0).sign(), const_interval!(0.0, 0.0));
assert_eq!(const_interval!(0.1, 10.0).sign(), const_interval!(1.0, 1.0));
assert_eq!(Interval::EMPTY.sign(), Interval::EMPTY);
assert_eq!(Interval::ENTIRE.sign(), const_interval!(-1.0, 1.0));
Rounds self
to the closest integer toward zero.
Domain | Range |
---|---|
$\R$ | $\Z$ |
Examples
use inari::*;
assert_eq!(const_interval!(0.2, 1.2).trunc(), const_interval!(0.0, 1.0));
assert_eq!(const_interval!(0.8, 1.8).trunc(), const_interval!(0.0, 1.0));
assert_eq!(const_interval!(-1.2, -0.2).trunc(), const_interval!(-1.0, 0.0));
assert_eq!(const_interval!(-1.8, -0.8).trunc(), const_interval!(-1.0, 0.0));
assert_eq!(Interval::EMPTY.trunc(), Interval::EMPTY);
assert_eq!(Interval::ENTIRE.trunc(), Interval::ENTIRE);
See also: Interval::ceil
, Interval::floor
.
Returns the lower bound of self
.
The lower bound of an interval $π$ is:
$$ \inf(π) = \begin{cases} +β & \if π = β , \\ a & \if π = [a, b]. \end{cases} $$
The exact value is returned.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).inf(), -2.0);
assert_eq!(Interval::EMPTY.inf(), f64::INFINITY);
assert_eq!(Interval::ENTIRE.inf(), f64::NEG_INFINITY);
See also: Interval::sup
.
Returns the magnitude of self
if it is nonempty; otherwise, a NaN.
The magnitude of a nonempty interval $π = [a, b]$ is:
$$ \begin{align*} \mag(π) &= \sup \set{|x| β£ x β π} \\ &= \max \set{|a|, |b|}. \end{align*} $$
The exact value is returned.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).mag(), 3.0);
assert!(Interval::EMPTY.mag().is_nan());
assert_eq!(Interval::ENTIRE.mag(), f64::INFINITY);
See also: Interval::mig
.
Returns the midpoint of self
if it is nonempty; otherwise, a NaN.
The midpoint of a nonempty interval $π = [a, b]$ is:
$$ \mid(π) = \frac{a + b}{2}. $$
As an approximation in f64
, it returns:
0.0
, if $\self = [-β, +β]$;f64::MIN
, if $\self = [-β, b]$, where $b β \R$;f64::MAX
, if $\self = [a, +β]$, where $a β \R$;- otherwise, the closest
f64
number to $\mid(\self)$, away from zero in case of ties.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).mid(), 0.5);
assert_eq!(const_interval!(f64::NEG_INFINITY, 3.0).mid(), f64::MIN);
assert_eq!(const_interval!(-2.0, f64::INFINITY).mid(), f64::MAX);
assert!(Interval::EMPTY.mid().is_nan());
assert_eq!(Interval::ENTIRE.mid(), 0.0);
See also: Interval::rad
.
Returns the mignitude of self
if it is nonempty; otherwise, a NaN.
The mignitude of a nonempty interval $π = [a, b]$ is:
$$ \begin{align*} \mig(π) &= \inf \set{|x| β£ x β π} \\ &= \begin{cases} \min \set{|a|, |b|} & \if \sgn(a) = \sgn(b), \\ 0 & \otherwise. \end{cases} \end{align*} $$
The exact value is returned.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).mig(), 0.0);
assert_eq!(const_interval!(2.0, 3.0).mig(), 2.0);
assert!(Interval::EMPTY.mig().is_nan());
assert_eq!(Interval::ENTIRE.mig(), 0.0);
See also: Interval::mag
.
Returns the radius of self
if it is nonempty; otherwise, a NaN.
The radius of a nonempty interval $π = [a, b]$ is:
$$ \rad(π) = \frac{b - a}{2}. $$
As an approximation in f64
, it returns the least f64
number r
that satisfies
$\self β [π - π, π + π]$, where m
is the midpoint returned by Self::mid
.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).rad(), 2.5);
assert!(Interval::EMPTY.rad().is_nan());
assert_eq!(Interval::ENTIRE.rad(), f64::INFINITY);
See also: Interval::mid
.
Returns the upper bound of self
.
The upper bound of an interval $π$ is:
$$ \sup(π) = \begin{cases} -β & \if π = β , \\ b & \if π = [a, b]. \end{cases} $$
The exact value is returned.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).sup(), 3.0);
assert_eq!(Interval::EMPTY.sup(), f64::NEG_INFINITY);
assert_eq!(Interval::ENTIRE.sup(), f64::INFINITY);
See also: Interval::inf
.
Returns the width of self
if it is nonempty; otherwise, a NaN.
The width of a nonempty interval $π = [a, b]$ is:
$$ \wid(π) = b - a. $$
As an approximation in f64
, it returns the closest f64
number toward $+β$.
Examples
use inari::*;
assert_eq!(const_interval!(-2.0, 3.0).wid(), 5.0);
assert_eq!(const_interval!(-1.0, f64::MAX).wid(), f64::INFINITY);
assert!(Interval::EMPTY.wid().is_nan());
assert_eq!(Interval::ENTIRE.wid(), f64::INFINITY);
Returns $\hull(\self βͺ \rhs)$, the tightest interval that contains both self
and rhs
as its subsets.
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | $β $ | $[c, d]$ |
$\self = [a, b]$ | $[a, b]$ | $[\min \set{a, c}, \max \set{b, d}]$ |
Returns $\self β© \rhs$, the intersection of self
and rhs
.
$\rhs = β $ | $\rhs = [c, d]$ | |
---|---|---|
$\self = β $ | $β $ | $β $ |
$\self = [a, b]$ | $β $ | $[\max \set{a, c}, \min \set{b, d}]$ |
Trait Implementations
Performs the +=
operation. Read more
Performs the /=
operation. Read more
Performs the *=
operation. Read more
Performs the -=
operation. Read more
Auto Trait Implementations
impl RefUnwindSafe for Interval
impl UnwindSafe for Interval
Blanket Implementations
Mutably borrows from an owned value. Read more
Casts the value.
Casts the value.
Casts the value.
Casts the value.
Casts the value.