use core::cmp::Ordering;
use core::num::FpCategory;
use core::ops::{Add, Div, Neg};
use core::f32;
use core::f64;
use crate::{Num, NumCast, ToPrimitive};
/// Generic trait for floating point numbers that works with `no_std`.
///
/// This trait implements a subset of the `Float` trait.
pub trait FloatCore: Num + NumCast + Neg<Output = Self> + PartialOrd + Copy {
/// Returns positive infinity.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T) {
/// assert!(T::infinity() == x);
/// }
///
/// check(f32::INFINITY);
/// check(f64::INFINITY);
/// ```
fn infinity() -> Self;
/// Returns negative infinity.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T) {
/// assert!(T::neg_infinity() == x);
/// }
///
/// check(f32::NEG_INFINITY);
/// check(f64::NEG_INFINITY);
/// ```
fn neg_infinity() -> Self;
/// Returns NaN.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
///
/// fn check<T: FloatCore>() {
/// let n = T::nan();
/// assert!(n != n);
/// }
///
/// check::<f32>();
/// check::<f64>();
/// ```
fn nan() -> Self;
/// Returns `-0.0`.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(n: T) {
/// let z = T::neg_zero();
/// assert!(z.is_zero());
/// assert!(T::one() / z == n);
/// }
///
/// check(f32::NEG_INFINITY);
/// check(f64::NEG_INFINITY);
/// ```
fn neg_zero() -> Self;
/// Returns the smallest finite value that this type can represent.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T) {
/// assert!(T::min_value() == x);
/// }
///
/// check(f32::MIN);
/// check(f64::MIN);
/// ```
fn min_value() -> Self;
/// Returns the smallest positive, normalized value that this type can represent.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T) {
/// assert!(T::min_positive_value() == x);
/// }
///
/// check(f32::MIN_POSITIVE);
/// check(f64::MIN_POSITIVE);
/// ```
fn min_positive_value() -> Self;
/// Returns epsilon, a small positive value.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T) {
/// assert!(T::epsilon() == x);
/// }
///
/// check(f32::EPSILON);
/// check(f64::EPSILON);
/// ```
fn epsilon() -> Self;
/// Returns the largest finite value that this type can represent.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T) {
/// assert!(T::max_value() == x);
/// }
///
/// check(f32::MAX);
/// check(f64::MAX);
/// ```
fn max_value() -> Self;
/// Returns `true` if the number is NaN.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, p: bool) {
/// assert!(x.is_nan() == p);
/// }
///
/// check(f32::NAN, true);
/// check(f32::INFINITY, false);
/// check(f64::NAN, true);
/// check(0.0f64, false);
/// ```
#[inline]
#[allow(clippy::eq_op)]
fn is_nan(self) -> bool {
self != self
}
/// Returns `true` if the number is infinite.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, p: bool) {
/// assert!(x.is_infinite() == p);
/// }
///
/// check(f32::INFINITY, true);
/// check(f32::NEG_INFINITY, true);
/// check(f32::NAN, false);
/// check(f64::INFINITY, true);
/// check(f64::NEG_INFINITY, true);
/// check(0.0f64, false);
/// ```
#[inline]
fn is_infinite(self) -> bool {
self == Self::infinity() || self == Self::neg_infinity()
}
/// Returns `true` if the number is neither infinite or NaN.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, p: bool) {
/// assert!(x.is_finite() == p);
/// }
///
/// check(f32::INFINITY, false);
/// check(f32::MAX, true);
/// check(f64::NEG_INFINITY, false);
/// check(f64::MIN_POSITIVE, true);
/// check(f64::NAN, false);
/// ```
#[inline]
fn is_finite(self) -> bool {
!(self.is_nan() || self.is_infinite())
}
/// Returns `true` if the number is neither zero, infinite, subnormal or NaN.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, p: bool) {
/// assert!(x.is_normal() == p);
/// }
///
/// check(f32::INFINITY, false);
/// check(f32::MAX, true);
/// check(f64::NEG_INFINITY, false);
/// check(f64::MIN_POSITIVE, true);
/// check(0.0f64, false);
/// ```
#[inline]
fn is_normal(self) -> bool {
self.classify() == FpCategory::Normal
}
/// Returns `true` if the number is [subnormal].
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::f64;
///
/// let min = f64::MIN_POSITIVE; // 2.2250738585072014e-308_f64
/// let max = f64::MAX;
/// let lower_than_min = 1.0e-308_f64;
/// let zero = 0.0_f64;
///
/// assert!(!min.is_subnormal());
/// assert!(!max.is_subnormal());
///
/// assert!(!zero.is_subnormal());
/// assert!(!f64::NAN.is_subnormal());
/// assert!(!f64::INFINITY.is_subnormal());
/// // Values between `0` and `min` are Subnormal.
/// assert!(lower_than_min.is_subnormal());
/// ```
/// [subnormal]: https://en.wikipedia.org/wiki/Subnormal_number
#[inline]
fn is_subnormal(self) -> bool {
self.classify() == FpCategory::Subnormal
}
/// Returns the floating point category of the number. If only one property
/// is going to be tested, it is generally faster to use the specific
/// predicate instead.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
/// use std::num::FpCategory;
///
/// fn check<T: FloatCore>(x: T, c: FpCategory) {
/// assert!(x.classify() == c);
/// }
///
/// check(f32::INFINITY, FpCategory::Infinite);
/// check(f32::MAX, FpCategory::Normal);
/// check(f64::NAN, FpCategory::Nan);
/// check(f64::MIN_POSITIVE, FpCategory::Normal);
/// check(f64::MIN_POSITIVE / 2.0, FpCategory::Subnormal);
/// check(0.0f64, FpCategory::Zero);
/// ```
fn classify(self) -> FpCategory;
/// Returns the largest integer less than or equal to a number.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.floor() == y);
/// }
///
/// check(f32::INFINITY, f32::INFINITY);
/// check(0.9f32, 0.0);
/// check(1.0f32, 1.0);
/// check(1.1f32, 1.0);
/// check(-0.0f64, 0.0);
/// check(-0.9f64, -1.0);
/// check(-1.0f64, -1.0);
/// check(-1.1f64, -2.0);
/// check(f64::MIN, f64::MIN);
/// ```
#[inline]
fn floor(self) -> Self {
let f = self.fract();
if f.is_nan() || f.is_zero() {
self
} else if self < Self::zero() {
self - f - Self::one()
} else {
self - f
}
}
/// Returns the smallest integer greater than or equal to a number.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.ceil() == y);
/// }
///
/// check(f32::INFINITY, f32::INFINITY);
/// check(0.9f32, 1.0);
/// check(1.0f32, 1.0);
/// check(1.1f32, 2.0);
/// check(-0.0f64, 0.0);
/// check(-0.9f64, -0.0);
/// check(-1.0f64, -1.0);
/// check(-1.1f64, -1.0);
/// check(f64::MIN, f64::MIN);
/// ```
#[inline]
fn ceil(self) -> Self {
let f = self.fract();
if f.is_nan() || f.is_zero() {
self
} else if self > Self::zero() {
self - f + Self::one()
} else {
self - f
}
}
/// Returns the nearest integer to a number. Round half-way cases away from `0.0`.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.round() == y);
/// }
///
/// check(f32::INFINITY, f32::INFINITY);
/// check(0.4f32, 0.0);
/// check(0.5f32, 1.0);
/// check(0.6f32, 1.0);
/// check(-0.4f64, 0.0);
/// check(-0.5f64, -1.0);
/// check(-0.6f64, -1.0);
/// check(f64::MIN, f64::MIN);
/// ```
#[inline]
fn round(self) -> Self {
let one = Self::one();
let h = Self::from(0.5).expect("Unable to cast from 0.5");
let f = self.fract();
if f.is_nan() || f.is_zero() {
self
} else if self > Self::zero() {
if f < h {
self - f
} else {
self - f + one
}
} else if -f < h {
self - f
} else {
self - f - one
}
}
/// Return the integer part of a number.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.trunc() == y);
/// }
///
/// check(f32::INFINITY, f32::INFINITY);
/// check(0.9f32, 0.0);
/// check(1.0f32, 1.0);
/// check(1.1f32, 1.0);
/// check(-0.0f64, 0.0);
/// check(-0.9f64, -0.0);
/// check(-1.0f64, -1.0);
/// check(-1.1f64, -1.0);
/// check(f64::MIN, f64::MIN);
/// ```
#[inline]
fn trunc(self) -> Self {
let f = self.fract();
if f.is_nan() {
self
} else {
self - f
}
}
/// Returns the fractional part of a number.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.fract() == y);
/// }
///
/// check(f32::MAX, 0.0);
/// check(0.75f32, 0.75);
/// check(1.0f32, 0.0);
/// check(1.25f32, 0.25);
/// check(-0.0f64, 0.0);
/// check(-0.75f64, -0.75);
/// check(-1.0f64, 0.0);
/// check(-1.25f64, -0.25);
/// check(f64::MIN, 0.0);
/// ```
#[inline]
fn fract(self) -> Self {
if self.is_zero() {
Self::zero()
} else {
self % Self::one()
}
}
/// Computes the absolute value of `self`. Returns `FloatCore::nan()` if the
/// number is `FloatCore::nan()`.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.abs() == y);
/// }
///
/// check(f32::INFINITY, f32::INFINITY);
/// check(1.0f32, 1.0);
/// check(0.0f64, 0.0);
/// check(-0.0f64, 0.0);
/// check(-1.0f64, 1.0);
/// check(f64::MIN, f64::MAX);
/// ```
#[inline]
fn abs(self) -> Self {
if self.is_sign_positive() {
return self;
}
if self.is_sign_negative() {
return -self;
}
Self::nan()
}
/// Returns a number that represents the sign of `self`.
///
/// - `1.0` if the number is positive, `+0.0` or `FloatCore::infinity()`
/// - `-1.0` if the number is negative, `-0.0` or `FloatCore::neg_infinity()`
/// - `FloatCore::nan()` if the number is `FloatCore::nan()`
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.signum() == y);
/// }
///
/// check(f32::INFINITY, 1.0);
/// check(3.0f32, 1.0);
/// check(0.0f32, 1.0);
/// check(-0.0f64, -1.0);
/// check(-3.0f64, -1.0);
/// check(f64::MIN, -1.0);
/// ```
#[inline]
fn signum(self) -> Self {
if self.is_nan() {
Self::nan()
} else if self.is_sign_negative() {
-Self::one()
} else {
Self::one()
}
}
/// Returns `true` if `self` is positive, including `+0.0` and
/// `FloatCore::infinity()`, and `FloatCore::nan()`.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, p: bool) {
/// assert!(x.is_sign_positive() == p);
/// }
///
/// check(f32::INFINITY, true);
/// check(f32::MAX, true);
/// check(0.0f32, true);
/// check(-0.0f64, false);
/// check(f64::NEG_INFINITY, false);
/// check(f64::MIN_POSITIVE, true);
/// check(f64::NAN, true);
/// check(-f64::NAN, false);
/// ```
#[inline]
fn is_sign_positive(self) -> bool {
!self.is_sign_negative()
}
/// Returns `true` if `self` is negative, including `-0.0` and
/// `FloatCore::neg_infinity()`, and `-FloatCore::nan()`.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, p: bool) {
/// assert!(x.is_sign_negative() == p);
/// }
///
/// check(f32::INFINITY, false);
/// check(f32::MAX, false);
/// check(0.0f32, false);
/// check(-0.0f64, true);
/// check(f64::NEG_INFINITY, true);
/// check(f64::MIN_POSITIVE, false);
/// check(f64::NAN, false);
/// check(-f64::NAN, true);
/// ```
#[inline]
fn is_sign_negative(self) -> bool {
let (_, _, sign) = self.integer_decode();
sign < 0
}
/// Returns the minimum of the two numbers.
///
/// If one of the arguments is NaN, then the other argument is returned.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T, min: T) {
/// assert!(x.min(y) == min);
/// }
///
/// check(1.0f32, 2.0, 1.0);
/// check(f32::NAN, 2.0, 2.0);
/// check(1.0f64, -2.0, -2.0);
/// check(1.0f64, f64::NAN, 1.0);
/// ```
#[inline]
fn min(self, other: Self) -> Self {
if self.is_nan() {
return other;
}
if other.is_nan() {
return self;
}
if self < other {
self
} else {
other
}
}
/// Returns the maximum of the two numbers.
///
/// If one of the arguments is NaN, then the other argument is returned.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T, max: T) {
/// assert!(x.max(y) == max);
/// }
///
/// check(1.0f32, 2.0, 2.0);
/// check(1.0f32, f32::NAN, 1.0);
/// check(-1.0f64, 2.0, 2.0);
/// check(-1.0f64, f64::NAN, -1.0);
/// ```
#[inline]
fn max(self, other: Self) -> Self {
if self.is_nan() {
return other;
}
if other.is_nan() {
return self;
}
if self > other {
self
} else {
other
}
}
/// Returns the reciprocal (multiplicative inverse) of the number.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, y: T) {
/// assert!(x.recip() == y);
/// assert!(y.recip() == x);
/// }
///
/// check(f32::INFINITY, 0.0);
/// check(2.0f32, 0.5);
/// check(-0.25f64, -4.0);
/// check(-0.0f64, f64::NEG_INFINITY);
/// ```
#[inline]
fn recip(self) -> Self {
Self::one() / self
}
/// Raise a number to an integer power.
///
/// Using this function is generally faster than using `powf`
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
///
/// fn check<T: FloatCore>(x: T, exp: i32, powi: T) {
/// assert!(x.powi(exp) == powi);
/// }
///
/// check(9.0f32, 2, 81.0);
/// check(1.0f32, -2, 1.0);
/// check(10.0f64, 20, 1e20);
/// check(4.0f64, -2, 0.0625);
/// check(-1.0f64, std::i32::MIN, 1.0);
/// ```
#[inline]
fn powi(mut self, mut exp: i32) -> Self {
if exp < 0 {
exp = exp.wrapping_neg();
self = self.recip();
}
// It should always be possible to convert a positive `i32` to a `usize`.
// Note, `i32::MIN` will wrap and still be negative, so we need to convert
// to `u32` without sign-extension before growing to `usize`.
super::pow(self, (exp as u32).to_usize().unwrap())
}
/// Converts to degrees, assuming the number is in radians.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(rad: T, deg: T) {
/// assert!(rad.to_degrees() == deg);
/// }
///
/// check(0.0f32, 0.0);
/// check(f32::consts::PI, 180.0);
/// check(f64::consts::FRAC_PI_4, 45.0);
/// check(f64::INFINITY, f64::INFINITY);
/// ```
fn to_degrees(self) -> Self;
/// Converts to radians, assuming the number is in degrees.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(deg: T, rad: T) {
/// assert!(deg.to_radians() == rad);
/// }
///
/// check(0.0f32, 0.0);
/// check(180.0, f32::consts::PI);
/// check(45.0, f64::consts::FRAC_PI_4);
/// check(f64::INFINITY, f64::INFINITY);
/// ```
fn to_radians(self) -> Self;
/// Returns the mantissa, base 2 exponent, and sign as integers, respectively.
/// The original number can be recovered by `sign * mantissa * 2 ^ exponent`.
///
/// # Examples
///
/// ```
/// use num_traits::float::FloatCore;
/// use std::{f32, f64};
///
/// fn check<T: FloatCore>(x: T, m: u64, e: i16, s:i8) {
/// let (mantissa, exponent, sign) = x.integer_decode();
/// assert_eq!(mantissa, m);
/// assert_eq!(exponent, e);
/// assert_eq!(sign, s);
/// }
///
/// check(2.0f32, 1 << 23, -22, 1);
/// check(-2.0f32, 1 << 23, -22, -1);
/// check(f32::INFINITY, 1 << 23, 105, 1);
/// check(f64::NEG_INFINITY, 1 << 52, 972, -1);
/// ```
fn integer_decode(self) -> (u64, i16, i8);
}
impl FloatCore for f32 {
constant! {
infinity() -> f32::INFINITY;
neg_infinity() -> f32::NEG_INFINITY;
nan() -> f32::NAN;
neg_zero() -> -0.0;
min_value() -> f32::MIN;
min_positive_value() -> f32::MIN_POSITIVE;
epsilon() -> f32::EPSILON;
max_value() -> f32::MAX;
}
#[inline]
fn integer_decode(self) -> (u64, i16, i8) {
integer_decode_f32(self)
}
forward! {
Self::is_nan(self) -> bool;
Self::is_infinite(self) -> bool;
Self::is_finite(self) -> bool;
Self::is_normal(self) -> bool;
Self::classify(self) -> FpCategory;
Self::is_sign_positive(self) -> bool;
Self::is_sign_negative(self) -> bool;
Self::min(self, other: Self) -> Self;
Self::max(self, other: Self) -> Self;
Self::recip(self) -> Self;
Self::to_degrees(self) -> Self;
Self::to_radians(self) -> Self;
}
#[cfg(has_is_subnormal)]
forward! {
Self::is_subnormal(self) -> bool;
}
#[cfg(feature = "std")]
forward! {
Self::floor(self) -> Self;
Self::ceil(self) -> Self;
Self::round(self) -> Self;
Self::trunc(self) -> Self;
Self::fract(self) -> Self;
Self::abs(self) -> Self;
Self::signum(self) -> Self;
Self::powi(self, n: i32) -> Self;
}
#[cfg(all(not(feature = "std"), feature = "libm"))]
forward! {
libm::floorf as floor(self) -> Self;
libm::ceilf as ceil(self) -> Self;
libm::roundf as round(self) -> Self;
libm::truncf as trunc(self) -> Self;
libm::fabsf as abs(self) -> Self;
}
#[cfg(all(not(feature = "std"), feature = "libm"))]
#[inline]
fn fract(self) -> Self {
self - libm::truncf(self)
}
}
impl FloatCore for f64 {
constant! {
infinity() -> f64::INFINITY;
neg_infinity() -> f64::NEG_INFINITY;
nan() -> f64::NAN;
neg_zero() -> -0.0;
min_value() -> f64::MIN;
min_positive_value() -> f64::MIN_POSITIVE;
epsilon() -> f64::EPSILON;
max_value() -> f64::MAX;
}
#[inline]
fn integer_decode(self) -> (u64, i16, i8) {
integer_decode_f64(self)
}
forward! {
Self::is_nan(self) -> bool;
Self::is_infinite(self) -> bool;
Self::is_finite(self) -> bool;
Self::is_normal(self) -> bool;
Self::classify(self) -> FpCategory;
Self::is_sign_positive(self) -> bool;
Self::is_sign_negative(self) -> bool;
Self::min(self, other: Self) -> Self;
Self::max(self, other: Self) -> Self;
Self::recip(self) -> Self;
Self::to_degrees(self) -> Self;
Self::to_radians(self) -> Self;
}
#[cfg(has_is_subnormal)]
forward! {
Self::is_subnormal(self) -> bool;
}
#[cfg(feature = "std")]
forward! {
Self::floor(self) -> Self;
Self::ceil(self) -> Self;
Self::round(self) -> Self;
Self::trunc(self) -> Self;
Self::fract(self) -> Self;
Self::abs(self) -> Self;
Self::signum(self) -> Self;
Self::powi(self, n: i32) -> Self;
}
#[cfg(all(not(feature = "std"), feature = "libm"))]
forward! {
libm::floor as floor(self) -> Self;
libm::ceil as ceil(self) -> Self;
libm::round as round(self) -> Self;
libm::trunc as trunc(self) -> Self;
libm::fabs as abs(self) -> Self;
}
#[cfg(all(not(feature = "std"), feature = "libm"))]
#[inline]
fn fract(self) -> Self {
self - libm::trunc(self)
}
}
// FIXME: these doctests aren't actually helpful, because they're using and
// testing the inherent methods directly, not going through `Float`.
/// Generic trait for floating point numbers
///
/// This trait is only available with the `std` feature, or with the `libm` feature otherwise.
#[cfg(any(feature = "std", feature = "libm"))]
pub trait Float: Num + Copy + NumCast + PartialOrd + Neg<Output = Self> {
/// Returns the `NaN` value.
///
/// ```
/// use num_traits::Float;
///
/// let nan: f32 = Float::nan();
///
/// assert!(nan.is_nan());
/// ```
fn nan() -> Self;
/// Returns the infinite value.
///
/// ```
/// use num_traits::Float;
/// use std::f32;
///
/// let infinity: f32 = Float::infinity();
///
/// assert!(infinity.is_infinite());
/// assert!(!infinity.is_finite());
/// assert!(infinity > f32::MAX);
/// ```
fn infinity() -> Self;
/// Returns the negative infinite value.
///
/// ```
/// use num_traits::Float;
/// use std::f32;
///
/// let neg_infinity: f32 = Float::neg_infinity();
///
/// assert!(neg_infinity.is_infinite());
/// assert!(!neg_infinity.is_finite());
/// assert!(neg_infinity < f32::MIN);
/// ```
fn neg_infinity() -> Self;
/// Returns `-0.0`.
///
/// ```
/// use num_traits::{Zero, Float};
///
/// let inf: f32 = Float::infinity();
/// let zero: f32 = Zero::zero();
/// let neg_zero: f32 = Float::neg_zero();
///
/// assert_eq!(zero, neg_zero);
/// assert_eq!(7.0f32/inf, zero);
/// assert_eq!(zero * 10.0, zero);
/// ```
fn neg_zero() -> Self;
/// Returns the smallest finite value that this type can represent.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x: f64 = Float::min_value();
///
/// assert_eq!(x, f64::MIN);
/// ```
fn min_value() -> Self;
/// Returns the smallest positive, normalized value that this type can represent.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x: f64 = Float::min_positive_value();
///
/// assert_eq!(x, f64::MIN_POSITIVE);
/// ```
fn min_positive_value() -> Self;
/// Returns epsilon, a small positive value.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x: f64 = Float::epsilon();
///
/// assert_eq!(x, f64::EPSILON);
/// ```
///
/// # Panics
///
/// The default implementation will panic if `f32::EPSILON` cannot
/// be cast to `Self`.
fn epsilon() -> Self {
Self::from(f32::EPSILON).expect("Unable to cast from f32::EPSILON")
}
/// Returns the largest finite value that this type can represent.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x: f64 = Float::max_value();
/// assert_eq!(x, f64::MAX);
/// ```
fn max_value() -> Self;
/// Returns `true` if this value is `NaN` and false otherwise.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let nan = f64::NAN;
/// let f = 7.0;
///
/// assert!(nan.is_nan());
/// assert!(!f.is_nan());
/// ```
fn is_nan(self) -> bool;
/// Returns `true` if this value is positive infinity or negative infinity and
/// false otherwise.
///
/// ```
/// use num_traits::Float;
/// use std::f32;
///
/// let f = 7.0f32;
/// let inf: f32 = Float::infinity();
/// let neg_inf: f32 = Float::neg_infinity();
/// let nan: f32 = f32::NAN;
///
/// assert!(!f.is_infinite());
/// assert!(!nan.is_infinite());
///
/// assert!(inf.is_infinite());
/// assert!(neg_inf.is_infinite());
/// ```
fn is_infinite(self) -> bool;
/// Returns `true` if this number is neither infinite nor `NaN`.
///
/// ```
/// use num_traits::Float;
/// use std::f32;
///
/// let f = 7.0f32;
/// let inf: f32 = Float::infinity();
/// let neg_inf: f32 = Float::neg_infinity();
/// let nan: f32 = f32::NAN;
///
/// assert!(f.is_finite());
///
/// assert!(!nan.is_finite());
/// assert!(!inf.is_finite());
/// assert!(!neg_inf.is_finite());
/// ```
fn is_finite(self) -> bool;
/// Returns `true` if the number is neither zero, infinite,
/// [subnormal][subnormal], or `NaN`.
///
/// ```
/// use num_traits::Float;
/// use std::f32;
///
/// let min = f32::MIN_POSITIVE; // 1.17549435e-38f32
/// let max = f32::MAX;
/// let lower_than_min = 1.0e-40_f32;
/// let zero = 0.0f32;
///
/// assert!(min.is_normal());
/// assert!(max.is_normal());
///
/// assert!(!zero.is_normal());
/// assert!(!f32::NAN.is_normal());
/// assert!(!f32::INFINITY.is_normal());
/// // Values between `0` and `min` are Subnormal.
/// assert!(!lower_than_min.is_normal());
/// ```
/// [subnormal]: http://en.wikipedia.org/wiki/Subnormal_number
fn is_normal(self) -> bool;
/// Returns `true` if the number is [subnormal].
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let min = f64::MIN_POSITIVE; // 2.2250738585072014e-308_f64
/// let max = f64::MAX;
/// let lower_than_min = 1.0e-308_f64;
/// let zero = 0.0_f64;
///
/// assert!(!min.is_subnormal());
/// assert!(!max.is_subnormal());
///
/// assert!(!zero.is_subnormal());
/// assert!(!f64::NAN.is_subnormal());
/// assert!(!f64::INFINITY.is_subnormal());
/// // Values between `0` and `min` are Subnormal.
/// assert!(lower_than_min.is_subnormal());
/// ```
/// [subnormal]: https://en.wikipedia.org/wiki/Subnormal_number
#[inline]
fn is_subnormal(self) -> bool {
self.classify() == FpCategory::Subnormal
}
/// Returns the floating point category of the number. If only one property
/// is going to be tested, it is generally faster to use the specific
/// predicate instead.
///
/// ```
/// use num_traits::Float;
/// use std::num::FpCategory;
/// use std::f32;
///
/// let num = 12.4f32;
/// let inf = f32::INFINITY;
///
/// assert_eq!(num.classify(), FpCategory::Normal);
/// assert_eq!(inf.classify(), FpCategory::Infinite);
/// ```
fn classify(self) -> FpCategory;
/// Returns the largest integer less than or equal to a number.
///
/// ```
/// use num_traits::Float;
///
/// let f = 3.99;
/// let g = 3.0;
///
/// assert_eq!(f.floor(), 3.0);
/// assert_eq!(g.floor(), 3.0);
/// ```
fn floor(self) -> Self;
/// Returns the smallest integer greater than or equal to a number.
///
/// ```
/// use num_traits::Float;
///
/// let f = 3.01;
/// let g = 4.0;
///
/// assert_eq!(f.ceil(), 4.0);
/// assert_eq!(g.ceil(), 4.0);
/// ```
fn ceil(self) -> Self;
/// Returns the nearest integer to a number. Round half-way cases away from
/// `0.0`.
///
/// ```
/// use num_traits::Float;
///
/// let f = 3.3;
/// let g = -3.3;
///
/// assert_eq!(f.round(), 3.0);
/// assert_eq!(g.round(), -3.0);
/// ```
fn round(self) -> Self;
/// Return the integer part of a number.
///
/// ```
/// use num_traits::Float;
///
/// let f = 3.3;
/// let g = -3.7;
///
/// assert_eq!(f.trunc(), 3.0);
/// assert_eq!(g.trunc(), -3.0);
/// ```
fn trunc(self) -> Self;
/// Returns the fractional part of a number.
///
/// ```
/// use num_traits::Float;
///
/// let x = 3.5;
/// let y = -3.5;
/// let abs_difference_x = (x.fract() - 0.5).abs();
/// let abs_difference_y = (y.fract() - (-0.5)).abs();
///
/// assert!(abs_difference_x < 1e-10);
/// assert!(abs_difference_y < 1e-10);
/// ```
fn fract(self) -> Self;
/// Computes the absolute value of `self`. Returns `Float::nan()` if the
/// number is `Float::nan()`.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x = 3.5;
/// let y = -3.5;
///
/// let abs_difference_x = (x.abs() - x).abs();
/// let abs_difference_y = (y.abs() - (-y)).abs();
///
/// assert!(abs_difference_x < 1e-10);
/// assert!(abs_difference_y < 1e-10);
///
/// assert!(f64::NAN.abs().is_nan());
/// ```
fn abs(self) -> Self;
/// Returns a number that represents the sign of `self`.
///
/// - `1.0` if the number is positive, `+0.0` or `Float::infinity()`
/// - `-1.0` if the number is negative, `-0.0` or `Float::neg_infinity()`
/// - `Float::nan()` if the number is `Float::nan()`
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let f = 3.5;
///
/// assert_eq!(f.signum(), 1.0);
/// assert_eq!(f64::NEG_INFINITY.signum(), -1.0);
///
/// assert!(f64::NAN.signum().is_nan());
/// ```
fn signum(self) -> Self;
/// Returns `true` if `self` is positive, including `+0.0`,
/// `Float::infinity()`, and `Float::nan()`.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let nan: f64 = f64::NAN;
/// let neg_nan: f64 = -f64::NAN;
///
/// let f = 7.0;
/// let g = -7.0;
///
/// assert!(f.is_sign_positive());
/// assert!(!g.is_sign_positive());
/// assert!(nan.is_sign_positive());
/// assert!(!neg_nan.is_sign_positive());
/// ```
fn is_sign_positive(self) -> bool;
/// Returns `true` if `self` is negative, including `-0.0`,
/// `Float::neg_infinity()`, and `-Float::nan()`.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let nan: f64 = f64::NAN;
/// let neg_nan: f64 = -f64::NAN;
///
/// let f = 7.0;
/// let g = -7.0;
///
/// assert!(!f.is_sign_negative());
/// assert!(g.is_sign_negative());
/// assert!(!nan.is_sign_negative());
/// assert!(neg_nan.is_sign_negative());
/// ```
fn is_sign_negative(self) -> bool;
/// Fused multiply-add. Computes `(self * a) + b` with only one rounding
/// error, yielding a more accurate result than an unfused multiply-add.
///
/// Using `mul_add` can be more performant than an unfused multiply-add if
/// the target architecture has a dedicated `fma` CPU instruction.
///
/// ```
/// use num_traits::Float;
///
/// let m = 10.0;
/// let x = 4.0;
/// let b = 60.0;
///
/// // 100.0
/// let abs_difference = (m.mul_add(x, b) - (m*x + b)).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn mul_add(self, a: Self, b: Self) -> Self;
/// Take the reciprocal (inverse) of a number, `1/x`.
///
/// ```
/// use num_traits::Float;
///
/// let x = 2.0;
/// let abs_difference = (x.recip() - (1.0/x)).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn recip(self) -> Self;
/// Raise a number to an integer power.
///
/// Using this function is generally faster than using `powf`
///
/// ```
/// use num_traits::Float;
///
/// let x = 2.0;
/// let abs_difference = (x.powi(2) - x*x).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn powi(self, n: i32) -> Self;
/// Raise a number to a floating point power.
///
/// ```
/// use num_traits::Float;
///
/// let x = 2.0;
/// let abs_difference = (x.powf(2.0) - x*x).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn powf(self, n: Self) -> Self;
/// Take the square root of a number.
///
/// Returns NaN if `self` is a negative number.
///
/// ```
/// use num_traits::Float;
///
/// let positive = 4.0;
/// let negative = -4.0;
///
/// let abs_difference = (positive.sqrt() - 2.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// assert!(negative.sqrt().is_nan());
/// ```
fn sqrt(self) -> Self;
/// Returns `e^(self)`, (the exponential function).
///
/// ```
/// use num_traits::Float;
///
/// let one = 1.0;
/// // e^1
/// let e = one.exp();
///
/// // ln(e) - 1 == 0
/// let abs_difference = (e.ln() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn exp(self) -> Self;
/// Returns `2^(self)`.
///
/// ```
/// use num_traits::Float;
///
/// let f = 2.0;
///
/// // 2^2 - 4 == 0
/// let abs_difference = (f.exp2() - 4.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn exp2(self) -> Self;
/// Returns the natural logarithm of the number.
///
/// ```
/// use num_traits::Float;
///
/// let one = 1.0;
/// // e^1
/// let e = one.exp();
///
/// // ln(e) - 1 == 0
/// let abs_difference = (e.ln() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn ln(self) -> Self;
/// Returns the logarithm of the number with respect to an arbitrary base.
///
/// ```
/// use num_traits::Float;
///
/// let ten = 10.0;
/// let two = 2.0;
///
/// // log10(10) - 1 == 0
/// let abs_difference_10 = (ten.log(10.0) - 1.0).abs();
///
/// // log2(2) - 1 == 0
/// let abs_difference_2 = (two.log(2.0) - 1.0).abs();
///
/// assert!(abs_difference_10 < 1e-10);
/// assert!(abs_difference_2 < 1e-10);
/// ```
fn log(self, base: Self) -> Self;
/// Returns the base 2 logarithm of the number.
///
/// ```
/// use num_traits::Float;
///
/// let two = 2.0;
///
/// // log2(2) - 1 == 0
/// let abs_difference = (two.log2() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn log2(self) -> Self;
/// Returns the base 10 logarithm of the number.
///
/// ```
/// use num_traits::Float;
///
/// let ten = 10.0;
///
/// // log10(10) - 1 == 0
/// let abs_difference = (ten.log10() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn log10(self) -> Self;
/// Converts radians to degrees.
///
/// ```
/// use std::f64::consts;
///
/// let angle = consts::PI;
///
/// let abs_difference = (angle.to_degrees() - 180.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
#[inline]
fn to_degrees(self) -> Self {
let halfpi = Self::zero().acos();
let ninety = Self::from(90u8).unwrap();
self * ninety / halfpi
}
/// Converts degrees to radians.
///
/// ```
/// use std::f64::consts;
///
/// let angle = 180.0_f64;
///
/// let abs_difference = (angle.to_radians() - consts::PI).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
#[inline]
fn to_radians(self) -> Self {
let halfpi = Self::zero().acos();
let ninety = Self::from(90u8).unwrap();
self * halfpi / ninety
}
/// Returns the maximum of the two numbers.
///
/// ```
/// use num_traits::Float;
///
/// let x = 1.0;
/// let y = 2.0;
///
/// assert_eq!(x.max(y), y);
/// ```
fn max(self, other: Self) -> Self;
/// Returns the minimum of the two numbers.
///
/// ```
/// use num_traits::Float;
///
/// let x = 1.0;
/// let y = 2.0;
///
/// assert_eq!(x.min(y), x);
/// ```
fn min(self, other: Self) -> Self;
/// The positive difference of two numbers.
///
/// * If `self <= other`: `0:0`
/// * Else: `self - other`
///
/// ```
/// use num_traits::Float;
///
/// let x = 3.0;
/// let y = -3.0;
///
/// let abs_difference_x = (x.abs_sub(1.0) - 2.0).abs();
/// let abs_difference_y = (y.abs_sub(1.0) - 0.0).abs();
///
/// assert!(abs_difference_x < 1e-10);
/// assert!(abs_difference_y < 1e-10);
/// ```
fn abs_sub(self, other: Self) -> Self;
/// Take the cubic root of a number.
///
/// ```
/// use num_traits::Float;
///
/// let x = 8.0;
///
/// // x^(1/3) - 2 == 0
/// let abs_difference = (x.cbrt() - 2.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn cbrt(self) -> Self;
/// Calculate the length of the hypotenuse of a right-angle triangle given
/// legs of length `x` and `y`.
///
/// ```
/// use num_traits::Float;
///
/// let x = 2.0;
/// let y = 3.0;
///
/// // sqrt(x^2 + y^2)
/// let abs_difference = (x.hypot(y) - (x.powi(2) + y.powi(2)).sqrt()).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn hypot(self, other: Self) -> Self;
/// Computes the sine of a number (in radians).
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x = f64::consts::PI/2.0;
///
/// let abs_difference = (x.sin() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn sin(self) -> Self;
/// Computes the cosine of a number (in radians).
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x = 2.0*f64::consts::PI;
///
/// let abs_difference = (x.cos() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn cos(self) -> Self;
/// Computes the tangent of a number (in radians).
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x = f64::consts::PI/4.0;
/// let abs_difference = (x.tan() - 1.0).abs();
///
/// assert!(abs_difference < 1e-14);
/// ```
fn tan(self) -> Self;
/// Computes the arcsine of a number. Return value is in radians in
/// the range [-pi/2, pi/2] or NaN if the number is outside the range
/// [-1, 1].
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let f = f64::consts::PI / 2.0;
///
/// // asin(sin(pi/2))
/// let abs_difference = (f.sin().asin() - f64::consts::PI / 2.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn asin(self) -> Self;
/// Computes the arccosine of a number. Return value is in radians in
/// the range [0, pi] or NaN if the number is outside the range
/// [-1, 1].
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let f = f64::consts::PI / 4.0;
///
/// // acos(cos(pi/4))
/// let abs_difference = (f.cos().acos() - f64::consts::PI / 4.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn acos(self) -> Self;
/// Computes the arctangent of a number. Return value is in radians in the
/// range [-pi/2, pi/2];
///
/// ```
/// use num_traits::Float;
///
/// let f = 1.0;
///
/// // atan(tan(1))
/// let abs_difference = (f.tan().atan() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn atan(self) -> Self;
/// Computes the four quadrant arctangent of `self` (`y`) and `other` (`x`).
///
/// * `x = 0`, `y = 0`: `0`
/// * `x >= 0`: `arctan(y/x)` -> `[-pi/2, pi/2]`
/// * `y >= 0`: `arctan(y/x) + pi` -> `(pi/2, pi]`
/// * `y < 0`: `arctan(y/x) - pi` -> `(-pi, -pi/2)`
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let pi = f64::consts::PI;
/// // All angles from horizontal right (+x)
/// // 45 deg counter-clockwise
/// let x1 = 3.0;
/// let y1 = -3.0;
///
/// // 135 deg clockwise
/// let x2 = -3.0;
/// let y2 = 3.0;
///
/// let abs_difference_1 = (y1.atan2(x1) - (-pi/4.0)).abs();
/// let abs_difference_2 = (y2.atan2(x2) - 3.0*pi/4.0).abs();
///
/// assert!(abs_difference_1 < 1e-10);
/// assert!(abs_difference_2 < 1e-10);
/// ```
fn atan2(self, other: Self) -> Self;
/// Simultaneously computes the sine and cosine of the number, `x`. Returns
/// `(sin(x), cos(x))`.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x = f64::consts::PI/4.0;
/// let f = x.sin_cos();
///
/// let abs_difference_0 = (f.0 - x.sin()).abs();
/// let abs_difference_1 = (f.1 - x.cos()).abs();
///
/// assert!(abs_difference_0 < 1e-10);
/// assert!(abs_difference_0 < 1e-10);
/// ```
fn sin_cos(self) -> (Self, Self);
/// Returns `e^(self) - 1` in a way that is accurate even if the
/// number is close to zero.
///
/// ```
/// use num_traits::Float;
///
/// let x = 7.0;
///
/// // e^(ln(7)) - 1
/// let abs_difference = (x.ln().exp_m1() - 6.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn exp_m1(self) -> Self;
/// Returns `ln(1+n)` (natural logarithm) more accurately than if
/// the operations were performed separately.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let x = f64::consts::E - 1.0;
///
/// // ln(1 + (e - 1)) == ln(e) == 1
/// let abs_difference = (x.ln_1p() - 1.0).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn ln_1p(self) -> Self;
/// Hyperbolic sine function.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let e = f64::consts::E;
/// let x = 1.0;
///
/// let f = x.sinh();
/// // Solving sinh() at 1 gives `(e^2-1)/(2e)`
/// let g = (e*e - 1.0)/(2.0*e);
/// let abs_difference = (f - g).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn sinh(self) -> Self;
/// Hyperbolic cosine function.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let e = f64::consts::E;
/// let x = 1.0;
/// let f = x.cosh();
/// // Solving cosh() at 1 gives this result
/// let g = (e*e + 1.0)/(2.0*e);
/// let abs_difference = (f - g).abs();
///
/// // Same result
/// assert!(abs_difference < 1.0e-10);
/// ```
fn cosh(self) -> Self;
/// Hyperbolic tangent function.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let e = f64::consts::E;
/// let x = 1.0;
///
/// let f = x.tanh();
/// // Solving tanh() at 1 gives `(1 - e^(-2))/(1 + e^(-2))`
/// let g = (1.0 - e.powi(-2))/(1.0 + e.powi(-2));
/// let abs_difference = (f - g).abs();
///
/// assert!(abs_difference < 1.0e-10);
/// ```
fn tanh(self) -> Self;
/// Inverse hyperbolic sine function.
///
/// ```
/// use num_traits::Float;
///
/// let x = 1.0;
/// let f = x.sinh().asinh();
///
/// let abs_difference = (f - x).abs();
///
/// assert!(abs_difference < 1.0e-10);
/// ```
fn asinh(self) -> Self;
/// Inverse hyperbolic cosine function.
///
/// ```
/// use num_traits::Float;
///
/// let x = 1.0;
/// let f = x.cosh().acosh();
///
/// let abs_difference = (f - x).abs();
///
/// assert!(abs_difference < 1.0e-10);
/// ```
fn acosh(self) -> Self;
/// Inverse hyperbolic tangent function.
///
/// ```
/// use num_traits::Float;
/// use std::f64;
///
/// let e = f64::consts::E;
/// let f = e.tanh().atanh();
///
/// let abs_difference = (f - e).abs();
///
/// assert!(abs_difference < 1.0e-10);
/// ```
fn atanh(self) -> Self;
/// Returns the mantissa, base 2 exponent, and sign as integers, respectively.
/// The original number can be recovered by `sign * mantissa * 2 ^ exponent`.
///
/// ```
/// use num_traits::Float;
///
/// let num = 2.0f32;
///
/// // (8388608, -22, 1)
/// let (mantissa, exponent, sign) = Float::integer_decode(num);
/// let sign_f = sign as f32;
/// let mantissa_f = mantissa as f32;
/// let exponent_f = num.powf(exponent as f32);
///
/// // 1 * 8388608 * 2^(-22) == 2
/// let abs_difference = (sign_f * mantissa_f * exponent_f - num).abs();
///
/// assert!(abs_difference < 1e-10);
/// ```
fn integer_decode(self) -> (u64, i16, i8);
/// Returns a number composed of the magnitude of `self` and the sign of
/// `sign`.
///
/// Equal to `self` if the sign of `self` and `sign` are the same, otherwise
/// equal to `-self`. If `self` is a `NAN`, then a `NAN` with the sign of
/// `sign` is returned.
///
/// # Examples
///
/// ```
/// use num_traits::Float;
///
/// let f = 3.5_f32;
///
/// assert_eq!(f.copysign(0.42), 3.5_f32);
/// assert_eq!(f.copysign(-0.42), -3.5_f32);
/// assert_eq!((-f).copysign(0.42), 3.5_f32);
/// assert_eq!((-f).copysign(-0.42), -3.5_f32);
///
/// assert!(f32::nan().copysign(1.0).is_nan());
/// ```
fn copysign(self, sign: Self) -> Self {
if self.is_sign_negative() == sign.is_sign_negative() {
self
} else {
self.neg()
}
}
}
#[cfg(feature = "std")]
macro_rules! float_impl_std {
($T:ident $decode:ident) => {
impl Float for $T {
constant! {
nan() -> $T::NAN;
infinity() -> $T::INFINITY;
neg_infinity() -> $T::NEG_INFINITY;
neg_zero() -> -0.0;
min_value() -> $T::MIN;
min_positive_value() -> $T::MIN_POSITIVE;
epsilon() -> $T::EPSILON;
max_value() -> $T::MAX;
}
#[inline]
#[allow(deprecated)]
fn abs_sub(self, other: Self) -> Self {
<$T>::abs_sub(self, other)
}
#[inline]
fn integer_decode(self) -> (u64, i16, i8) {
$decode(self)
}
forward! {
Self::is_nan(self) -> bool;
Self::is_infinite(self) -> bool;
Self::is_finite(self) -> bool;
Self::is_normal(self) -> bool;
Self::classify(self) -> FpCategory;
Self::floor(self) -> Self;
Self::ceil(self) -> Self;
Self::round(self) -> Self;
Self::trunc(self) -> Self;
Self::fract(self) -> Self;
Self::abs(self) -> Self;
Self::signum(self) -> Self;
Self::is_sign_positive(self) -> bool;
Self::is_sign_negative(self) -> bool;
Self::mul_add(self, a: Self, b: Self) -> Self;
Self::recip(self) -> Self;
Self::powi(self, n: i32) -> Self;
Self::powf(self, n: Self) -> Self;
Self::sqrt(self) -> Self;
Self::exp(self) -> Self;
Self::exp2(self) -> Self;
Self::ln(self) -> Self;
Self::log(self, base: Self) -> Self;
Self::log2(self) -> Self;
Self::log10(self) -> Self;
Self::to_degrees(self) -> Self;
Self::to_radians(self) -> Self;
Self::max(self, other: Self) -> Self;
Self::min(self, other: Self) -> Self;
Self::cbrt(self) -> Self;
Self::hypot(self, other: Self) -> Self;
Self::sin(self) -> Self;
Self::cos(self) -> Self;
Self::tan(self) -> Self;
Self::asin(self) -> Self;
Self::acos(self) -> Self;
Self::atan(self) -> Self;
Self::atan2(self, other: Self) -> Self;
Self::sin_cos(self) -> (Self, Self);
Self::exp_m1(self) -> Self;
Self::ln_1p(self) -> Self;
Self::sinh(self) -> Self;
Self::cosh(self) -> Self;
Self::tanh(self) -> Self;
Self::asinh(self) -> Self;
Self::acosh(self) -> Self;
Self::atanh(self) -> Self;
}
#[cfg(has_copysign)]
forward! {
Self::copysign(self, sign: Self) -> Self;
}
#[cfg(has_is_subnormal)]
forward! {
Self::is_subnormal(self) -> bool;
}
}
};
}
#[cfg(all(not(feature = "std"), feature = "libm"))]
macro_rules! float_impl_libm {
($T:ident $decode:ident) => {
constant! {
nan() -> $T::NAN;
infinity() -> $T::INFINITY;
neg_infinity() -> $T::NEG_INFINITY;
neg_zero() -> -0.0;
min_value() -> $T::MIN;
min_positive_value() -> $T::MIN_POSITIVE;
epsilon() -> $T::EPSILON;
max_value() -> $T::MAX;
}
#[inline]
fn integer_decode(self) -> (u64, i16, i8) {
$decode(self)
}
#[inline]
fn fract(self) -> Self {
self - Float::trunc(self)
}
#[inline]
fn log(self, base: Self) -> Self {
self.ln() / base.ln()
}
forward! {
Self::is_nan(self) -> bool;
Self::is_infinite(self) -> bool;
Self::is_finite(self) -> bool;
Self::is_normal(self) -> bool;
Self::classify(self) -> FpCategory;
Self::is_sign_positive(self) -> bool;
Self::is_sign_negative(self) -> bool;
Self::min(self, other: Self) -> Self;
Self::max(self, other: Self) -> Self;
Self::recip(self) -> Self;
Self::to_degrees(self) -> Self;
Self::to_radians(self) -> Self;
}
#[cfg(has_is_subnormal)]
forward! {
Self::is_subnormal(self) -> bool;
}
forward! {
FloatCore::signum(self) -> Self;
FloatCore::powi(self, n: i32) -> Self;
}
};
}
fn integer_decode_f32(f: f32) -> (u64, i16, i8) {
let bits: u32 = f.to_bits();
let sign: i8 = if bits >> 31 == 0 { 1 } else { -1 };
let mut exponent: i16 = ((bits >> 23) & 0xff) as i16;
let mantissa = if exponent == 0 {
(bits & 0x7fffff) << 1
} else {
(bits & 0x7fffff) | 0x800000
};
// Exponent bias + mantissa shift
exponent -= 127 + 23;
(mantissa as u64, exponent, sign)
}
fn integer_decode_f64(f: f64) -> (u64, i16, i8) {
let bits: u64 = f.to_bits();
let sign: i8 = if bits >> 63 == 0 { 1 } else { -1 };
let mut exponent: i16 = ((bits >> 52) & 0x7ff) as i16;
let mantissa = if exponent == 0 {
(bits & 0xfffffffffffff) << 1
} else {
(bits & 0xfffffffffffff) | 0x10000000000000
};
// Exponent bias + mantissa shift
exponent -= 1023 + 52;
(mantissa, exponent, sign)
}
#[cfg(feature = "std")]
float_impl_std!(f32 integer_decode_f32);
#[cfg(feature = "std")]
float_impl_std!(f64 integer_decode_f64);
#[cfg(all(not(feature = "std"), feature = "libm"))]
impl Float for f32 {
float_impl_libm!(f32 integer_decode_f32);
#[inline]
#[allow(deprecated)]
fn abs_sub(self, other: Self) -> Self {
libm::fdimf(self, other)
}
forward! {
libm::floorf as floor(self) -> Self;
libm::ceilf as ceil(self) -> Self;
libm::roundf as round(self) -> Self;
libm::truncf as trunc(self) -> Self;
libm::fabsf as abs(self) -> Self;
libm::fmaf as mul_add(self, a: Self, b: Self) -> Self;
libm::powf as powf(self, n: Self) -> Self;
libm::sqrtf as sqrt(self) -> Self;
libm::expf as exp(self) -> Self;
libm::exp2f as exp2(self) -> Self;
libm::logf as ln(self) -> Self;
libm::log2f as log2(self) -> Self;
libm::log10f as log10(self) -> Self;
libm::cbrtf as cbrt(self) -> Self;
libm::hypotf as hypot(self, other: Self) -> Self;
libm::sinf as sin(self) -> Self;
libm::cosf as cos(self) -> Self;
libm::tanf as tan(self) -> Self;
libm::asinf as asin(self) -> Self;
libm::acosf as acos(self) -> Self;
libm::atanf as atan(self) -> Self;
libm::atan2f as atan2(self, other: Self) -> Self;
libm::sincosf as sin_cos(self) -> (Self, Self);
libm::expm1f as exp_m1(self) -> Self;
libm::log1pf as ln_1p(self) -> Self;
libm::sinhf as sinh(self) -> Self;
libm::coshf as cosh(self) -> Self;
libm::tanhf as tanh(self) -> Self;
libm::asinhf as asinh(self) -> Self;
libm::acoshf as acosh(self) -> Self;
libm::atanhf as atanh(self) -> Self;
libm::copysignf as copysign(self, other: Self) -> Self;
}
}
#[cfg(all(not(feature = "std"), feature = "libm"))]
impl Float for f64 {
float_impl_libm!(f64 integer_decode_f64);
#[inline]
#[allow(deprecated)]
fn abs_sub(self, other: Self) -> Self {
libm::fdim(self, other)
}
forward! {
libm::floor as floor(self) -> Self;
libm::ceil as ceil(self) -> Self;
libm::round as round(self) -> Self;
libm::trunc as trunc(self) -> Self;
libm::fabs as abs(self) -> Self;
libm::fma as mul_add(self, a: Self, b: Self) -> Self;
libm::pow as powf(self, n: Self) -> Self;
libm::sqrt as sqrt(self) -> Self;
libm::exp as exp(self) -> Self;
libm::exp2 as exp2(self) -> Self;
libm::log as ln(self) -> Self;
libm::log2 as log2(self) -> Self;
libm::log10 as log10(self) -> Self;
libm::cbrt as cbrt(self) -> Self;
libm::hypot as hypot(self, other: Self) -> Self;
libm::sin as sin(self) -> Self;
libm::cos as cos(self) -> Self;
libm::tan as tan(self) -> Self;
libm::asin as asin(self) -> Self;
libm::acos as acos(self) -> Self;
libm::atan as atan(self) -> Self;
libm::atan2 as atan2(self, other: Self) -> Self;
libm::sincos as sin_cos(self) -> (Self, Self);
libm::expm1 as exp_m1(self) -> Self;
libm::log1p as ln_1p(self) -> Self;
libm::sinh as sinh(self) -> Self;
libm::cosh as cosh(self) -> Self;
libm::tanh as tanh(self) -> Self;
libm::asinh as asinh(self) -> Self;
libm::acosh as acosh(self) -> Self;
libm::atanh as atanh(self) -> Self;
libm::copysign as copysign(self, sign: Self) -> Self;
}
}
macro_rules! float_const_impl {
($(#[$doc:meta] $constant:ident,)+) => (
#[allow(non_snake_case)]
pub trait FloatConst {
$(#[$doc] fn $constant() -> Self;)+
#[doc = "Return the full circle constant `Ï„`."]
#[inline]
fn TAU() -> Self where Self: Sized + Add<Self, Output = Self> {
Self::PI() + Self::PI()
}
#[doc = "Return `log10(2.0)`."]
#[inline]
fn LOG10_2() -> Self where Self: Sized + Div<Self, Output = Self> {
Self::LN_2() / Self::LN_10()
}
#[doc = "Return `log2(10.0)`."]
#[inline]
fn LOG2_10() -> Self where Self: Sized + Div<Self, Output = Self> {
Self::LN_10() / Self::LN_2()
}
}
float_const_impl! { @float f32, $($constant,)+ }
float_const_impl! { @float f64, $($constant,)+ }
);
(@float $T:ident, $($constant:ident,)+) => (
impl FloatConst for $T {
constant! {
$( $constant() -> $T::consts::$constant; )+
TAU() -> 6.28318530717958647692528676655900577;
LOG10_2() -> 0.301029995663981195213738894724493027;
LOG2_10() -> 3.32192809488736234787031942948939018;
}
}
);
}
float_const_impl! {
#[doc = "Return Euler’s number."]
E,
#[doc = "Return `1.0 / π`."]
FRAC_1_PI,
#[doc = "Return `1.0 / sqrt(2.0)`."]
FRAC_1_SQRT_2,
#[doc = "Return `2.0 / π`."]
FRAC_2_PI,
#[doc = "Return `2.0 / sqrt(Ï€)`."]
FRAC_2_SQRT_PI,
#[doc = "Return `Ï€ / 2.0`."]
FRAC_PI_2,
#[doc = "Return `Ï€ / 3.0`."]
FRAC_PI_3,
#[doc = "Return `Ï€ / 4.0`."]
FRAC_PI_4,
#[doc = "Return `Ï€ / 6.0`."]
FRAC_PI_6,
#[doc = "Return `Ï€ / 8.0`."]
FRAC_PI_8,
#[doc = "Return `ln(10.0)`."]
LN_10,
#[doc = "Return `ln(2.0)`."]
LN_2,
#[doc = "Return `log10(e)`."]
LOG10_E,
#[doc = "Return `log2(e)`."]
LOG2_E,
#[doc = "Return Archimedes’ constant `π`."]
PI,
#[doc = "Return `sqrt(2.0)`."]
SQRT_2,
}
/// Trait for floating point numbers that provide an implementation
/// of the `totalOrder` predicate as defined in the IEEE 754 (2008 revision)
/// floating point standard.
pub trait TotalOrder {
/// Return the ordering between `self` and `other`.
///
/// Unlike the standard partial comparison between floating point numbers,
/// this comparison always produces an ordering in accordance to
/// the `totalOrder` predicate as defined in the IEEE 754 (2008 revision)
/// floating point standard. The values are ordered in the following sequence:
///
/// - negative quiet NaN
/// - negative signaling NaN
/// - negative infinity
/// - negative numbers
/// - negative subnormal numbers
/// - negative zero
/// - positive zero
/// - positive subnormal numbers
/// - positive numbers
/// - positive infinity
/// - positive signaling NaN
/// - positive quiet NaN.
///
/// The ordering established by this function does not always agree with the
/// [`PartialOrd`] and [`PartialEq`] implementations. For example,
/// they consider negative and positive zero equal, while `total_cmp`
/// doesn't.
///
/// The interpretation of the signaling NaN bit follows the definition in
/// the IEEE 754 standard, which may not match the interpretation by some of
/// the older, non-conformant (e.g. MIPS) hardware implementations.
///
/// # Examples
/// ```
/// use num_traits::float::TotalOrder;
/// use std::cmp::Ordering;
/// use std::{f32, f64};
///
/// fn check_eq<T: TotalOrder>(x: T, y: T) {
/// assert_eq!(x.total_cmp(&y), Ordering::Equal);
/// }
///
/// check_eq(f64::NAN, f64::NAN);
/// check_eq(f32::NAN, f32::NAN);
///
/// fn check_lt<T: TotalOrder>(x: T, y: T) {
/// assert_eq!(x.total_cmp(&y), Ordering::Less);
/// }
///
/// check_lt(-f64::NAN, f64::NAN);
/// check_lt(f64::INFINITY, f64::NAN);
/// check_lt(-0.0_f64, 0.0_f64);
/// ```
fn total_cmp(&self, other: &Self) -> Ordering;
}
macro_rules! totalorder_impl {
($T:ident, $I:ident, $U:ident, $bits:expr) => {
impl TotalOrder for $T {
#[inline]
#[cfg(has_total_cmp)]
fn total_cmp(&self, other: &Self) -> Ordering {
// Forward to the core implementation
Self::total_cmp(&self, other)
}
#[inline]
#[cfg(not(has_total_cmp))]
fn total_cmp(&self, other: &Self) -> Ordering {
// Backport the core implementation (since 1.62)
let mut left = self.to_bits() as $I;
let mut right = other.to_bits() as $I;
left ^= (((left >> ($bits - 1)) as $U) >> 1) as $I;
right ^= (((right >> ($bits - 1)) as $U) >> 1) as $I;
left.cmp(&right)
}
}
};
}
totalorder_impl!(f64, i64, u64, 64);
totalorder_impl!(f32, i32, u32, 32);
#[cfg(test)]
mod tests {
use core::f64::consts;
const DEG_RAD_PAIRS: [(f64, f64); 7] = [
(0.0, 0.),
(22.5, consts::FRAC_PI_8),
(30.0, consts::FRAC_PI_6),
(45.0, consts::FRAC_PI_4),
(60.0, consts::FRAC_PI_3),
(90.0, consts::FRAC_PI_2),
(180.0, consts::PI),
];
#[test]
fn convert_deg_rad() {
use crate::float::FloatCore;
for &(deg, rad) in &DEG_RAD_PAIRS {
assert!((FloatCore::to_degrees(rad) - deg).abs() < 1e-6);
assert!((FloatCore::to_radians(deg) - rad).abs() < 1e-6);
let (deg, rad) = (deg as f32, rad as f32);
assert!((FloatCore::to_degrees(rad) - deg).abs() < 1e-5);
assert!((FloatCore::to_radians(deg) - rad).abs() < 1e-5);
}
}
#[cfg(any(feature = "std", feature = "libm"))]
#[test]
fn convert_deg_rad_std() {
for &(deg, rad) in &DEG_RAD_PAIRS {
use crate::Float;
assert!((Float::to_degrees(rad) - deg).abs() < 1e-6);
assert!((Float::to_radians(deg) - rad).abs() < 1e-6);
let (deg, rad) = (deg as f32, rad as f32);
assert!((Float::to_degrees(rad) - deg).abs() < 1e-5);
assert!((Float::to_radians(deg) - rad).abs() < 1e-5);
}
}
#[test]
fn to_degrees_rounding() {
use crate::float::FloatCore;
assert_eq!(
FloatCore::to_degrees(1_f32),
57.2957795130823208767981548141051703
);
}
#[test]
#[cfg(any(feature = "std", feature = "libm"))]
fn extra_logs() {
use crate::float::{Float, FloatConst};
fn check<F: Float + FloatConst>(diff: F) {
let _2 = F::from(2.0).unwrap();
assert!((F::LOG10_2() - F::log10(_2)).abs() < diff);
assert!((F::LOG10_2() - F::LN_2() / F::LN_10()).abs() < diff);
let _10 = F::from(10.0).unwrap();
assert!((F::LOG2_10() - F::log2(_10)).abs() < diff);
assert!((F::LOG2_10() - F::LN_10() / F::LN_2()).abs() < diff);
}
check::<f32>(1e-6);
check::<f64>(1e-12);
}
#[test]
#[cfg(any(feature = "std", feature = "libm"))]
fn copysign() {
use crate::float::Float;
test_copysign_generic(2.0_f32, -2.0_f32, f32::nan());
test_copysign_generic(2.0_f64, -2.0_f64, f64::nan());
test_copysignf(2.0_f32, -2.0_f32, f32::nan());
}
#[cfg(any(feature = "std", feature = "libm"))]
fn test_copysignf(p: f32, n: f32, nan: f32) {
use crate::float::Float;
use core::ops::Neg;
assert!(p.is_sign_positive());
assert!(n.is_sign_negative());
assert!(nan.is_nan());
assert_eq!(p, Float::copysign(p, p));
assert_eq!(p.neg(), Float::copysign(p, n));
assert_eq!(n, Float::copysign(n, n));
assert_eq!(n.neg(), Float::copysign(n, p));
assert!(Float::copysign(nan, p).is_sign_positive());
assert!(Float::copysign(nan, n).is_sign_negative());
}
#[cfg(any(feature = "std", feature = "libm"))]
fn test_copysign_generic<F: crate::float::Float + ::core::fmt::Debug>(p: F, n: F, nan: F) {
assert!(p.is_sign_positive());
assert!(n.is_sign_negative());
assert!(nan.is_nan());
assert!(!nan.is_subnormal());
assert_eq!(p, p.copysign(p));
assert_eq!(p.neg(), p.copysign(n));
assert_eq!(n, n.copysign(n));
assert_eq!(n.neg(), n.copysign(p));
assert!(nan.copysign(p).is_sign_positive());
assert!(nan.copysign(n).is_sign_negative());
}
#[cfg(any(feature = "std", feature = "libm"))]
fn test_subnormal<F: crate::float::Float + ::core::fmt::Debug>() {
let min_positive = F::min_positive_value();
let lower_than_min = min_positive / F::from(2.0f32).unwrap();
assert!(!min_positive.is_subnormal());
assert!(lower_than_min.is_subnormal());
}
#[test]
#[cfg(any(feature = "std", feature = "libm"))]
fn subnormal() {
test_subnormal::<f64>();
test_subnormal::<f32>();
}
#[test]
fn total_cmp() {
use crate::float::TotalOrder;
use core::cmp::Ordering;
use core::{f32, f64};
fn check_eq<T: TotalOrder>(x: T, y: T) {
assert_eq!(x.total_cmp(&y), Ordering::Equal);
}
fn check_lt<T: TotalOrder>(x: T, y: T) {
assert_eq!(x.total_cmp(&y), Ordering::Less);
}
fn check_gt<T: TotalOrder>(x: T, y: T) {
assert_eq!(x.total_cmp(&y), Ordering::Greater);
}
check_eq(f64::NAN, f64::NAN);
check_eq(f32::NAN, f32::NAN);
check_lt(-0.0_f64, 0.0_f64);
check_lt(-0.0_f32, 0.0_f32);
// x87 registers don't preserve the exact value of signaling NaN:
// https://github.com/rust-lang/rust/issues/115567
#[cfg(not(target_arch = "x86"))]
{
let s_nan = f64::from_bits(0x7ff4000000000000);
let q_nan = f64::from_bits(0x7ff8000000000000);
check_lt(s_nan, q_nan);
let neg_s_nan = f64::from_bits(0xfff4000000000000);
let neg_q_nan = f64::from_bits(0xfff8000000000000);
check_lt(neg_q_nan, neg_s_nan);
let s_nan = f32::from_bits(0x7fa00000);
let q_nan = f32::from_bits(0x7fc00000);
check_lt(s_nan, q_nan);
let neg_s_nan = f32::from_bits(0xffa00000);
let neg_q_nan = f32::from_bits(0xffc00000);
check_lt(neg_q_nan, neg_s_nan);
}
check_lt(-f64::NAN, f64::NEG_INFINITY);
check_gt(1.0_f64, -f64::NAN);
check_lt(f64::INFINITY, f64::NAN);
check_gt(f64::NAN, 1.0_f64);
check_lt(-f32::NAN, f32::NEG_INFINITY);
check_gt(1.0_f32, -f32::NAN);
check_lt(f32::INFINITY, f32::NAN);
check_gt(f32::NAN, 1.0_f32);
}
}