core/num/

f16.rs

//! Constants for the `f16` half-precision floating point type.
//!
//! *[See also the `f16` primitive type][f16].*
//!
//! Mathematically significant numbers are provided in the `consts` sub-module.
//!
//! For the constants defined directly in this module
//! (as distinct from those defined in the `consts` sub-module),
//! new code should instead use the associated constants
//! defined directly on the `f16` type.

#![unstable(feature = "f16", issue = "116909")]

use crate::convert::FloatToInt;
use crate::num::FpCategory;
#[cfg(not(test))]
use crate::num::libm;
use crate::panic::const_assert;
use crate::{intrinsics, mem};

/// Basic mathematical constants.
#[unstable(feature = "f16", issue = "116909")]
#[rustc_diagnostic_item = "f16_consts_mod"]
pub mod consts {
    // FIXME: replace with mathematical constants from cmath.

    /// Archimedes' constant (π)
    #[unstable(feature = "f16", issue = "116909")]
    pub const PI: f16 = 3.14159265358979323846264338327950288_f16;

    /// The full circle constant (τ)
    ///
    /// Equal to 2π.
    #[unstable(feature = "f16", issue = "116909")]
    pub const TAU: f16 = 6.28318530717958647692528676655900577_f16;

    /// The golden ratio (φ)
    #[unstable(feature = "f16", issue = "116909")]
    pub const GOLDEN_RATIO: f16 = 1.618033988749894848204586834365638118_f16;

    /// The Euler-Mascheroni constant (γ)
    #[unstable(feature = "f16", issue = "116909")]
    pub const EULER_GAMMA: f16 = 0.577215664901532860606512090082402431_f16;

    /// π/2
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_PI_2: f16 = 1.57079632679489661923132169163975144_f16;

    /// π/3
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_PI_3: f16 = 1.04719755119659774615421446109316763_f16;

    /// π/4
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_PI_4: f16 = 0.785398163397448309615660845819875721_f16;

    /// π/6
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_PI_6: f16 = 0.52359877559829887307710723054658381_f16;

    /// π/8
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_PI_8: f16 = 0.39269908169872415480783042290993786_f16;

    /// 1/π
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_1_PI: f16 = 0.318309886183790671537767526745028724_f16;

    /// 1/sqrt(π)
    #[unstable(feature = "f16", issue = "116909")]
    // Also, #[unstable(feature = "more_float_constants", issue = "146939")]
    pub const FRAC_1_SQRT_PI: f16 = 0.564189583547756286948079451560772586_f16;

    /// 1/sqrt(2π)
    #[doc(alias = "FRAC_1_SQRT_TAU")]
    #[unstable(feature = "f16", issue = "116909")]
    // Also, #[unstable(feature = "more_float_constants", issue = "146939")]
    pub const FRAC_1_SQRT_2PI: f16 = 0.398942280401432677939946059934381868_f16;

    /// 2/π
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_2_PI: f16 = 0.636619772367581343075535053490057448_f16;

    /// 2/sqrt(π)
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_2_SQRT_PI: f16 = 1.12837916709551257389615890312154517_f16;

    /// sqrt(2)
    #[unstable(feature = "f16", issue = "116909")]
    pub const SQRT_2: f16 = 1.41421356237309504880168872420969808_f16;

    /// 1/sqrt(2)
    #[unstable(feature = "f16", issue = "116909")]
    pub const FRAC_1_SQRT_2: f16 = 0.707106781186547524400844362104849039_f16;

    /// sqrt(3)
    #[unstable(feature = "f16", issue = "116909")]
    // Also, #[unstable(feature = "more_float_constants", issue = "146939")]
    pub const SQRT_3: f16 = 1.732050807568877293527446341505872367_f16;

    /// 1/sqrt(3)
    #[unstable(feature = "f16", issue = "116909")]
    // Also, #[unstable(feature = "more_float_constants", issue = "146939")]
    pub const FRAC_1_SQRT_3: f16 = 0.577350269189625764509148780501957456_f16;

    /// Euler's number (e)
    #[unstable(feature = "f16", issue = "116909")]
    pub const E: f16 = 2.71828182845904523536028747135266250_f16;

    /// log<sub>2</sub>(10)
    #[unstable(feature = "f16", issue = "116909")]
    pub const LOG2_10: f16 = 3.32192809488736234787031942948939018_f16;

    /// log<sub>2</sub>(e)
    #[unstable(feature = "f16", issue = "116909")]
    pub const LOG2_E: f16 = 1.44269504088896340735992468100189214_f16;

    /// log<sub>10</sub>(2)
    #[unstable(feature = "f16", issue = "116909")]
    pub const LOG10_2: f16 = 0.301029995663981195213738894724493027_f16;

    /// log<sub>10</sub>(e)
    #[unstable(feature = "f16", issue = "116909")]
    pub const LOG10_E: f16 = 0.434294481903251827651128918916605082_f16;

    /// ln(2)
    #[unstable(feature = "f16", issue = "116909")]
    pub const LN_2: f16 = 0.693147180559945309417232121458176568_f16;

    /// ln(10)
    #[unstable(feature = "f16", issue = "116909")]
    pub const LN_10: f16 = 2.30258509299404568401799145468436421_f16;
}

#[doc(test(attr(feature(cfg_target_has_reliable_f16_f128), allow(internal_features))))]
impl f16 {
    /// The radix or base of the internal representation of `f16`.
    #[unstable(feature = "f16", issue = "116909")]
    pub const RADIX: u32 = 2;

    /// Number of significant digits in base 2.
    ///
    /// Note that the size of the mantissa in the bitwise representation is one
    /// smaller than this since the leading 1 is not stored explicitly.
    #[unstable(feature = "f16", issue = "116909")]
    pub const MANTISSA_DIGITS: u32 = 11;

    /// Approximate number of significant digits in base 10.
    ///
    /// This is the maximum <i>x</i> such that any decimal number with <i>x</i>
    /// significant digits can be converted to `f16` and back without loss.
    ///
    /// Equal to floor(log<sub>10</sub>&nbsp;2<sup>[`MANTISSA_DIGITS`]&nbsp;&minus;&nbsp;1</sup>).
    ///
    /// [`MANTISSA_DIGITS`]: f16::MANTISSA_DIGITS
    #[unstable(feature = "f16", issue = "116909")]
    pub const DIGITS: u32 = 3;

    /// [Machine epsilon] value for `f16`.
    ///
    /// This is the difference between `1.0` and the next larger representable number.
    ///
    /// Equal to 2<sup>1&nbsp;&minus;&nbsp;[`MANTISSA_DIGITS`]</sup>.
    ///
    /// [Machine epsilon]: https://en.wikipedia.org/wiki/Machine_epsilon
    /// [`MANTISSA_DIGITS`]: f16::MANTISSA_DIGITS
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_diagnostic_item = "f16_epsilon"]
    pub const EPSILON: f16 = 9.7656e-4_f16;

    /// Smallest finite `f16` value.
    ///
    /// Equal to &minus;[`MAX`].
    ///
    /// [`MAX`]: f16::MAX
    #[unstable(feature = "f16", issue = "116909")]
    pub const MIN: f16 = -6.5504e+4_f16;
    /// Smallest positive normal `f16` value.
    ///
    /// Equal to 2<sup>[`MIN_EXP`]&nbsp;&minus;&nbsp;1</sup>.
    ///
    /// [`MIN_EXP`]: f16::MIN_EXP
    #[unstable(feature = "f16", issue = "116909")]
    pub const MIN_POSITIVE: f16 = 6.1035e-5_f16;
    /// Largest finite `f16` value.
    ///
    /// Equal to
    /// (1&nbsp;&minus;&nbsp;2<sup>&minus;[`MANTISSA_DIGITS`]</sup>)&nbsp;2<sup>[`MAX_EXP`]</sup>.
    ///
    /// [`MANTISSA_DIGITS`]: f16::MANTISSA_DIGITS
    /// [`MAX_EXP`]: f16::MAX_EXP
    #[unstable(feature = "f16", issue = "116909")]
    pub const MAX: f16 = 6.5504e+4_f16;

    /// One greater than the minimum possible *normal* power of 2 exponent
    /// for a significand bounded by 1 ≤ x < 2 (i.e. the IEEE definition).
    ///
    /// This corresponds to the exact minimum possible *normal* power of 2 exponent
    /// for a significand bounded by 0.5 ≤ x < 1 (i.e. the C definition).
    /// In other words, all normal numbers representable by this type are
    /// greater than or equal to 0.5&nbsp;×&nbsp;2<sup><i>MIN_EXP</i></sup>.
    #[unstable(feature = "f16", issue = "116909")]
    pub const MIN_EXP: i32 = -13;
    /// One greater than the maximum possible power of 2 exponent
    /// for a significand bounded by 1 ≤ x < 2 (i.e. the IEEE definition).
    ///
    /// This corresponds to the exact maximum possible power of 2 exponent
    /// for a significand bounded by 0.5 ≤ x < 1 (i.e. the C definition).
    /// In other words, all numbers representable by this type are
    /// strictly less than 2<sup><i>MAX_EXP</i></sup>.
    #[unstable(feature = "f16", issue = "116909")]
    pub const MAX_EXP: i32 = 16;

    /// Minimum <i>x</i> for which 10<sup><i>x</i></sup> is normal.
    ///
    /// Equal to ceil(log<sub>10</sub>&nbsp;[`MIN_POSITIVE`]).
    ///
    /// [`MIN_POSITIVE`]: f16::MIN_POSITIVE
    #[unstable(feature = "f16", issue = "116909")]
    pub const MIN_10_EXP: i32 = -4;
    /// Maximum <i>x</i> for which 10<sup><i>x</i></sup> is normal.
    ///
    /// Equal to floor(log<sub>10</sub>&nbsp;[`MAX`]).
    ///
    /// [`MAX`]: f16::MAX
    #[unstable(feature = "f16", issue = "116909")]
    pub const MAX_10_EXP: i32 = 4;

    /// Not a Number (NaN).
    ///
    /// Note that IEEE 754 doesn't define just a single NaN value; a plethora of bit patterns are
    /// considered to be NaN. Furthermore, the standard makes a difference between a "signaling" and
    /// a "quiet" NaN, and allows inspecting its "payload" (the unspecified bits in the bit pattern)
    /// and its sign. See the [specification of NaN bit patterns](f32#nan-bit-patterns) for more
    /// info.
    ///
    /// This constant is guaranteed to be a quiet NaN (on targets that follow the Rust assumptions
    /// that the quiet/signaling bit being set to 1 indicates a quiet NaN). Beyond that, nothing is
    /// guaranteed about the specific bit pattern chosen here: both payload and sign are arbitrary.
    /// The concrete bit pattern may change across Rust versions and target platforms.
    #[allow(clippy::eq_op)]
    #[rustc_diagnostic_item = "f16_nan"]
    #[unstable(feature = "f16", issue = "116909")]
    pub const NAN: f16 = 0.0_f16 / 0.0_f16;

    /// Infinity (∞).
    #[unstable(feature = "f16", issue = "116909")]
    pub const INFINITY: f16 = 1.0_f16 / 0.0_f16;

    /// Negative infinity (−∞).
    #[unstable(feature = "f16", issue = "116909")]
    pub const NEG_INFINITY: f16 = -1.0_f16 / 0.0_f16;

    /// Sign bit
    pub(crate) const SIGN_MASK: u16 = 0x8000;

    /// Exponent mask
    pub(crate) const EXP_MASK: u16 = 0x7c00;

    /// Mantissa mask
    pub(crate) const MAN_MASK: u16 = 0x03ff;

    /// Minimum representable positive value (min subnormal)
    const TINY_BITS: u16 = 0x1;

    /// Minimum representable negative value (min negative subnormal)
    const NEG_TINY_BITS: u16 = Self::TINY_BITS | Self::SIGN_MASK;

    /// Returns `true` if this value is NaN.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let nan = f16::NAN;
    /// let f = 7.0_f16;
    ///
    /// assert!(nan.is_nan());
    /// assert!(!f.is_nan());
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    #[allow(clippy::eq_op)] // > if you intended to check if the operand is NaN, use `.is_nan()` instead :)
    pub const fn is_nan(self) -> bool {
        self != self
    }

    /// Returns `true` if this value is positive infinity or negative infinity, and
    /// `false` otherwise.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 7.0f16;
    /// let inf = f16::INFINITY;
    /// let neg_inf = f16::NEG_INFINITY;
    /// let nan = f16::NAN;
    ///
    /// assert!(!f.is_infinite());
    /// assert!(!nan.is_infinite());
    ///
    /// assert!(inf.is_infinite());
    /// assert!(neg_inf.is_infinite());
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn is_infinite(self) -> bool {
        (self == f16::INFINITY) | (self == f16::NEG_INFINITY)
    }

    /// Returns `true` if this number is neither infinite nor NaN.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 7.0f16;
    /// let inf: f16 = f16::INFINITY;
    /// let neg_inf: f16 = f16::NEG_INFINITY;
    /// let nan: f16 = f16::NAN;
    ///
    /// assert!(f.is_finite());
    ///
    /// assert!(!nan.is_finite());
    /// assert!(!inf.is_finite());
    /// assert!(!neg_inf.is_finite());
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    pub const fn is_finite(self) -> bool {
        // There's no need to handle NaN separately: if self is NaN,
        // the comparison is not true, exactly as desired.
        self.abs() < Self::INFINITY
    }

    /// Returns `true` if the number is [subnormal].
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let min = f16::MIN_POSITIVE; // 6.1035e-5
    /// let max = f16::MAX;
    /// let lower_than_min = 1.0e-7_f16;
    /// let zero = 0.0_f16;
    ///
    /// assert!(!min.is_subnormal());
    /// assert!(!max.is_subnormal());
    ///
    /// assert!(!zero.is_subnormal());
    /// assert!(!f16::NAN.is_subnormal());
    /// assert!(!f16::INFINITY.is_subnormal());
    /// // Values between `0` and `min` are Subnormal.
    /// assert!(lower_than_min.is_subnormal());
    /// # }
    /// ```
    /// [subnormal]: https://en.wikipedia.org/wiki/Denormal_number
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn is_subnormal(self) -> bool {
        matches!(self.classify(), FpCategory::Subnormal)
    }

    /// Returns `true` if the number is neither zero, infinite, [subnormal], or NaN.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let min = f16::MIN_POSITIVE; // 6.1035e-5
    /// let max = f16::MAX;
    /// let lower_than_min = 1.0e-7_f16;
    /// let zero = 0.0_f16;
    ///
    /// assert!(min.is_normal());
    /// assert!(max.is_normal());
    ///
    /// assert!(!zero.is_normal());
    /// assert!(!f16::NAN.is_normal());
    /// assert!(!f16::INFINITY.is_normal());
    /// // Values between `0` and `min` are Subnormal.
    /// assert!(!lower_than_min.is_normal());
    /// # }
    /// ```
    /// [subnormal]: https://en.wikipedia.org/wiki/Denormal_number
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn is_normal(self) -> bool {
        matches!(self.classify(), FpCategory::Normal)
    }

    /// 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.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// use std::num::FpCategory;
    ///
    /// let num = 12.4_f16;
    /// let inf = f16::INFINITY;
    ///
    /// assert_eq!(num.classify(), FpCategory::Normal);
    /// assert_eq!(inf.classify(), FpCategory::Infinite);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn classify(self) -> FpCategory {
        let b = self.to_bits();
        match (b & Self::MAN_MASK, b & Self::EXP_MASK) {
            (0, Self::EXP_MASK) => FpCategory::Infinite,
            (_, Self::EXP_MASK) => FpCategory::Nan,
            (0, 0) => FpCategory::Zero,
            (_, 0) => FpCategory::Subnormal,
            _ => FpCategory::Normal,
        }
    }

    /// Returns `true` if `self` has a positive sign, including `+0.0`, NaNs with
    /// positive sign bit and positive infinity.
    ///
    /// Note that IEEE 754 doesn't assign any meaning to the sign bit in case of
    /// a NaN, and as Rust doesn't guarantee that the bit pattern of NaNs are
    /// conserved over arithmetic operations, the result of `is_sign_positive` on
    /// a NaN might produce an unexpected or non-portable result. See the [specification
    /// of NaN bit patterns](f32#nan-bit-patterns) for more info. Use `self.signum() == 1.0`
    /// if you need fully portable behavior (will return `false` for all NaNs).
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 7.0_f16;
    /// let g = -7.0_f16;
    ///
    /// assert!(f.is_sign_positive());
    /// assert!(!g.is_sign_positive());
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn is_sign_positive(self) -> bool {
        !self.is_sign_negative()
    }

    /// Returns `true` if `self` has a negative sign, including `-0.0`, NaNs with
    /// negative sign bit and negative infinity.
    ///
    /// Note that IEEE 754 doesn't assign any meaning to the sign bit in case of
    /// a NaN, and as Rust doesn't guarantee that the bit pattern of NaNs are
    /// conserved over arithmetic operations, the result of `is_sign_negative` on
    /// a NaN might produce an unexpected or non-portable result. See the [specification
    /// of NaN bit patterns](f32#nan-bit-patterns) for more info. Use `self.signum() == -1.0`
    /// if you need fully portable behavior (will return `false` for all NaNs).
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 7.0_f16;
    /// let g = -7.0_f16;
    ///
    /// assert!(!f.is_sign_negative());
    /// assert!(g.is_sign_negative());
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn is_sign_negative(self) -> bool {
        // IEEE754 says: isSignMinus(x) is true if and only if x has negative sign. isSignMinus
        // applies to zeros and NaNs as well.
        // SAFETY: This is just transmuting to get the sign bit, it's fine.
        (self.to_bits() & (1 << 15)) != 0
    }

    /// Returns the least number greater than `self`.
    ///
    /// Let `TINY` be the smallest representable positive `f16`. Then,
    ///  - if `self.is_nan()`, this returns `self`;
    ///  - if `self` is [`NEG_INFINITY`], this returns [`MIN`];
    ///  - if `self` is `-TINY`, this returns -0.0;
    ///  - if `self` is -0.0 or +0.0, this returns `TINY`;
    ///  - if `self` is [`MAX`] or [`INFINITY`], this returns [`INFINITY`];
    ///  - otherwise the unique least value greater than `self` is returned.
    ///
    /// The identity `x.next_up() == -(-x).next_down()` holds for all non-NaN `x`. When `x`
    /// is finite `x == x.next_up().next_down()` also holds.
    ///
    /// ```rust
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// // f16::EPSILON is the difference between 1.0 and the next number up.
    /// assert_eq!(1.0f16.next_up(), 1.0 + f16::EPSILON);
    /// // But not for most numbers.
    /// assert!(0.1f16.next_up() < 0.1 + f16::EPSILON);
    /// assert_eq!(4356f16.next_up(), 4360.0);
    /// # }
    /// ```
    ///
    /// This operation corresponds to IEEE-754 `nextUp`.
    ///
    /// [`NEG_INFINITY`]: Self::NEG_INFINITY
    /// [`INFINITY`]: Self::INFINITY
    /// [`MIN`]: Self::MIN
    /// [`MAX`]: Self::MAX
    #[inline]
    #[doc(alias = "nextUp")]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn next_up(self) -> Self {
        // Some targets violate Rust's assumption of IEEE semantics, e.g. by flushing
        // denormals to zero. This is in general unsound and unsupported, but here
        // we do our best to still produce the correct result on such targets.
        let bits = self.to_bits();
        if self.is_nan() || bits == Self::INFINITY.to_bits() {
            return self;
        }

        let abs = bits & !Self::SIGN_MASK;
        let next_bits = if abs == 0 {
            Self::TINY_BITS
        } else if bits == abs {
            bits + 1
        } else {
            bits - 1
        };
        Self::from_bits(next_bits)
    }

    /// Returns the greatest number less than `self`.
    ///
    /// Let `TINY` be the smallest representable positive `f16`. Then,
    ///  - if `self.is_nan()`, this returns `self`;
    ///  - if `self` is [`INFINITY`], this returns [`MAX`];
    ///  - if `self` is `TINY`, this returns 0.0;
    ///  - if `self` is -0.0 or +0.0, this returns `-TINY`;
    ///  - if `self` is [`MIN`] or [`NEG_INFINITY`], this returns [`NEG_INFINITY`];
    ///  - otherwise the unique greatest value less than `self` is returned.
    ///
    /// The identity `x.next_down() == -(-x).next_up()` holds for all non-NaN `x`. When `x`
    /// is finite `x == x.next_down().next_up()` also holds.
    ///
    /// ```rust
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 1.0f16;
    /// // Clamp value into range [0, 1).
    /// let clamped = x.clamp(0.0, 1.0f16.next_down());
    /// assert!(clamped < 1.0);
    /// assert_eq!(clamped.next_up(), 1.0);
    /// # }
    /// ```
    ///
    /// This operation corresponds to IEEE-754 `nextDown`.
    ///
    /// [`NEG_INFINITY`]: Self::NEG_INFINITY
    /// [`INFINITY`]: Self::INFINITY
    /// [`MIN`]: Self::MIN
    /// [`MAX`]: Self::MAX
    #[inline]
    #[doc(alias = "nextDown")]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn next_down(self) -> Self {
        // Some targets violate Rust's assumption of IEEE semantics, e.g. by flushing
        // denormals to zero. This is in general unsound and unsupported, but here
        // we do our best to still produce the correct result on such targets.
        let bits = self.to_bits();
        if self.is_nan() || bits == Self::NEG_INFINITY.to_bits() {
            return self;
        }

        let abs = bits & !Self::SIGN_MASK;
        let next_bits = if abs == 0 {
            Self::NEG_TINY_BITS
        } else if bits == abs {
            bits - 1
        } else {
            bits + 1
        };
        Self::from_bits(next_bits)
    }

    /// Takes the reciprocal (inverse) of a number, `1/x`.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 2.0_f16;
    /// let abs_difference = (x.recip() - (1.0 / x)).abs();
    ///
    /// assert!(abs_difference <= f16::EPSILON);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub const fn recip(self) -> Self {
        1.0 / self
    }

    /// Converts radians to degrees.
    ///
    /// # Unspecified precision
    ///
    /// The precision of this function is non-deterministic. This means it varies by platform,
    /// Rust version, and can even differ within the same execution from one invocation to the next.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let angle = std::f16::consts::PI;
    ///
    /// let abs_difference = (angle.to_degrees() - 180.0).abs();
    /// assert!(abs_difference <= 0.5);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub const fn to_degrees(self) -> Self {
        // Use a literal to avoid double rounding, consts::PI is already rounded,
        // and dividing would round again.
        const PIS_IN_180: f16 = 57.2957795130823208767981548141051703_f16;
        self * PIS_IN_180
    }

    /// Converts degrees to radians.
    ///
    /// # Unspecified precision
    ///
    /// The precision of this function is non-deterministic. This means it varies by platform,
    /// Rust version, and can even differ within the same execution from one invocation to the next.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let angle = 180.0f16;
    ///
    /// let abs_difference = (angle.to_radians() - std::f16::consts::PI).abs();
    ///
    /// assert!(abs_difference <= 0.01);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub const fn to_radians(self) -> f16 {
        // Use a literal to avoid double rounding, consts::PI is already rounded,
        // and dividing would round again.
        const RADS_PER_DEG: f16 = 0.017453292519943295769236907684886_f16;
        self * RADS_PER_DEG
    }

    /// Returns the maximum of the two numbers, ignoring NaN.
    ///
    /// If exactly one of the arguments is NaN (quiet or signaling), then the other argument is
    /// returned. If both arguments are NaN, the return value is NaN, with the bit pattern picked
    /// using the usual [rules for arithmetic operations](f32#nan-bit-patterns). If the inputs
    /// compare equal (such as for the case of `+0.0` and `-0.0`), either input may be returned
    /// non-deterministically.
    ///
    /// The handling of NaNs follows the IEEE 754-2019 semantics for `maximumNumber`, treating all
    /// NaNs the same way to ensure the operation is associative. The handling of signed zeros
    /// follows the IEEE 754-2008 semantics for `maxNum`.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 1.0f16;
    /// let y = 2.0f16;
    ///
    /// assert_eq!(x.max(y), y);
    /// assert_eq!(x.max(f16::NAN), x);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the comparison, without modifying either input"]
    pub const fn max(self, other: f16) -> f16 {
        intrinsics::maxnumf16(self, other)
    }

    /// Returns the minimum of the two numbers, ignoring NaN.
    ///
    /// If exactly one of the arguments is NaN (quiet or signaling), then the other argument is
    /// returned. If both arguments are NaN, the return value is NaN, with the bit pattern picked
    /// using the usual [rules for arithmetic operations](f32#nan-bit-patterns). If the inputs
    /// compare equal (such as for the case of `+0.0` and `-0.0`), either input may be returned
    /// non-deterministically.
    ///
    /// The handling of NaNs follows the IEEE 754-2019 semantics for `minimumNumber`, treating all
    /// NaNs the same way to ensure the operation is associative. The handling of signed zeros
    /// follows the IEEE 754-2008 semantics for `minNum`.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 1.0f16;
    /// let y = 2.0f16;
    ///
    /// assert_eq!(x.min(y), x);
    /// assert_eq!(x.min(f16::NAN), x);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the comparison, without modifying either input"]
    pub const fn min(self, other: f16) -> f16 {
        intrinsics::minnumf16(self, other)
    }

    /// Returns the maximum of the two numbers, propagating NaN.
    ///
    /// If at least one of the arguments is NaN, the return value is NaN, with the bit pattern
    /// picked using the usual [rules for arithmetic operations](f32#nan-bit-patterns). Furthermore,
    /// `-0.0` is considered to be less than `+0.0`, making this function fully deterministic for
    /// non-NaN inputs.
    ///
    /// This is in contrast to [`f16::max`] which only returns NaN when *both* arguments are NaN,
    /// and which does not reliably order `-0.0` and `+0.0`.
    ///
    /// This follows the IEEE 754-2019 semantics for `maximum`.
    ///
    /// ```
    /// #![feature(f16)]
    /// #![feature(float_minimum_maximum)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 1.0f16;
    /// let y = 2.0f16;
    ///
    /// assert_eq!(x.maximum(y), y);
    /// assert!(x.maximum(f16::NAN).is_nan());
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    // #[unstable(feature = "float_minimum_maximum", issue = "91079")]
    #[must_use = "this returns the result of the comparison, without modifying either input"]
    pub const fn maximum(self, other: f16) -> f16 {
        intrinsics::maximumf16(self, other)
    }

    /// Returns the minimum of the two numbers, propagating NaN.
    ///
    /// If at least one of the arguments is NaN, the return value is NaN, with the bit pattern
    /// picked using the usual [rules for arithmetic operations](f32#nan-bit-patterns). Furthermore,
    /// `-0.0` is considered to be less than `+0.0`, making this function fully deterministic for
    /// non-NaN inputs.
    ///
    /// This is in contrast to [`f16::min`] which only returns NaN when *both* arguments are NaN,
    /// and which does not reliably order `-0.0` and `+0.0`.
    ///
    /// This follows the IEEE 754-2019 semantics for `minimum`.
    ///
    /// ```
    /// #![feature(f16)]
    /// #![feature(float_minimum_maximum)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 1.0f16;
    /// let y = 2.0f16;
    ///
    /// assert_eq!(x.minimum(y), x);
    /// assert!(x.minimum(f16::NAN).is_nan());
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    // #[unstable(feature = "float_minimum_maximum", issue = "91079")]
    #[must_use = "this returns the result of the comparison, without modifying either input"]
    pub const fn minimum(self, other: f16) -> f16 {
        intrinsics::minimumf16(self, other)
    }

    /// Calculates the midpoint (average) between `self` and `rhs`.
    ///
    /// This returns NaN when *either* argument is NaN or if a combination of
    /// +inf and -inf is provided as arguments.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// assert_eq!(1f16.midpoint(4.0), 2.5);
    /// assert_eq!((-5.5f16).midpoint(8.0), 1.25);
    /// # }
    /// ```
    #[inline]
    #[doc(alias = "average")]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    pub const fn midpoint(self, other: f16) -> f16 {
        const HI: f16 = f16::MAX / 2.;

        let (a, b) = (self, other);
        let abs_a = a.abs();
        let abs_b = b.abs();

        if abs_a <= HI && abs_b <= HI {
            // Overflow is impossible
            (a + b) / 2.
        } else {
            (a / 2.) + (b / 2.)
        }
    }

    /// Rounds toward zero and converts to any primitive integer type,
    /// assuming that the value is finite and fits in that type.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let value = 4.6_f16;
    /// let rounded = unsafe { value.to_int_unchecked::<u16>() };
    /// assert_eq!(rounded, 4);
    ///
    /// let value = -128.9_f16;
    /// let rounded = unsafe { value.to_int_unchecked::<i8>() };
    /// assert_eq!(rounded, i8::MIN);
    /// # }
    /// ```
    ///
    /// # Safety
    ///
    /// The value must:
    ///
    /// * Not be `NaN`
    /// * Not be infinite
    /// * Be representable in the return type `Int`, after truncating off its fractional part
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub unsafe fn to_int_unchecked<Int>(self) -> Int
    where
        Self: FloatToInt<Int>,
    {
        // SAFETY: the caller must uphold the safety contract for
        // `FloatToInt::to_int_unchecked`.
        unsafe { FloatToInt::<Int>::to_int_unchecked(self) }
    }

    /// Raw transmutation to `u16`.
    ///
    /// This is currently identical to `transmute::<f16, u16>(self)` on all platforms.
    ///
    /// See [`from_bits`](#method.from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// Note that this function is distinct from `as` casting, which attempts to
    /// preserve the *numeric* value, and not the bitwise value.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// assert_ne!((1f16).to_bits(), 1f16 as u16); // to_bits() is not casting!
    /// assert_eq!((12.5f16).to_bits(), 0x4a40);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    #[allow(unnecessary_transmutes)]
    pub const fn to_bits(self) -> u16 {
        // SAFETY: `u16` is a plain old datatype so we can always transmute to it.
        unsafe { mem::transmute(self) }
    }

    /// Raw transmutation from `u16`.
    ///
    /// This is currently identical to `transmute::<u16, f16>(v)` on all platforms.
    /// It turns out this is incredibly portable, for two reasons:
    ///
    /// * Floats and Ints have the same endianness on all supported platforms.
    /// * IEEE 754 very precisely specifies the bit layout of floats.
    ///
    /// However there is one caveat: prior to the 2008 version of IEEE 754, how
    /// to interpret the NaN signaling bit wasn't actually specified. Most platforms
    /// (notably x86 and ARM) picked the interpretation that was ultimately
    /// standardized in 2008, but some didn't (notably MIPS). As a result, all
    /// signaling NaNs on MIPS are quiet NaNs on x86, and vice-versa.
    ///
    /// Rather than trying to preserve signaling-ness cross-platform, this
    /// implementation favors preserving the exact bits. This means that
    /// any payloads encoded in NaNs will be preserved even if the result of
    /// this method is sent over the network from an x86 machine to a MIPS one.
    ///
    /// If the results of this method are only manipulated by the same
    /// architecture that produced them, then there is no portability concern.
    ///
    /// If the input isn't NaN, then there is no portability concern.
    ///
    /// If you don't care about signalingness (very likely), then there is no
    /// portability concern.
    ///
    /// Note that this function is distinct from `as` casting, which attempts to
    /// preserve the *numeric* value, and not the bitwise value.
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let v = f16::from_bits(0x4a40);
    /// assert_eq!(v, 12.5);
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    #[allow(unnecessary_transmutes)]
    pub const fn from_bits(v: u16) -> Self {
        // It turns out the safety issues with sNaN were overblown! Hooray!
        // SAFETY: `u16` is a plain old datatype so we can always transmute from it.
        unsafe { mem::transmute(v) }
    }

    /// Returns the memory representation of this floating point number as a byte array in
    /// big-endian (network) byte order.
    ///
    /// See [`from_bits`](Self::from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let bytes = 12.5f16.to_be_bytes();
    /// assert_eq!(bytes, [0x4a, 0x40]);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub const fn to_be_bytes(self) -> [u8; 2] {
        self.to_bits().to_be_bytes()
    }

    /// Returns the memory representation of this floating point number as a byte array in
    /// little-endian byte order.
    ///
    /// See [`from_bits`](Self::from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let bytes = 12.5f16.to_le_bytes();
    /// assert_eq!(bytes, [0x40, 0x4a]);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub const fn to_le_bytes(self) -> [u8; 2] {
        self.to_bits().to_le_bytes()
    }

    /// Returns the memory representation of this floating point number as a byte array in
    /// native byte order.
    ///
    /// As the target platform's native endianness is used, portable code
    /// should use [`to_be_bytes`] or [`to_le_bytes`], as appropriate, instead.
    ///
    /// [`to_be_bytes`]: f16::to_be_bytes
    /// [`to_le_bytes`]: f16::to_le_bytes
    ///
    /// See [`from_bits`](Self::from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let bytes = 12.5f16.to_ne_bytes();
    /// assert_eq!(
    ///     bytes,
    ///     if cfg!(target_endian = "big") {
    ///         [0x4a, 0x40]
    ///     } else {
    ///         [0x40, 0x4a]
    ///     }
    /// );
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "this returns the result of the operation, without modifying the original"]
    pub const fn to_ne_bytes(self) -> [u8; 2] {
        self.to_bits().to_ne_bytes()
    }

    /// Creates a floating point value from its representation as a byte array in big endian.
    ///
    /// See [`from_bits`](Self::from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let value = f16::from_be_bytes([0x4a, 0x40]);
    /// assert_eq!(value, 12.5);
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn from_be_bytes(bytes: [u8; 2]) -> Self {
        Self::from_bits(u16::from_be_bytes(bytes))
    }

    /// Creates a floating point value from its representation as a byte array in little endian.
    ///
    /// See [`from_bits`](Self::from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let value = f16::from_le_bytes([0x40, 0x4a]);
    /// assert_eq!(value, 12.5);
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn from_le_bytes(bytes: [u8; 2]) -> Self {
        Self::from_bits(u16::from_le_bytes(bytes))
    }

    /// Creates a floating point value from its representation as a byte array in native endian.
    ///
    /// As the target platform's native endianness is used, portable code
    /// likely wants to use [`from_be_bytes`] or [`from_le_bytes`], as
    /// appropriate instead.
    ///
    /// [`from_be_bytes`]: f16::from_be_bytes
    /// [`from_le_bytes`]: f16::from_le_bytes
    ///
    /// See [`from_bits`](Self::from_bits) for some discussion of the
    /// portability of this operation (there are almost no issues).
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let value = f16::from_ne_bytes(if cfg!(target_endian = "big") {
    ///     [0x4a, 0x40]
    /// } else {
    ///     [0x40, 0x4a]
    /// });
    /// assert_eq!(value, 12.5);
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    pub const fn from_ne_bytes(bytes: [u8; 2]) -> Self {
        Self::from_bits(u16::from_ne_bytes(bytes))
    }

    /// Returns 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 of `f16`. 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.
    ///
    /// # Example
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// struct GoodBoy {
    ///     name: &'static str,
    ///     weight: f16,
    /// }
    ///
    /// let mut bois = vec![
    ///     GoodBoy { name: "Pucci", weight: 0.1 },
    ///     GoodBoy { name: "Woofer", weight: 99.0 },
    ///     GoodBoy { name: "Yapper", weight: 10.0 },
    ///     GoodBoy { name: "Chonk", weight: f16::INFINITY },
    ///     GoodBoy { name: "Abs. Unit", weight: f16::NAN },
    ///     GoodBoy { name: "Floaty", weight: -5.0 },
    /// ];
    ///
    /// bois.sort_by(|a, b| a.weight.total_cmp(&b.weight));
    ///
    /// // `f16::NAN` could be positive or negative, which will affect the sort order.
    /// if f16::NAN.is_sign_negative() {
    ///     bois.into_iter().map(|b| b.weight)
    ///         .zip([f16::NAN, -5.0, 0.1, 10.0, 99.0, f16::INFINITY].iter())
    ///         .for_each(|(a, b)| assert_eq!(a.to_bits(), b.to_bits()))
    /// } else {
    ///     bois.into_iter().map(|b| b.weight)
    ///         .zip([-5.0, 0.1, 10.0, 99.0, f16::INFINITY, f16::NAN].iter())
    ///         .for_each(|(a, b)| assert_eq!(a.to_bits(), b.to_bits()))
    /// }
    /// # }
    /// ```
    #[inline]
    #[must_use]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "const_cmp", issue = "143800")]
    pub const fn total_cmp(&self, other: &Self) -> crate::cmp::Ordering {
        let mut left = self.to_bits() as i16;
        let mut right = other.to_bits() as i16;

        // In case of negatives, flip all the bits except the sign
        // to achieve a similar layout as two's complement integers
        //
        // Why does this work? IEEE 754 floats consist of three fields:
        // Sign bit, exponent and mantissa. The set of exponent and mantissa
        // fields as a whole have the property that their bitwise order is
        // equal to the numeric magnitude where the magnitude is defined.
        // The magnitude is not normally defined on NaN values, but
        // IEEE 754 totalOrder defines the NaN values also to follow the
        // bitwise order. This leads to order explained in the doc comment.
        // However, the representation of magnitude is the same for negative
        // and positive numbers – only the sign bit is different.
        // To easily compare the floats as signed integers, we need to
        // flip the exponent and mantissa bits in case of negative numbers.
        // We effectively convert the numbers to "two's complement" form.
        //
        // To do the flipping, we construct a mask and XOR against it.
        // We branchlessly calculate an "all-ones except for the sign bit"
        // mask from negative-signed values: right shifting sign-extends
        // the integer, so we "fill" the mask with sign bits, and then
        // convert to unsigned to push one more zero bit.
        // On positive values, the mask is all zeros, so it's a no-op.
        left ^= (((left >> 15) as u16) >> 1) as i16;
        right ^= (((right >> 15) as u16) >> 1) as i16;

        left.cmp(&right)
    }

    /// Restrict a value to a certain interval unless it is NaN.
    ///
    /// Returns `max` if `self` is greater than `max`, and `min` if `self` is
    /// less than `min`. Otherwise this returns `self`.
    ///
    /// Note that this function returns NaN if the initial value was NaN as
    /// well. If the result is zero and among the three inputs `self`, `min`, and `max` there are
    /// zeros with different sign, either `0.0` or `-0.0` is returned non-deterministically.
    ///
    /// # Panics
    ///
    /// Panics if `min > max`, `min` is NaN, or `max` is NaN.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// assert!((-3.0f16).clamp(-2.0, 1.0) == -2.0);
    /// assert!((0.0f16).clamp(-2.0, 1.0) == 0.0);
    /// assert!((2.0f16).clamp(-2.0, 1.0) == 1.0);
    /// assert!((f16::NAN).clamp(-2.0, 1.0).is_nan());
    ///
    /// // These always returns zero, but the sign (which is ignored by `==`) is non-deterministic.
    /// assert!((0.0f16).clamp(-0.0, -0.0) == 0.0);
    /// assert!((1.0f16).clamp(-0.0, 0.0) == 0.0);
    /// // This is definitely a negative zero.
    /// assert!((-1.0f16).clamp(-0.0, 1.0).is_sign_negative());
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn clamp(mut self, min: f16, max: f16) -> f16 {
        const_assert!(
            min <= max,
            "min > max, or either was NaN",
            "min > max, or either was NaN. min = {min:?}, max = {max:?}",
            min: f16,
            max: f16,
        );

        if self < min {
            self = min;
        }
        if self > max {
            self = max;
        }
        self
    }

    /// Clamps this number to a symmetric range centered around zero.
    ///
    /// The method clamps the number's magnitude (absolute value) to be at most `limit`.
    ///
    /// This is functionally equivalent to `self.clamp(-limit, limit)`, but is more
    /// explicit about the intent.
    ///
    /// # Panics
    ///
    /// Panics if `limit` is negative or NaN, as this indicates a logic error.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// #![feature(clamp_magnitude)]
    /// # #[cfg(target_has_reliable_f16)] {
    /// assert_eq!(5.0f16.clamp_magnitude(3.0), 3.0);
    /// assert_eq!((-5.0f16).clamp_magnitude(3.0), -3.0);
    /// assert_eq!(2.0f16.clamp_magnitude(3.0), 2.0);
    /// assert_eq!((-2.0f16).clamp_magnitude(3.0), -2.0);
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "clamp_magnitude", issue = "148519")]
    #[must_use = "this returns the clamped value and does not modify the original"]
    pub fn clamp_magnitude(self, limit: f16) -> f16 {
        assert!(limit >= 0.0, "limit must be non-negative");
        let limit = limit.abs(); // Canonicalises -0.0 to 0.0
        self.clamp(-limit, limit)
    }

    /// Computes the absolute value of `self`.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16_math)] {
    ///
    /// let x = 3.5_f16;
    /// let y = -3.5_f16;
    ///
    /// assert_eq!(x.abs(), x);
    /// assert_eq!(y.abs(), -y);
    ///
    /// assert!(f16::NAN.abs().is_nan());
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn abs(self) -> Self {
        intrinsics::fabsf16(self)
    }

    /// Returns a number that represents the sign of `self`.
    ///
    /// - `1.0` if the number is positive, `+0.0` or `INFINITY`
    /// - `-1.0` if the number is negative, `-0.0` or `NEG_INFINITY`
    /// - NaN if the number is NaN
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 3.5_f16;
    ///
    /// assert_eq!(f.signum(), 1.0);
    /// assert_eq!(f16::NEG_INFINITY.signum(), -1.0);
    ///
    /// assert!(f16::NAN.signum().is_nan());
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn signum(self) -> f16 {
        if self.is_nan() { Self::NAN } else { 1.0_f16.copysign(self) }
    }

    /// 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 same payload as `self` and the sign bit of `sign` is
    /// returned.
    ///
    /// If `sign` is a NaN, then this operation will still carry over its sign into the result. Note
    /// that IEEE 754 doesn't assign any meaning to the sign bit in case of a NaN, and as Rust
    /// doesn't guarantee that the bit pattern of NaNs are conserved over arithmetic operations, the
    /// result of `copysign` with `sign` being a NaN might produce an unexpected or non-portable
    /// result. See the [specification of NaN bit patterns](primitive@f32#nan-bit-patterns) for more
    /// info.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(target_has_reliable_f16_math)] {
    ///
    /// let f = 3.5_f16;
    ///
    /// assert_eq!(f.copysign(0.42), 3.5_f16);
    /// assert_eq!(f.copysign(-0.42), -3.5_f16);
    /// assert_eq!((-f).copysign(0.42), 3.5_f16);
    /// assert_eq!((-f).copysign(-0.42), -3.5_f16);
    ///
    /// assert!(f16::NAN.copysign(1.0).is_nan());
    /// # }
    /// ```
    #[inline]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn copysign(self, sign: f16) -> f16 {
        intrinsics::copysignf16(self, sign)
    }

    /// Float addition that allows optimizations based on algebraic rules.
    ///
    /// See [algebraic operators](primitive@f32#algebraic-operators) for more info.
    #[must_use = "method returns a new number and does not mutate the original value"]
    #[unstable(feature = "float_algebraic", issue = "136469")]
    #[rustc_const_unstable(feature = "float_algebraic", issue = "136469")]
    #[inline]
    pub const fn algebraic_add(self, rhs: f16) -> f16 {
        intrinsics::fadd_algebraic(self, rhs)
    }

    /// Float subtraction that allows optimizations based on algebraic rules.
    ///
    /// See [algebraic operators](primitive@f32#algebraic-operators) for more info.
    #[must_use = "method returns a new number and does not mutate the original value"]
    #[unstable(feature = "float_algebraic", issue = "136469")]
    #[rustc_const_unstable(feature = "float_algebraic", issue = "136469")]
    #[inline]
    pub const fn algebraic_sub(self, rhs: f16) -> f16 {
        intrinsics::fsub_algebraic(self, rhs)
    }

    /// Float multiplication that allows optimizations based on algebraic rules.
    ///
    /// See [algebraic operators](primitive@f32#algebraic-operators) for more info.
    #[must_use = "method returns a new number and does not mutate the original value"]
    #[unstable(feature = "float_algebraic", issue = "136469")]
    #[rustc_const_unstable(feature = "float_algebraic", issue = "136469")]
    #[inline]
    pub const fn algebraic_mul(self, rhs: f16) -> f16 {
        intrinsics::fmul_algebraic(self, rhs)
    }

    /// Float division that allows optimizations based on algebraic rules.
    ///
    /// See [algebraic operators](primitive@f32#algebraic-operators) for more info.
    #[must_use = "method returns a new number and does not mutate the original value"]
    #[unstable(feature = "float_algebraic", issue = "136469")]
    #[rustc_const_unstable(feature = "float_algebraic", issue = "136469")]
    #[inline]
    pub const fn algebraic_div(self, rhs: f16) -> f16 {
        intrinsics::fdiv_algebraic(self, rhs)
    }

    /// Float remainder that allows optimizations based on algebraic rules.
    ///
    /// See [algebraic operators](primitive@f32#algebraic-operators) for more info.
    #[must_use = "method returns a new number and does not mutate the original value"]
    #[unstable(feature = "float_algebraic", issue = "136469")]
    #[rustc_const_unstable(feature = "float_algebraic", issue = "136469")]
    #[inline]
    pub const fn algebraic_rem(self, rhs: f16) -> f16 {
        intrinsics::frem_algebraic(self, rhs)
    }
}

// Functions in this module fall into `core_float_math`
// #[unstable(feature = "core_float_math", issue = "137578")]
#[cfg(not(test))]
#[doc(test(attr(feature(cfg_target_has_reliable_f16_f128), expect(internal_features))))]
impl f16 {
    /// Returns the largest integer less than or equal to `self`.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 3.7_f16;
    /// let g = 3.0_f16;
    /// let h = -3.7_f16;
    ///
    /// assert_eq!(f.floor(), 3.0);
    /// assert_eq!(g.floor(), 3.0);
    /// assert_eq!(h.floor(), -4.0);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn floor(self) -> f16 {
        intrinsics::floorf16(self)
    }

    /// Returns the smallest integer greater than or equal to `self`.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 3.01_f16;
    /// let g = 4.0_f16;
    ///
    /// assert_eq!(f.ceil(), 4.0);
    /// assert_eq!(g.ceil(), 4.0);
    /// # }
    /// ```
    #[inline]
    #[doc(alias = "ceiling")]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn ceil(self) -> f16 {
        intrinsics::ceilf16(self)
    }

    /// Returns the nearest integer to `self`. If a value is half-way between two
    /// integers, round away from `0.0`.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 3.3_f16;
    /// let g = -3.3_f16;
    /// let h = -3.7_f16;
    /// let i = 3.5_f16;
    /// let j = 4.5_f16;
    ///
    /// assert_eq!(f.round(), 3.0);
    /// assert_eq!(g.round(), -3.0);
    /// assert_eq!(h.round(), -4.0);
    /// assert_eq!(i.round(), 4.0);
    /// assert_eq!(j.round(), 5.0);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn round(self) -> f16 {
        intrinsics::roundf16(self)
    }

    /// Returns the nearest integer to a number. Rounds half-way cases to the number
    /// with an even least significant digit.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 3.3_f16;
    /// let g = -3.3_f16;
    /// let h = 3.5_f16;
    /// let i = 4.5_f16;
    ///
    /// assert_eq!(f.round_ties_even(), 3.0);
    /// assert_eq!(g.round_ties_even(), -3.0);
    /// assert_eq!(h.round_ties_even(), 4.0);
    /// assert_eq!(i.round_ties_even(), 4.0);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn round_ties_even(self) -> f16 {
        intrinsics::round_ties_even_f16(self)
    }

    /// Returns the integer part of `self`.
    /// This means that non-integer numbers are always truncated towards zero.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let f = 3.7_f16;
    /// let g = 3.0_f16;
    /// let h = -3.7_f16;
    ///
    /// assert_eq!(f.trunc(), 3.0);
    /// assert_eq!(g.trunc(), 3.0);
    /// assert_eq!(h.trunc(), -3.0);
    /// # }
    /// ```
    #[inline]
    #[doc(alias = "truncate")]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn trunc(self) -> f16 {
        intrinsics::truncf16(self)
    }

    /// Returns the fractional part of `self`.
    ///
    /// This function always returns the precise result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 3.6_f16;
    /// let y = -3.6_f16;
    /// let abs_difference_x = (x.fract() - 0.6).abs();
    /// let abs_difference_y = (y.fract() - (-0.6)).abs();
    ///
    /// assert!(abs_difference_x <= f16::EPSILON);
    /// assert!(abs_difference_y <= f16::EPSILON);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[rustc_const_unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn fract(self) -> f16 {
        self - self.trunc()
    }

    /// 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` *may* be more performant than an unfused multiply-add if
    /// the target architecture has a dedicated `fma` CPU instruction. However,
    /// this is not always true, and will be heavily dependant on designing
    /// algorithms with specific target hardware in mind.
    ///
    /// # Precision
    ///
    /// The result of this operation is guaranteed to be the rounded
    /// infinite-precision result. It is specified by IEEE 754 as
    /// `fusedMultiplyAdd` and guaranteed not to change.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let m = 10.0_f16;
    /// let x = 4.0_f16;
    /// let b = 60.0_f16;
    ///
    /// assert_eq!(m.mul_add(x, b), 100.0);
    /// assert_eq!(m * x + b, 100.0);
    ///
    /// let one_plus_eps = 1.0_f16 + f16::EPSILON;
    /// let one_minus_eps = 1.0_f16 - f16::EPSILON;
    /// let minus_one = -1.0_f16;
    ///
    /// // The exact result (1 + eps) * (1 - eps) = 1 - eps * eps.
    /// assert_eq!(one_plus_eps.mul_add(one_minus_eps, minus_one), -f16::EPSILON * f16::EPSILON);
    /// // Different rounding with the non-fused multiply and add.
    /// assert_eq!(one_plus_eps * one_minus_eps + minus_one, 0.0);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[doc(alias = "fmaf16", alias = "fusedMultiplyAdd")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub const fn mul_add(self, a: f16, b: f16) -> f16 {
        intrinsics::fmaf16(self, a, b)
    }

    /// Calculates Euclidean division, the matching method for `rem_euclid`.
    ///
    /// This computes the integer `n` such that
    /// `self = n * rhs + self.rem_euclid(rhs)`.
    /// In other words, the result is `self / rhs` rounded to the integer `n`
    /// such that `self >= n * rhs`.
    ///
    /// # Precision
    ///
    /// The result of this operation is guaranteed to be the rounded
    /// infinite-precision result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let a: f16 = 7.0;
    /// let b = 4.0;
    /// assert_eq!(a.div_euclid(b), 1.0); // 7.0 > 4.0 * 1.0
    /// assert_eq!((-a).div_euclid(b), -2.0); // -7.0 >= 4.0 * -2.0
    /// assert_eq!(a.div_euclid(-b), -1.0); // 7.0 >= -4.0 * -1.0
    /// assert_eq!((-a).div_euclid(-b), 2.0); // -7.0 >= -4.0 * 2.0
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub fn div_euclid(self, rhs: f16) -> f16 {
        let q = (self / rhs).trunc();
        if self % rhs < 0.0 {
            return if rhs > 0.0 { q - 1.0 } else { q + 1.0 };
        }
        q
    }

    /// Calculates the least nonnegative remainder of `self` when
    /// divided by `rhs`.
    ///
    /// In particular, the return value `r` satisfies `0.0 <= r < rhs.abs()` in
    /// most cases. However, due to a floating point round-off error it can
    /// result in `r == rhs.abs()`, violating the mathematical definition, if
    /// `self` is much smaller than `rhs.abs()` in magnitude and `self < 0.0`.
    /// This result is not an element of the function's codomain, but it is the
    /// closest floating point number in the real numbers and thus fulfills the
    /// property `self == self.div_euclid(rhs) * rhs + self.rem_euclid(rhs)`
    /// approximately.
    ///
    /// # Precision
    ///
    /// The result of this operation is guaranteed to be the rounded
    /// infinite-precision result.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let a: f16 = 7.0;
    /// let b = 4.0;
    /// assert_eq!(a.rem_euclid(b), 3.0);
    /// assert_eq!((-a).rem_euclid(b), 1.0);
    /// assert_eq!(a.rem_euclid(-b), 3.0);
    /// assert_eq!((-a).rem_euclid(-b), 1.0);
    /// // limitation due to round-off error
    /// assert!((-f16::EPSILON).rem_euclid(3.0) != 0.0);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[doc(alias = "modulo", alias = "mod")]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub fn rem_euclid(self, rhs: f16) -> f16 {
        let r = self % rhs;
        if r < 0.0 { r + rhs.abs() } else { r }
    }

    /// Raises a number to an integer power.
    ///
    /// Using this function is generally faster than using `powf`.
    /// It might have a different sequence of rounding operations than `powf`,
    /// so the results are not guaranteed to agree.
    ///
    /// Note that this function is special in that it can return non-NaN results for NaN inputs. For
    /// example, `f16::powi(f16::NAN, 0)` returns `1.0`. However, if an input is a *signaling*
    /// NaN, then the result is non-deterministically either a NaN or the result that the
    /// corresponding quiet NaN would produce.
    ///
    /// # Unspecified precision
    ///
    /// The precision of this function is non-deterministic. This means it varies by platform,
    /// Rust version, and can even differ within the same execution from one invocation to the next.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 2.0_f16;
    /// let abs_difference = (x.powi(2) - (x * x)).abs();
    /// assert!(abs_difference <= f16::EPSILON);
    ///
    /// assert_eq!(f16::powi(f16::NAN, 0), 1.0);
    /// assert_eq!(f16::powi(0.0, 0), 1.0);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub fn powi(self, n: i32) -> f16 {
        intrinsics::powif16(self, n)
    }

    /// Returns the square root of a number.
    ///
    /// Returns NaN if `self` is a negative number other than `-0.0`.
    ///
    /// # Precision
    ///
    /// The result of this operation is guaranteed to be the rounded
    /// infinite-precision result. It is specified by IEEE 754 as `squareRoot`
    /// and guaranteed not to change.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let positive = 4.0_f16;
    /// let negative = -4.0_f16;
    /// let negative_zero = -0.0_f16;
    ///
    /// assert_eq!(positive.sqrt(), 2.0);
    /// assert!(negative.sqrt().is_nan());
    /// assert!(negative_zero.sqrt() == negative_zero);
    /// # }
    /// ```
    #[inline]
    #[doc(alias = "squareRoot")]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub fn sqrt(self) -> f16 {
        intrinsics::sqrtf16(self)
    }

    /// Returns the cube root of a number.
    ///
    /// # Unspecified precision
    ///
    /// The precision of this function is non-deterministic. This means it varies by platform,
    /// Rust version, and can even differ within the same execution from one invocation to the next.
    ///
    /// This function currently corresponds to the `cbrtf` from libc on Unix
    /// and Windows. Note that this might change in the future.
    ///
    /// # Examples
    ///
    /// ```
    /// #![feature(f16)]
    /// # #[cfg(not(miri))]
    /// # #[cfg(target_has_reliable_f16)] {
    ///
    /// let x = 8.0f16;
    ///
    /// // x^(1/3) - 2 == 0
    /// let abs_difference = (x.cbrt() - 2.0).abs();
    ///
    /// assert!(abs_difference <= f16::EPSILON);
    /// # }
    /// ```
    #[inline]
    #[rustc_allow_incoherent_impl]
    #[unstable(feature = "f16", issue = "116909")]
    #[must_use = "method returns a new number and does not mutate the original value"]
    pub fn cbrt(self) -> f16 {
        libm::cbrtf(self as f32) as f16
    }
}