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    //! This module provides constants which are specific to the implementation
    //! of the `f64` floating point data type.
    //!
    //! *[See also the `f64` primitive type](../../std/primitive.f64.html).*
    //!
    //! Mathematically significant numbers are provided in the `consts` sub-module.

    #![stable(feature = "rust1", since = "1.0.0")]

    use mem;
    use num::FpCategory;

    /// The radix or base of the internal representation of `f64`.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const RADIX: u32 = 2;

    /// Number of significant digits in base 2.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MANTISSA_DIGITS: u32 = 53;
    /// Approximate number of significant digits in base 10.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const DIGITS: u32 = 15;

    /// [Machine epsilon] value for `f64`.
    ///
    /// This is the difference between `1.0` and the next largest representable number.
    ///
    /// [Machine epsilon]: https://en.wikipedia.org/wiki/Machine_epsilon
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const EPSILON: f64 = 2.2204460492503131e-16_f64;

    /// Smallest finite `f64` value.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MIN: f64 = -1.7976931348623157e+308_f64;
    /// Smallest positive normal `f64` value.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MIN_POSITIVE: f64 = 2.2250738585072014e-308_f64;
    /// Largest finite `f64` value.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MAX: f64 = 1.7976931348623157e+308_f64;

    /// One greater than the minimum possible normal power of 2 exponent.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MIN_EXP: i32 = -1021;
    /// Maximum possible power of 2 exponent.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MAX_EXP: i32 = 1024;

    /// Minimum possible normal power of 10 exponent.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MIN_10_EXP: i32 = -307;
    /// Maximum possible power of 10 exponent.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const MAX_10_EXP: i32 = 308;

    /// Not a Number (NaN).
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const NAN: f64 = 0.0_f64 / 0.0_f64;
    /// Infinity (∞).
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const INFINITY: f64 = 1.0_f64 / 0.0_f64;
    /// Negative infinity (-∞).
    #[stable(feature = "rust1", since = "1.0.0")]
    pub const NEG_INFINITY: f64 = -1.0_f64 / 0.0_f64;

    /// Basic mathematical constants.
    #[stable(feature = "rust1", since = "1.0.0")]
    pub mod consts {
        // FIXME: replace with mathematical constants from cmath.

        /// Archimedes' constant (π)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const PI: f64 = 3.14159265358979323846264338327950288_f64;

        /// π/2
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_PI_2: f64 = 1.57079632679489661923132169163975144_f64;

        /// π/3
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_PI_3: f64 = 1.04719755119659774615421446109316763_f64;

        /// π/4
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_PI_4: f64 = 0.785398163397448309615660845819875721_f64;

        /// π/6
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_PI_6: f64 = 0.52359877559829887307710723054658381_f64;

        /// π/8
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_PI_8: f64 = 0.39269908169872415480783042290993786_f64;

        /// 1/π
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_1_PI: f64 = 0.318309886183790671537767526745028724_f64;

        /// 2/π
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_2_PI: f64 = 0.636619772367581343075535053490057448_f64;

        /// 2/sqrt(π)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_2_SQRT_PI: f64 = 1.12837916709551257389615890312154517_f64;

        /// sqrt(2)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const SQRT_2: f64 = 1.41421356237309504880168872420969808_f64;

        /// 1/sqrt(2)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const FRAC_1_SQRT_2: f64 = 0.707106781186547524400844362104849039_f64;

        /// Euler's number (e)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const E: f64 = 2.71828182845904523536028747135266250_f64;

        /// log<sub>2</sub>(10)
        #[unstable(feature = "extra_log_consts", issue = "50540")]
        pub const LOG2_10: f64 = 3.32192809488736234787031942948939018_f64;

        /// log<sub>2</sub>(e)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const LOG2_E: f64 = 1.44269504088896340735992468100189214_f64;

        /// log<sub>10</sub>(2)
        #[unstable(feature = "extra_log_consts", issue = "50540")]
        pub const LOG10_2: f64 = 0.301029995663981195213738894724493027_f64;

        /// log<sub>10</sub>(e)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const LOG10_E: f64 = 0.434294481903251827651128918916605082_f64;

        /// ln(2)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const LN_2: f64 = 0.693147180559945309417232121458176568_f64;

        /// ln(10)
        #[stable(feature = "rust1", since = "1.0.0")]
        pub const LN_10: f64 = 2.30258509299404568401799145468436421_f64;
    }

    #[lang = "f64"]
    #[cfg(not(test))]
    impl f64 {
        /// Returns `true` if this value is `NaN` and false otherwise.
        ///
        /// ```
        /// use std::f64;
        ///
        /// let nan = f64::NAN;
        /// let f = 7.0_f64;
        ///
        /// assert!(nan.is_nan());
        /// assert!(!f.is_nan());
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn is_nan(self) -> bool {
            self != self
        }

        // FIXME(#50145): `abs` is publicly unavailable in libcore due to
        // concerns about portability, so this implementation is for
        // private use internally.
        #[inline]
        fn abs_private(self) -> f64 {
            f64::from_bits(self.to_bits() & 0x7fff_ffff_ffff_ffff)
        }

        /// Returns `true` if this value is positive infinity or negative infinity and
        /// false otherwise.
        ///
        /// ```
        /// use std::f64;
        ///
        /// let f = 7.0f64;
        /// let inf = f64::INFINITY;
        /// let neg_inf = f64::NEG_INFINITY;
        /// let nan = f64::NAN;
        ///
        /// assert!(!f.is_infinite());
        /// assert!(!nan.is_infinite());
        ///
        /// assert!(inf.is_infinite());
        /// assert!(neg_inf.is_infinite());
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn is_infinite(self) -> bool {
            self.abs_private() == INFINITY
        }

        /// Returns `true` if this number is neither infinite nor `NaN`.
        ///
        /// ```
        /// use std::f64;
        ///
        /// let f = 7.0f64;
        /// let inf: f64 = f64::INFINITY;
        /// let neg_inf: f64 = f64::NEG_INFINITY;
        /// let nan: f64 = f64::NAN;
        ///
        /// assert!(f.is_finite());
        ///
        /// assert!(!nan.is_finite());
        /// assert!(!inf.is_finite());
        /// assert!(!neg_inf.is_finite());
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub 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_private() < INFINITY
        }

        /// Returns `true` if the number is neither zero, infinite,
        /// [subnormal][subnormal], or `NaN`.
        ///
        /// ```
        /// use std::f64;
        ///
        /// let min = f64::MIN_POSITIVE; // 2.2250738585072014e-308f64
        /// let max = f64::MAX;
        /// let lower_than_min = 1.0e-308_f64;
        /// let zero = 0.0f64;
        ///
        /// assert!(min.is_normal());
        /// assert!(max.is_normal());
        ///
        /// assert!(!zero.is_normal());
        /// assert!(!f64::NAN.is_normal());
        /// assert!(!f64::INFINITY.is_normal());
        /// // Values between `0` and `min` are Subnormal.
        /// assert!(!lower_than_min.is_normal());
        /// ```
        /// [subnormal]: https://en.wikipedia.org/wiki/Denormal_number
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn is_normal(self) -> bool {
            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.
        ///
        /// ```
        /// use std::num::FpCategory;
        /// use std::f64;
        ///
        /// let num = 12.4_f64;
        /// let inf = f64::INFINITY;
        ///
        /// assert_eq!(num.classify(), FpCategory::Normal);
        /// assert_eq!(inf.classify(), FpCategory::Infinite);
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        pub fn classify(self) -> FpCategory {
            const EXP_MASK: u64 = 0x7ff0000000000000;
            const MAN_MASK: u64 = 0x000fffffffffffff;

            let bits = self.to_bits();
            match (bits & MAN_MASK, bits & EXP_MASK) {
                (0, 0) => FpCategory::Zero,
                (_, 0) => FpCategory::Subnormal,
                (0, EXP_MASK) => FpCategory::Infinite,
                (_, EXP_MASK) => FpCategory::Nan,
                _ => FpCategory::Normal,
            }
        }

        /// Returns `true` if and only if `self` has a positive sign, including `+0.0`, `NaN`s with
        /// positive sign bit and positive infinity.
        ///
        /// ```
        /// let f = 7.0_f64;
        /// let g = -7.0_f64;
        ///
        /// assert!(f.is_sign_positive());
        /// assert!(!g.is_sign_positive());
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn is_sign_positive(self) -> bool {
            !self.is_sign_negative()
        }

        #[stable(feature = "rust1", since = "1.0.0")]
        #[rustc_deprecated(since = "1.0.0", reason = "renamed to is_sign_positive")]
        #[inline]
        #[doc(hidden)]
        pub fn is_positive(self) -> bool {
            self.is_sign_positive()
        }

        /// Returns `true` if and only if `self` has a negative sign, including `-0.0`, `NaN`s with
        /// negative sign bit and negative infinity.
        ///
        /// ```
        /// let f = 7.0_f64;
        /// let g = -7.0_f64;
        ///
        /// assert!(!f.is_sign_negative());
        /// assert!(g.is_sign_negative());
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn is_sign_negative(self) -> bool {
            self.to_bits() & 0x8000_0000_0000_0000 != 0
        }

        #[stable(feature = "rust1", since = "1.0.0")]
        #[rustc_deprecated(since = "1.0.0", reason = "renamed to is_sign_negative")]
        #[inline]
        #[doc(hidden)]
        pub fn is_negative(self) -> bool {
            self.is_sign_negative()
        }

        /// Takes the reciprocal (inverse) of a number, `1/x`.
        ///
        /// ```
        /// let x = 2.0_f64;
        /// let abs_difference = (x.recip() - (1.0/x)).abs();
        ///
        /// assert!(abs_difference < 1e-10);
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn recip(self) -> f64 {
            1.0 / 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);
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn to_degrees(self) -> f64 {
            // The division here is correctly rounded with respect to the true
            // value of 180/π. (This differs from f32, where a constant must be
            // used to ensure a correctly rounded result.)
            self * (180.0f64 / consts::PI)
        }

        /// 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);
        /// ```
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn to_radians(self) -> f64 {
            let value: f64 = consts::PI;
            self * (value / 180.0)
        }

        /// Returns the maximum of the two numbers.
        ///
        /// ```
        /// let x = 1.0_f64;
        /// let y = 2.0_f64;
        ///
        /// assert_eq!(x.max(y), y);
        /// ```
        ///
        /// If one of the arguments is NaN, then the other argument is returned.
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn max(self, other: f64) -> f64 {
            // IEEE754 says: maxNum(x, y) is the canonicalized number y if x < y, x if y < x, the
            // canonicalized number if one operand is a number and the other a quiet NaN. Otherwise it
            // is either x or y, canonicalized (this means results might differ among implementations).
            // When either x or y is a signalingNaN, then the result is according to 6.2.
            //
            // Since we do not support sNaN in Rust yet, we do not need to handle them.
            // FIXME(nagisa): due to https://bugs.llvm.org/show_bug.cgi?id=33303 we canonicalize by
            // multiplying by 1.0. Should switch to the `canonicalize` when it works.
            (if self.is_nan() || self < other { other } else { self }) * 1.0
        }

        /// Returns the minimum of the two numbers.
        ///
        /// ```
        /// let x = 1.0_f64;
        /// let y = 2.0_f64;
        ///
        /// assert_eq!(x.min(y), x);
        /// ```
        ///
        /// If one of the arguments is NaN, then the other argument is returned.
        #[stable(feature = "rust1", since = "1.0.0")]
        #[inline]
        pub fn min(self, other: f64) -> f64 {
            // IEEE754 says: minNum(x, y) is the canonicalized number x if x < y, y if y < x, the
            // canonicalized number if one operand is a number and the other a quiet NaN. Otherwise it
            // is either x or y, canonicalized (this means results might differ among implementations).
            // When either x or y is a signalingNaN, then the result is according to 6.2.
            //
            // Since we do not support sNaN in Rust yet, we do not need to handle them.
            // FIXME(nagisa): due to https://bugs.llvm.org/show_bug.cgi?id=33303 we canonicalize by
            // multiplying by 1.0. Should switch to the `canonicalize` when it works.
            (if other.is_nan() || self < other { self } else { other }) * 1.0
        }

        /// Raw transmutation to `u64`.
        ///
        /// This is currently identical to `transmute::<f64, u64>(self)` on all platforms.
        ///
        /// See `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.
        ///
        /// # Examples
        ///
        /// ```
        /// assert!((1f64).to_bits() != 1f64 as u64); // to_bits() is not casting!
        /// assert_eq!((12.5f64).to_bits(), 0x4029000000000000);
        ///
        /// ```
        #[stable(feature = "float_bits_conv", since = "1.20.0")]
        #[inline]
        pub fn to_bits(self) -> u64 {
            unsafe { mem::transmute(self) }
        }

        /// Raw transmutation from `u64`.
        ///
        /// This is currently identical to `transmute::<u64, f64>(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 favours 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.
        ///
        /// # Examples
        ///
        /// ```
        /// use std::f64;
        /// let v = f64::from_bits(0x4029000000000000);
        /// let difference = (v - 12.5).abs();
        /// assert!(difference <= 1e-5);
        /// ```
        #[stable(feature = "float_bits_conv", since = "1.20.0")]
        #[inline]
        pub fn from_bits(v: u64) -> Self {
            // It turns out the safety issues with sNaN were overblown! Hooray!
            unsafe { mem::transmute(v) }
        }
    }

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