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    // Copyright 2018 Developers of the Rand project.
    //
    // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
    // https://www.apache.org/licenses/LICENSE-2.0> or the MIT license
    // <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your
    // option. This file may not be copied, modified, or distributed
    // except according to those terms.

    //! Math helper functions

    #[cfg(feature="simd_support")]
    use packed_simd::*;
    #[cfg(feature="std")]
    use crate::distributions::ziggurat_tables;
    #[cfg(feature="std")]
    use crate::Rng;

    pub trait WideningMultiply<RHS = Self> {
        type Output;

        fn wmul(self, x: RHS) -> Self::Output;
    }

    macro_rules! wmul_impl {
        ($ty:ty, $wide:ty, $shift:expr) => {
            impl WideningMultiply for $ty {
                type Output = ($ty, $ty);

                #[inline(always)]
                fn wmul(self, x: $ty) -> Self::Output {
                    let tmp = (self as $wide) * (x as $wide);
                    ((tmp >> $shift) as $ty, tmp as $ty)
                }
            }
        };

        // simd bulk implementation
        ($(($ty:ident, $wide:ident),)+, $shift:expr) => {
            $(
                impl WideningMultiply for $ty {
                    type Output = ($ty, $ty);

                    #[inline(always)]
                    fn wmul(self, x: $ty) -> Self::Output {
                        // For supported vectors, this should compile to a couple
                        // supported multiply & swizzle instructions (no actual
                        // casting).
                        // TODO: optimize
                        let y: $wide = self.cast();
                        let x: $wide = x.cast();
                        let tmp = y * x;
                        let hi: $ty = (tmp >> $shift).cast();
                        let lo: $ty = tmp.cast();
                        (hi, lo)
                    }
                }
            )+
        };
    }
    wmul_impl! { u8, u16, 8 }
    wmul_impl! { u16, u32, 16 }
    wmul_impl! { u32, u64, 32 }
    #[cfg(not(target_os = "emscripten"))]
    wmul_impl! { u64, u128, 64 }

    // This code is a translation of the __mulddi3 function in LLVM's
    // compiler-rt. It is an optimised variant of the common method
    // `(a + b) * (c + d) = ac + ad + bc + bd`.
    //
    // For some reason LLVM can optimise the C version very well, but
    // keeps shuffling registers in this Rust translation.
    macro_rules! wmul_impl_large {
        ($ty:ty, $half:expr) => {
            impl WideningMultiply for $ty {
                type Output = ($ty, $ty);

                #[inline(always)]
                fn wmul(self, b: $ty) -> Self::Output {
                    const LOWER_MASK: $ty = !0 >> $half;
                    let mut low = (self & LOWER_MASK).wrapping_mul(b & LOWER_MASK);
                    let mut t = low >> $half;
                    low &= LOWER_MASK;
                    t += (self >> $half).wrapping_mul(b & LOWER_MASK);
                    low += (t & LOWER_MASK) << $half;
                    let mut high = t >> $half;
                    t = low >> $half;
                    low &= LOWER_MASK;
                    t += (b >> $half).wrapping_mul(self & LOWER_MASK);
                    low += (t & LOWER_MASK) << $half;
                    high += t >> $half;
                    high += (self >> $half).wrapping_mul(b >> $half);

                    (high, low)
                }
            }
        };

        // simd bulk implementation
        (($($ty:ty,)+) $scalar:ty, $half:expr) => {
            $(
                impl WideningMultiply for $ty {
                    type Output = ($ty, $ty);

                    #[inline(always)]
                    fn wmul(self, b: $ty) -> Self::Output {
                        // needs wrapping multiplication
                        const LOWER_MASK: $scalar = !0 >> $half;
                        let mut low = (self & LOWER_MASK) * (b & LOWER_MASK);
                        let mut t = low >> $half;
                        low &= LOWER_MASK;
                        t += (self >> $half) * (b & LOWER_MASK);
                        low += (t & LOWER_MASK) << $half;
                        let mut high = t >> $half;
                        t = low >> $half;
                        low &= LOWER_MASK;
                        t += (b >> $half) * (self & LOWER_MASK);
                        low += (t & LOWER_MASK) << $half;
                        high += t >> $half;
                        high += (self >> $half) * (b >> $half);

                        (high, low)
                    }
                }
            )+
        };
    }
    #[cfg(target_os = "emscripten")]
    wmul_impl_large! { u64, 32 }
    #[cfg(not(target_os = "emscripten"))]
    wmul_impl_large! { u128, 64 }

    macro_rules! wmul_impl_usize {
        ($ty:ty) => {
            impl WideningMultiply for usize {
                type Output = (usize, usize);

                #[inline(always)]
                fn wmul(self, x: usize) -> Self::Output {
                    let (high, low) = (self as $ty).wmul(x as $ty);
                    (high as usize, low as usize)
                }
            }
        }
    }
    #[cfg(target_pointer_width = "32")]
    wmul_impl_usize! { u32 }
    #[cfg(target_pointer_width = "64")]
    wmul_impl_usize! { u64 }

    #[cfg(all(feature = "simd_support", feature = "nightly"))]
    mod simd_wmul {
        #[cfg(target_arch = "x86")]
        use core::arch::x86::*;
        #[cfg(target_arch = "x86_64")]
        use core::arch::x86_64::*;
        use super::*;

        wmul_impl! {
            (u8x2, u16x2),
            (u8x4, u16x4),
            (u8x8, u16x8),
            (u8x16, u16x16),
            (u8x32, u16x32),,
            8
        }

        wmul_impl! { (u16x2, u32x2),, 16 }
        #[cfg(not(target_feature = "sse2"))]
        wmul_impl! { (u16x4, u32x4),, 16 }
        #[cfg(not(target_feature = "sse4.2"))]
        wmul_impl! { (u16x8, u32x8),, 16 }
        #[cfg(not(target_feature = "avx2"))]
        wmul_impl! { (u16x16, u32x16),, 16 }

        // 16-bit lane widths allow use of the x86 `mulhi` instructions, which
        // means `wmul` can be implemented with only two instructions.
        #[allow(unused_macros)]
        macro_rules! wmul_impl_16 {
            ($ty:ident, $intrinsic:ident, $mulhi:ident, $mullo:ident) => {
                impl WideningMultiply for $ty {
                    type Output = ($ty, $ty);

                    #[inline(always)]
                    fn wmul(self, x: $ty) -> Self::Output {
                        let b = $intrinsic::from_bits(x);
                        let a = $intrinsic::from_bits(self);
                        let hi = $ty::from_bits(unsafe { $mulhi(a, b) });
                        let lo = $ty::from_bits(unsafe { $mullo(a, b) });
                        (hi, lo)
                    }
                }
            };
        }

        #[cfg(target_feature = "sse2")]
        wmul_impl_16! { u16x4, __m64, _mm_mulhi_pu16, _mm_mullo_pi16 }
        #[cfg(target_feature = "sse4.2")]
        wmul_impl_16! { u16x8, __m128i, _mm_mulhi_epu16, _mm_mullo_epi16 }
        #[cfg(target_feature = "avx2")]
        wmul_impl_16! { u16x16, __m256i, _mm256_mulhi_epu16, _mm256_mullo_epi16 }
        // FIXME: there are no `__m512i` types in stdsimd yet, so `wmul::<u16x32>`
        // cannot use the same implementation.

        wmul_impl! {
            (u32x2, u64x2),
            (u32x4, u64x4),
            (u32x8, u64x8),,
            32
        }

        // TODO: optimize, this seems to seriously slow things down
        wmul_impl_large! { (u8x64,) u8, 4 }
        wmul_impl_large! { (u16x32,) u16, 8 }
        wmul_impl_large! { (u32x16,) u32, 16 }
        wmul_impl_large! { (u64x2, u64x4, u64x8,) u64, 32 }
    }
    #[cfg(all(feature = "simd_support", feature = "nightly"))]
    pub use self::simd_wmul::*;

    /// Helper trait when dealing with scalar and SIMD floating point types.
    pub(crate) trait FloatSIMDUtils {
        // `PartialOrd` for vectors compares lexicographically. We want to compare all
        // the individual SIMD lanes instead, and get the combined result over all
        // lanes. This is possible using something like `a.lt(b).all()`, but we
        // implement it as a trait so we can write the same code for `f32` and `f64`.
        // Only the comparison functions we need are implemented.
        fn all_lt(self, other: Self) -> bool;
        fn all_le(self, other: Self) -> bool;
        fn all_finite(self) -> bool;

        type Mask;
        fn finite_mask(self) -> Self::Mask;
        fn gt_mask(self, other: Self) -> Self::Mask;
        fn ge_mask(self, other: Self) -> Self::Mask;

        // Decrease all lanes where the mask is `true` to the next lower value
        // representable by the floating-point type. At least one of the lanes
        // must be set.
        fn decrease_masked(self, mask: Self::Mask) -> Self;

        // Convert from int value. Conversion is done while retaining the numerical
        // value, not by retaining the binary representation.
        type UInt;
        fn cast_from_int(i: Self::UInt) -> Self;
    }

    /// Implement functions available in std builds but missing from core primitives
    #[cfg(not(std))]
    pub(crate) trait Float : Sized {
        type Bits;

        fn is_nan(self) -> bool;
        fn is_infinite(self) -> bool;
        fn is_finite(self) -> bool;
        fn to_bits(self) -> Self::Bits;
        fn from_bits(v: Self::Bits) -> Self;
    }

    /// Implement functions on f32/f64 to give them APIs similar to SIMD types
    pub(crate) trait FloatAsSIMD : Sized {
        #[inline(always)]
        fn lanes() -> usize { 1 }
        #[inline(always)]
        fn splat(scalar: Self) -> Self { scalar }
        #[inline(always)]
        fn extract(self, index: usize) -> Self { debug_assert_eq!(index, 0); self }
        #[inline(always)]
        fn replace(self, index: usize, new_value: Self) -> Self { debug_assert_eq!(index, 0); new_value }
    }

    pub(crate) trait BoolAsSIMD : Sized {
        fn any(self) -> bool;
        fn all(self) -> bool;
        fn none(self) -> bool;
    }

    impl BoolAsSIMD for bool {
        #[inline(always)]
        fn any(self) -> bool { self }
        #[inline(always)]
        fn all(self) -> bool { self }
        #[inline(always)]
        fn none(self) -> bool { !self }
    }

    macro_rules! scalar_float_impl {
        ($ty:ident, $uty:ident) => {
            #[cfg(not(std))]
            impl Float for $ty {
                type Bits = $uty;

                #[inline]
                fn is_nan(self) -> bool {
                    self != self
                }

                #[inline]
                fn is_infinite(self) -> bool {
                    self == ::core::$ty::INFINITY || self == ::core::$ty::NEG_INFINITY
                }

                #[inline]
                fn is_finite(self) -> bool {
                    !(self.is_nan() || self.is_infinite())
                }

                #[inline]
                fn to_bits(self) -> Self::Bits {
                    unsafe { ::core::mem::transmute(self) }
                }

                #[inline]
                fn from_bits(v: Self::Bits) -> Self {
                    // It turns out the safety issues with sNaN were overblown! Hooray!
                    unsafe { ::core::mem::transmute(v) }
                }
            }

            impl FloatSIMDUtils for $ty {
                type Mask = bool;
                #[inline(always)]
                fn all_lt(self, other: Self) -> bool { self < other }
                #[inline(always)]
                fn all_le(self, other: Self) -> bool { self <= other }
                #[inline(always)]
                fn all_finite(self) -> bool { self.is_finite() }
                #[inline(always)]
                fn finite_mask(self) -> Self::Mask { self.is_finite() }
                #[inline(always)]
                fn gt_mask(self, other: Self) -> Self::Mask { self > other }
                #[inline(always)]
                fn ge_mask(self, other: Self) -> Self::Mask { self >= other }
                #[inline(always)]
                fn decrease_masked(self, mask: Self::Mask) -> Self {
                    debug_assert!(mask, "At least one lane must be set");
                    <$ty>::from_bits(self.to_bits() - 1)
                }
                type UInt = $uty;
                fn cast_from_int(i: Self::UInt) -> Self { i as $ty }
            }

            impl FloatAsSIMD for $ty {}
        }
    }

    scalar_float_impl!(f32, u32);
    scalar_float_impl!(f64, u64);

    #[cfg(feature="simd_support")]
    macro_rules! simd_impl {
        ($ty:ident, $f_scalar:ident, $mty:ident, $uty:ident) => {
            impl FloatSIMDUtils for $ty {
                type Mask = $mty;
                #[inline(always)]
                fn all_lt(self, other: Self) -> bool { self.lt(other).all() }
                #[inline(always)]
                fn all_le(self, other: Self) -> bool { self.le(other).all() }
                #[inline(always)]
                fn all_finite(self) -> bool { self.finite_mask().all() }
                #[inline(always)]
                fn finite_mask(self) -> Self::Mask {
                    // This can possibly be done faster by checking bit patterns
                    let neg_inf = $ty::splat(::core::$f_scalar::NEG_INFINITY);
                    let pos_inf = $ty::splat(::core::$f_scalar::INFINITY);
                    self.gt(neg_inf) & self.lt(pos_inf)
                }
                #[inline(always)]
                fn gt_mask(self, other: Self) -> Self::Mask { self.gt(other) }
                #[inline(always)]
                fn ge_mask(self, other: Self) -> Self::Mask { self.ge(other) }
                #[inline(always)]
                fn decrease_masked(self, mask: Self::Mask) -> Self {
                    // Casting a mask into ints will produce all bits set for
                    // true, and 0 for false. Adding that to the binary
                    // representation of a float means subtracting one from
                    // the binary representation, resulting in the next lower
                    // value representable by $ty. This works even when the
                    // current value is infinity.
                    debug_assert!(mask.any(), "At least one lane must be set");
                    <$ty>::from_bits(<$uty>::from_bits(self) + <$uty>::from_bits(mask))
                }
                type UInt = $uty;
                #[inline]
                fn cast_from_int(i: Self::UInt) -> Self { i.cast() }
            }
        }
    }

    #[cfg(feature="simd_support")] simd_impl! { f32x2, f32, m32x2, u32x2 }
    #[cfg(feature="simd_support")] simd_impl! { f32x4, f32, m32x4, u32x4 }
    #[cfg(feature="simd_support")] simd_impl! { f32x8, f32, m32x8, u32x8 }
    #[cfg(feature="simd_support")] simd_impl! { f32x16, f32, m32x16, u32x16 }
    #[cfg(feature="simd_support")] simd_impl! { f64x2, f64, m64x2, u64x2 }
    #[cfg(feature="simd_support")] simd_impl! { f64x4, f64, m64x4, u64x4 }
    #[cfg(feature="simd_support")] simd_impl! { f64x8, f64, m64x8, u64x8 }

    /// Calculates ln(gamma(x)) (natural logarithm of the gamma
    /// function) using the Lanczos approximation.
    ///
    /// The approximation expresses the gamma function as:
    /// `gamma(z+1) = sqrt(2*pi)*(z+g+0.5)^(z+0.5)*exp(-z-g-0.5)*Ag(z)`
    /// `g` is an arbitrary constant; we use the approximation with `g=5`.
    ///
    /// Noting that `gamma(z+1) = z*gamma(z)` and applying `ln` to both sides:
    /// `ln(gamma(z)) = (z+0.5)*ln(z+g+0.5)-(z+g+0.5) + ln(sqrt(2*pi)*Ag(z)/z)`
    ///
    /// `Ag(z)` is an infinite series with coefficients that can be calculated
    /// ahead of time - we use just the first 6 terms, which is good enough
    /// for most purposes.
    #[cfg(feature="std")]
    pub fn log_gamma(x: f64) -> f64 {
        // precalculated 6 coefficients for the first 6 terms of the series
        let coefficients: [f64; 6] = [
            76.18009172947146,
            -86.50532032941677,
            24.01409824083091,
            -1.231739572450155,
            0.1208650973866179e-2,
            -0.5395239384953e-5,
        ];

        // (x+0.5)*ln(x+g+0.5)-(x+g+0.5)
        let tmp = x + 5.5;
        let log = (x + 0.5) * tmp.ln() - tmp;

        // the first few terms of the series for Ag(x)
        let mut a = 1.000000000190015;
        let mut denom = x;
        for coeff in &coefficients {
            denom += 1.0;
            a += coeff / denom;
        }

        // get everything together
        // a is Ag(x)
        // 2.5066... is sqrt(2pi)
        log + (2.5066282746310005 * a / x).ln()
    }

    /// Sample a random number using the Ziggurat method (specifically the
    /// ZIGNOR variant from Doornik 2005). Most of the arguments are
    /// directly from the paper:
    ///
    /// * `rng`: source of randomness
    /// * `symmetric`: whether this is a symmetric distribution, or one-sided with P(x < 0) = 0.
    /// * `X`: the $x_i$ abscissae.
    /// * `F`: precomputed values of the PDF at the $x_i$, (i.e. $f(x_i)$)
    /// * `F_DIFF`: precomputed values of $f(x_i) - f(x_{i+1})$
    /// * `pdf`: the probability density function
    /// * `zero_case`: manual sampling from the tail when we chose the
    ///    bottom box (i.e. i == 0)

    // the perf improvement (25-50%) is definitely worth the extra code
    // size from force-inlining.
    #[cfg(feature="std")]
    #[inline(always)]
    pub fn ziggurat<R: Rng + ?Sized, P, Z>(
                rng: &mut R,
                symmetric: bool,
                x_tab: ziggurat_tables::ZigTable,
                f_tab: ziggurat_tables::ZigTable,
                mut pdf: P,
                mut zero_case: Z)
                -> f64 where P: FnMut(f64) -> f64, Z: FnMut(&mut R, f64) -> f64 {
        use crate::distributions::float::IntoFloat;
        loop {
            // As an optimisation we re-implement the conversion to a f64.
            // From the remaining 12 most significant bits we use 8 to construct `i`.
            // This saves us generating a whole extra random number, while the added
            // precision of using 64 bits for f64 does not buy us much.
            let bits = rng.next_u64();
            let i = bits as usize & 0xff;

            let u = if symmetric {
                // Convert to a value in the range [2,4) and substract to get [-1,1)
                // We can't convert to an open range directly, that would require
                // substracting `3.0 - EPSILON`, which is not representable.
                // It is possible with an extra step, but an open range does not
                // seem neccesary for the ziggurat algorithm anyway.
                (bits >> 12).into_float_with_exponent(1) - 3.0
            } else {
                // Convert to a value in the range [1,2) and substract to get (0,1)
                (bits >> 12).into_float_with_exponent(0)
                - (1.0 - ::core::f64::EPSILON / 2.0)
            };
            let x = u * x_tab[i];

            let test_x = if symmetric { x.abs() } else {x};

            // algebraically equivalent to |u| < x_tab[i+1]/x_tab[i] (or u < x_tab[i+1]/x_tab[i])
            if test_x < x_tab[i + 1] {
                return x;
            }
            if i == 0 {
                return zero_case(rng, u);
            }
            // algebraically equivalent to f1 + DRanU()*(f0 - f1) < 1
            if f_tab[i + 1] + (f_tab[i] - f_tab[i + 1]) * rng.gen::<f64>() < pdf(x) {
                return x;
            }
        }
    }

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