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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 distributions::ziggurat_tables;
+#[cfg(feature="std")]
+use 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: $ty = tmp.cast();
+ (hi, lo)
+ }
+ }
+ )+
+ };
+}
+wmul_impl! { u8, u16, 8 }
+wmul_impl! { u16, u32, 16 }
+wmul_impl! { u32, u64, 32 }
+#[cfg(rust_1_26)]
+wmul_impl! { u64, u128, 64 }
+
+// This code is a translation of the __mulddi3 function in LLVM's
+// compiler-rt. It is an optimised variant of the common method
+// `(a + b) * (c + d) = ac + ad + bc + bd`.
+//
+// For some reason LLVM can optimise the C version very well, but
+// keeps shuffling registers in this Rust translation.
+macro_rules! wmul_impl_large {
+ ($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 >> $half);
+
+ (high, low)
+ }
+ }
+ )+
+ };
+}
+#[cfg(not(rust_1_26))]
+wmul_impl_large! { u64, 32 }
+#[cfg(rust_1_26)]
+wmul_impl_large! { u128, 64 }
+
+macro_rules! wmul_impl_usize {
+ ($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;
+ 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 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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