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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
-
-use rand::Rng;
-use crate::ziggurat_tables;
-use rand::distributions::hidden_export::IntoFloat;
-use core::{cmp, ops};
-
-/// Trait for floating-point scalar types
-///
-/// This allows many distributions to work with `f32` or `f64` parameters and is
-/// potentially extensible. Note however that the `Exp1` and `StandardNormal`
-/// distributions are implemented exclusively for `f32` and `f64`.
-///
-/// The bounds and methods are based purely on internal
-/// requirements, and will change as needed.
-pub trait Float: Copy + Sized + cmp::PartialOrd
- + ops::Neg<Output = Self>
- + ops::Add<Output = Self>
- + ops::Sub<Output = Self>
- + ops::Mul<Output = Self>
- + ops::Div<Output = Self>
- + ops::AddAssign + ops::SubAssign + ops::MulAssign + ops::DivAssign
-{
- /// The constant π
- fn pi() -> Self;
- /// Support approximate representation of a f64 value
- fn from(x: f64) -> Self;
- /// Support converting to an unsigned integer.
- fn to_u64(self) -> Option<u64>;
-
- /// Take the absolute value of self
- fn abs(self) -> Self;
- /// Take the largest integer less than or equal to self
- fn floor(self) -> Self;
-
- /// Take the exponential of self
- fn exp(self) -> Self;
- /// Take the natural logarithm of self
- fn ln(self) -> Self;
- /// Take square root of self
- fn sqrt(self) -> Self;
- /// Take self to a floating-point power
- fn powf(self, power: Self) -> Self;
-
- /// Take the tangent of self
- fn tan(self) -> Self;
- /// Take the logarithm of the gamma function of self
- fn log_gamma(self) -> Self;
-}
-
-impl Float for f32 {
- #[inline]
- fn pi() -> Self { core::f32::consts::PI }
- #[inline]
- fn from(x: f64) -> Self { x as f32 }
- #[inline]
- fn to_u64(self) -> Option<u64> {
- if self >= 0. && self <= ::core::u64::MAX as f32 {
- Some(self as u64)
- } else {
- None
- }
- }
-
- #[inline]
- fn abs(self) -> Self { self.abs() }
- #[inline]
- fn floor(self) -> Self { self.floor() }
-
- #[inline]
- fn exp(self) -> Self { self.exp() }
- #[inline]
- fn ln(self) -> Self { self.ln() }
- #[inline]
- fn sqrt(self) -> Self { self.sqrt() }
- #[inline]
- fn powf(self, power: Self) -> Self { self.powf(power) }
-
- #[inline]
- fn tan(self) -> Self { self.tan() }
- #[inline]
- fn log_gamma(self) -> Self {
- let result = log_gamma(self.into());
- assert!(result <= ::core::f32::MAX.into());
- assert!(result >= ::core::f32::MIN.into());
- result as f32
- }
-}
-
-impl Float for f64 {
- #[inline]
- fn pi() -> Self { core::f64::consts::PI }
- #[inline]
- fn from(x: f64) -> Self { x }
- #[inline]
- fn to_u64(self) -> Option<u64> {
- if self >= 0. && self <= ::core::u64::MAX as f64 {
- Some(self as u64)
- } else {
- None
- }
- }
-
- #[inline]
- fn abs(self) -> Self { self.abs() }
- #[inline]
- fn floor(self) -> Self { self.floor() }
-
- #[inline]
- fn exp(self) -> Self { self.exp() }
- #[inline]
- fn ln(self) -> Self { self.ln() }
- #[inline]
- fn sqrt(self) -> Self { self.sqrt() }
- #[inline]
- fn powf(self, power: Self) -> Self { self.powf(power) }
-
- #[inline]
- fn tan(self) -> Self { self.tan() }
- #[inline]
- fn log_gamma(self) -> Self { log_gamma(self) }
-}
-
-/// 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.
-pub(crate) 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.
-#[inline(always)]
-pub(crate) 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 {
- 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 - std::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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