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 // Copyright 2018 Developers of the Rand project. // // Licensed under the Apache License, Version 2.0 or the MIT license // , at your // option. This file may not be copied, modified, or distributed // except according to those terms. //! Math helper functions use crate::ziggurat_tables; use rand::distributions::hidden_export::IntoFloat; use rand::Rng; /// 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: F) -> F { // precalculated 6 coefficients for the first 6 terms of the series let coefficients: [F; 6] = [ F::from(76.18009172947146).unwrap(), F::from(-86.50532032941677).unwrap(), F::from(24.01409824083091).unwrap(), F::from(-1.231739572450155).unwrap(), F::from(0.1208650973866179e-2).unwrap(), F::from(-0.5395239384953e-5).unwrap(), ]; // (x+0.5)*ln(x+g+0.5)-(x+g+0.5) let tmp = x + F::from(5.5).unwrap(); let log = (x + F::from(0.5).unwrap()) * tmp.ln() - tmp; // the first few terms of the series for Ag(x) let mut a = F::from(1.000000000190015).unwrap(); let mut denom = x; for &coeff in &coefficients { denom = denom + F::one(); a = a + (coeff / denom); } // get everything together // a is Ag(x) // 2.5066... is sqrt(2pi) log + (F::from(2.5066282746310005).unwrap() * 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( 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 - 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::() < pdf(x) { return x; } } }