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// Copyright 2018 Developers of the Rand project.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// 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;
use num_traits::Float;
/// 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<F: Float>(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<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 subtract to get [-1,1)
// We can't convert to an open range directly, that would require
// subtracting `3.0 - EPSILON`, which is not representable.
// It is possible with an extra step, but an open range does not
// seem necessary 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 subtract 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;
}
}
}