utils.rs

// 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;
        }
    }
}