diff options
Diffstat (limited to 'rand/rand_distr/src')
| -rw-r--r-- | rand/rand_distr/src/binomial.rs | 329 | ||||
| -rw-r--r-- | rand/rand_distr/src/cauchy.rs | 120 | ||||
| -rw-r--r-- | rand/rand_distr/src/dirichlet.rs | 154 | ||||
| -rw-r--r-- | rand/rand_distr/src/exponential.rs | 145 | ||||
| -rw-r--r-- | rand/rand_distr/src/gamma.rs | 485 | ||||
| -rw-r--r-- | rand/rand_distr/src/lib.rs | 134 | ||||
| -rw-r--r-- | rand/rand_distr/src/normal.rs | 219 | ||||
| -rw-r--r-- | rand/rand_distr/src/pareto.rs | 89 | ||||
| -rw-r--r-- | rand/rand_distr/src/pert.rs | 132 | ||||
| -rw-r--r-- | rand/rand_distr/src/poisson.rs | 233 | ||||
| -rw-r--r-- | rand/rand_distr/src/triangular.rs | 125 | ||||
| -rw-r--r-- | rand/rand_distr/src/unit_ball.rs | 69 | ||||
| -rw-r--r-- | rand/rand_distr/src/unit_circle.rs | 99 | ||||
| -rw-r--r-- | rand/rand_distr/src/unit_disc.rs | 66 | ||||
| -rw-r--r-- | rand/rand_distr/src/unit_sphere.rs | 94 | ||||
| -rw-r--r-- | rand/rand_distr/src/utils.rs | 234 | ||||
| -rw-r--r-- | rand/rand_distr/src/weibull.rs | 86 | ||||
| -rw-r--r-- | rand/rand_distr/src/ziggurat_tables.rs | 279 |
18 files changed, 3092 insertions, 0 deletions
diff --git a/rand/rand_distr/src/binomial.rs b/rand/rand_distr/src/binomial.rs new file mode 100644 index 0000000..0e6bf9a --- /dev/null +++ b/rand/rand_distr/src/binomial.rs @@ -0,0 +1,329 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2016-2017 The Rust Project Developers. +// +// 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. + +//! The binomial distribution. + +use rand::Rng; +use crate::{Distribution, Uniform}; + +/// The binomial distribution `Binomial(n, p)`. +/// +/// This distribution has density function: +/// `f(k) = n!/(k! (n-k)!) p^k (1-p)^(n-k)` for `k >= 0`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Binomial, Distribution}; +/// +/// let bin = Binomial::new(20, 0.3).unwrap(); +/// let v = bin.sample(&mut rand::thread_rng()); +/// println!("{} is from a binomial distribution", v); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Binomial { + /// Number of trials. + n: u64, + /// Probability of success. + p: f64, +} + +/// Error type returned from `Binomial::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `p < 0` or `nan`. + ProbabilityTooSmall, + /// `p > 1`. + ProbabilityTooLarge, +} + +impl Binomial { + /// Construct a new `Binomial` with the given shape parameters `n` (number + /// of trials) and `p` (probability of success). + pub fn new(n: u64, p: f64) -> Result<Binomial, Error> { + if !(p >= 0.0) { + return Err(Error::ProbabilityTooSmall); + } + if !(p <= 1.0) { + return Err(Error::ProbabilityTooLarge); + } + Ok(Binomial { n, p }) + } +} + +/// Convert a `f64` to an `i64`, panicing on overflow. +// In the future (Rust 1.34), this might be replaced with `TryFrom`. +fn f64_to_i64(x: f64) -> i64 { + assert!(x < (::std::i64::MAX as f64)); + x as i64 +} + +impl Distribution<u64> for Binomial { + #[allow(clippy::many_single_char_names)] // Same names as in the reference. + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 { + // Handle these values directly. + if self.p == 0.0 { + return 0; + } else if self.p == 1.0 { + return self.n; + } + + // The binomial distribution is symmetrical with respect to p -> 1-p, + // k -> n-k switch p so that it is less than 0.5 - this allows for lower + // expected values we will just invert the result at the end + let p = if self.p <= 0.5 { + self.p + } else { + 1.0 - self.p + }; + + let result; + let q = 1. - p; + + // For small n * min(p, 1 - p), the BINV algorithm based on the inverse + // transformation of the binomial distribution is efficient. Otherwise, + // the BTPE algorithm is used. + // + // Voratas Kachitvichyanukul and Bruce W. Schmeiser. 1988. Binomial + // random variate generation. Commun. ACM 31, 2 (February 1988), + // 216-222. http://dx.doi.org/10.1145/42372.42381 + + // Threshold for prefering the BINV algorithm. The paper suggests 10, + // Ranlib uses 30, and GSL uses 14. + const BINV_THRESHOLD: f64 = 10.; + + if (self.n as f64) * p < BINV_THRESHOLD && + self.n <= (::std::i32::MAX as u64) { + // Use the BINV algorithm. + let s = p / q; + let a = ((self.n + 1) as f64) * s; + let mut r = q.powi(self.n as i32); + let mut u: f64 = rng.gen(); + let mut x = 0; + while u > r as f64 { + u -= r; + x += 1; + r *= a / (x as f64) - s; + } + result = x; + } else { + // Use the BTPE algorithm. + + // Threshold for using the squeeze algorithm. This can be freely + // chosen based on performance. Ranlib and GSL use 20. + const SQUEEZE_THRESHOLD: i64 = 20; + + // Step 0: Calculate constants as functions of `n` and `p`. + let n = self.n as f64; + let np = n * p; + let npq = np * q; + let f_m = np + p; + let m = f64_to_i64(f_m); + // radius of triangle region, since height=1 also area of region + let p1 = (2.195 * npq.sqrt() - 4.6 * q).floor() + 0.5; + // tip of triangle + let x_m = (m as f64) + 0.5; + // left edge of triangle + let x_l = x_m - p1; + // right edge of triangle + let x_r = x_m + p1; + let c = 0.134 + 20.5 / (15.3 + (m as f64)); + // p1 + area of parallelogram region + let p2 = p1 * (1. + 2. * c); + + fn lambda(a: f64) -> f64 { + a * (1. + 0.5 * a) + } + + let lambda_l = lambda((f_m - x_l) / (f_m - x_l * p)); + let lambda_r = lambda((x_r - f_m) / (x_r * q)); + // p1 + area of left tail + let p3 = p2 + c / lambda_l; + // p1 + area of right tail + let p4 = p3 + c / lambda_r; + + // return value + let mut y: i64; + + let gen_u = Uniform::new(0., p4); + let gen_v = Uniform::new(0., 1.); + + loop { + // Step 1: Generate `u` for selecting the region. If region 1 is + // selected, generate a triangularly distributed variate. + let u = gen_u.sample(rng); + let mut v = gen_v.sample(rng); + if !(u > p1) { + y = f64_to_i64(x_m - p1 * v + u); + break; + } + + if !(u > p2) { + // Step 2: Region 2, parallelograms. Check if region 2 is + // used. If so, generate `y`. + let x = x_l + (u - p1) / c; + v = v * c + 1.0 - (x - x_m).abs() / p1; + if v > 1. { + continue; + } else { + y = f64_to_i64(x); + } + } else if !(u > p3) { + // Step 3: Region 3, left exponential tail. + y = f64_to_i64(x_l + v.ln() / lambda_l); + if y < 0 { + continue; + } else { + v *= (u - p2) * lambda_l; + } + } else { + // Step 4: Region 4, right exponential tail. + y = f64_to_i64(x_r - v.ln() / lambda_r); + if y > 0 && (y as u64) > self.n { + continue; + } else { + v *= (u - p3) * lambda_r; + } + } + + // Step 5: Acceptance/rejection comparison. + + // Step 5.0: Test for appropriate method of evaluating f(y). + let k = (y - m).abs(); + if !(k > SQUEEZE_THRESHOLD && (k as f64) < 0.5 * npq - 1.) { + // Step 5.1: Evaluate f(y) via the recursive relationship. Start the + // search from the mode. + let s = p / q; + let a = s * (n + 1.); + let mut f = 1.0; + if m < y { + let mut i = m; + loop { + i += 1; + f *= a / (i as f64) - s; + if i == y { + break; + } + } + } else if m > y { + let mut i = y; + loop { + i += 1; + f /= a / (i as f64) - s; + if i == m { + break; + } + } + } + if v > f { + continue; + } else { + break; + } + } + + // Step 5.2: Squeezing. Check the value of ln(v) againts upper and + // lower bound of ln(f(y)). + let k = k as f64; + let rho = (k / npq) * ((k * (k / 3. + 0.625) + 1./6.) / npq + 0.5); + let t = -0.5 * k*k / npq; + let alpha = v.ln(); + if alpha < t - rho { + break; + } + if alpha > t + rho { + continue; + } + + // Step 5.3: Final acceptance/rejection test. + let x1 = (y + 1) as f64; + let f1 = (m + 1) as f64; + let z = (f64_to_i64(n) + 1 - m) as f64; + let w = (f64_to_i64(n) - y + 1) as f64; + + fn stirling(a: f64) -> f64 { + let a2 = a * a; + (13860. - (462. - (132. - (99. - 140. / a2) / a2) / a2) / a2) / a / 166320. + } + + if alpha > x_m * (f1 / x1).ln() + + (n - (m as f64) + 0.5) * (z / w).ln() + + ((y - m) as f64) * (w * p / (x1 * q)).ln() + // We use the signs from the GSL implementation, which are + // different than the ones in the reference. According to + // the GSL authors, the new signs were verified to be + // correct by one of the original designers of the + // algorithm. + + stirling(f1) + stirling(z) - stirling(x1) - stirling(w) + { + continue; + } + + break; + } + assert!(y >= 0); + result = y as u64; + } + + // Invert the result for p < 0.5. + if p != self.p { + self.n - result + } else { + result + } + } +} + +#[cfg(test)] +mod test { + use rand::Rng; + use crate::Distribution; + use super::Binomial; + + fn test_binomial_mean_and_variance<R: Rng>(n: u64, p: f64, rng: &mut R) { + let binomial = Binomial::new(n, p).unwrap(); + + let expected_mean = n as f64 * p; + let expected_variance = n as f64 * p * (1.0 - p); + + let mut results = [0.0; 1000]; + for i in results.iter_mut() { *i = binomial.sample(rng) as f64; } + + let mean = results.iter().sum::<f64>() / results.len() as f64; + assert!((mean as f64 - expected_mean).abs() < expected_mean / 50.0); + + let variance = + results.iter().map(|x| (x - mean) * (x - mean)).sum::<f64>() + / results.len() as f64; + assert!((variance - expected_variance).abs() < expected_variance / 10.0); + } + + #[test] + fn test_binomial() { + let mut rng = crate::test::rng(351); + test_binomial_mean_and_variance(150, 0.1, &mut rng); + test_binomial_mean_and_variance(70, 0.6, &mut rng); + test_binomial_mean_and_variance(40, 0.5, &mut rng); + test_binomial_mean_and_variance(20, 0.7, &mut rng); + test_binomial_mean_and_variance(20, 0.5, &mut rng); + } + + #[test] + fn test_binomial_end_points() { + let mut rng = crate::test::rng(352); + assert_eq!(rng.sample(Binomial::new(20, 0.0).unwrap()), 0); + assert_eq!(rng.sample(Binomial::new(20, 1.0).unwrap()), 20); + } + + #[test] + #[should_panic] + fn test_binomial_invalid_lambda_neg() { + Binomial::new(20, -10.0).unwrap(); + } +} diff --git a/rand/rand_distr/src/cauchy.rs b/rand/rand_distr/src/cauchy.rs new file mode 100644 index 0000000..6b0e7c6 --- /dev/null +++ b/rand/rand_distr/src/cauchy.rs @@ -0,0 +1,120 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2016-2017 The Rust Project Developers. +// +// 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. + +//! The Cauchy distribution. + +use rand::Rng; +use crate::{Distribution, Standard}; +use crate::utils::Float; + +/// The Cauchy distribution `Cauchy(median, scale)`. +/// +/// This distribution has a density function: +/// `f(x) = 1 / (pi * scale * (1 + ((x - median) / scale)^2))` +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Cauchy, Distribution}; +/// +/// let cau = Cauchy::new(2.0, 5.0).unwrap(); +/// let v = cau.sample(&mut rand::thread_rng()); +/// println!("{} is from a Cauchy(2, 5) distribution", v); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Cauchy<N> { + median: N, + scale: N, +} + +/// Error type returned from `Cauchy::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `scale <= 0` or `nan`. + ScaleTooSmall, +} + +impl<N: Float> Cauchy<N> +where Standard: Distribution<N> +{ + /// Construct a new `Cauchy` with the given shape parameters + /// `median` the peak location and `scale` the scale factor. + pub fn new(median: N, scale: N) -> Result<Cauchy<N>, Error> { + if !(scale > N::from(0.0)) { + return Err(Error::ScaleTooSmall); + } + Ok(Cauchy { + median, + scale + }) + } +} + +impl<N: Float> Distribution<N> for Cauchy<N> +where Standard: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + // sample from [0, 1) + let x = Standard.sample(rng); + // get standard cauchy random number + // note that π/2 is not exactly representable, even if x=0.5 the result is finite + let comp_dev = (N::pi() * x).tan(); + // shift and scale according to parameters + self.median + self.scale * comp_dev + } +} + +#[cfg(test)] +mod test { + use crate::Distribution; + use super::Cauchy; + + fn median(mut numbers: &mut [f64]) -> f64 { + sort(&mut numbers); + let mid = numbers.len() / 2; + numbers[mid] + } + + fn sort(numbers: &mut [f64]) { + numbers.sort_by(|a, b| a.partial_cmp(b).unwrap()); + } + + #[test] + fn test_cauchy_averages() { + // NOTE: given that the variance and mean are undefined, + // this test does not have any rigorous statistical meaning. + let cauchy = Cauchy::new(10.0, 5.0).unwrap(); + let mut rng = crate::test::rng(123); + let mut numbers: [f64; 1000] = [0.0; 1000]; + let mut sum = 0.0; + for i in 0..1000 { + numbers[i] = cauchy.sample(&mut rng); + sum += numbers[i]; + } + let median = median(&mut numbers); + println!("Cauchy median: {}", median); + assert!((median - 10.0).abs() < 0.4); // not 100% certain, but probable enough + let mean = sum / 1000.0; + println!("Cauchy mean: {}", mean); + // for a Cauchy distribution the mean should not converge + assert!((mean - 10.0).abs() > 0.4); // not 100% certain, but probable enough + } + + #[test] + #[should_panic] + fn test_cauchy_invalid_scale_zero() { + Cauchy::new(0.0, 0.0).unwrap(); + } + + #[test] + #[should_panic] + fn test_cauchy_invalid_scale_neg() { + Cauchy::new(0.0, -10.0).unwrap(); + } +} diff --git a/rand/rand_distr/src/dirichlet.rs b/rand/rand_distr/src/dirichlet.rs new file mode 100644 index 0000000..71cf73c --- /dev/null +++ b/rand/rand_distr/src/dirichlet.rs @@ -0,0 +1,154 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2013 The Rust Project Developers. +// +// 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. + +//! The dirichlet distribution. + +use rand::Rng; +use crate::{Distribution, Gamma, StandardNormal, Exp1, Open01}; +use crate::utils::Float; + +/// The dirichelet distribution `Dirichlet(alpha)`. +/// +/// The Dirichlet distribution is a family of continuous multivariate +/// probability distributions parameterized by a vector alpha of positive reals. +/// It is a multivariate generalization of the beta distribution. +/// +/// # Example +/// +/// ``` +/// use rand::prelude::*; +/// use rand_distr::Dirichlet; +/// +/// let dirichlet = Dirichlet::new(vec![1.0, 2.0, 3.0]).unwrap(); +/// let samples = dirichlet.sample(&mut rand::thread_rng()); +/// println!("{:?} is from a Dirichlet([1.0, 2.0, 3.0]) distribution", samples); +/// ``` +#[derive(Clone, Debug)] +pub struct Dirichlet<N> { + /// Concentration parameters (alpha) + alpha: Vec<N>, +} + +/// Error type returned from `Dirchlet::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `alpha.len() < 2`. + AlphaTooShort, + /// `alpha <= 0.0` or `nan`. + AlphaTooSmall, + /// `size < 2`. + SizeTooSmall, +} + +impl<N: Float> Dirichlet<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Construct a new `Dirichlet` with the given alpha parameter `alpha`. + /// + /// Requires `alpha.len() >= 2`. + #[inline] + pub fn new<V: Into<Vec<N>>>(alpha: V) -> Result<Dirichlet<N>, Error> { + let a = alpha.into(); + if a.len() < 2 { + return Err(Error::AlphaTooShort); + } + for &ai in &a { + if !(ai > N::from(0.0)) { + return Err(Error::AlphaTooSmall); + } + } + + Ok(Dirichlet { alpha: a }) + } + + /// Construct a new `Dirichlet` with the given shape parameter `alpha` and `size`. + /// + /// Requires `size >= 2`. + #[inline] + pub fn new_with_size(alpha: N, size: usize) -> Result<Dirichlet<N>, Error> { + if !(alpha > N::from(0.0)) { + return Err(Error::AlphaTooSmall); + } + if size < 2 { + return Err(Error::SizeTooSmall); + } + Ok(Dirichlet { + alpha: vec![alpha; size], + }) + } +} + +impl<N: Float> Distribution<Vec<N>> for Dirichlet<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Vec<N> { + let n = self.alpha.len(); + let mut samples = vec![N::from(0.0); n]; + let mut sum = N::from(0.0); + + for (s, &a) in samples.iter_mut().zip(self.alpha.iter()) { + let g = Gamma::new(a, N::from(1.0)).unwrap(); + *s = g.sample(rng); + sum += *s; + } + let invacc = N::from(1.0) / sum; + for s in samples.iter_mut() { + *s *= invacc; + } + samples + } +} + +#[cfg(test)] +mod test { + use super::Dirichlet; + use crate::Distribution; + + #[test] + fn test_dirichlet() { + let d = Dirichlet::new(vec![1.0, 2.0, 3.0]).unwrap(); + let mut rng = crate::test::rng(221); + let samples = d.sample(&mut rng); + let _: Vec<f64> = samples + .into_iter() + .map(|x| { + assert!(x > 0.0); + x + }) + .collect(); + } + + #[test] + fn test_dirichlet_with_param() { + let alpha = 0.5f64; + let size = 2; + let d = Dirichlet::new_with_size(alpha, size).unwrap(); + let mut rng = crate::test::rng(221); + let samples = d.sample(&mut rng); + let _: Vec<f64> = samples + .into_iter() + .map(|x| { + assert!(x > 0.0); + x + }) + .collect(); + } + + #[test] + #[should_panic] + fn test_dirichlet_invalid_length() { + Dirichlet::new_with_size(0.5f64, 1).unwrap(); + } + + #[test] + #[should_panic] + fn test_dirichlet_invalid_alpha() { + Dirichlet::new_with_size(0.0f64, 2).unwrap(); + } +} diff --git a/rand/rand_distr/src/exponential.rs b/rand/rand_distr/src/exponential.rs new file mode 100644 index 0000000..8322489 --- /dev/null +++ b/rand/rand_distr/src/exponential.rs @@ -0,0 +1,145 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2013 The Rust Project Developers. +// +// 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. + +//! The exponential distribution. + +use rand::Rng; +use crate::{ziggurat_tables, Distribution}; +use crate::utils::{ziggurat, Float}; + +/// Samples floating-point numbers according to the exponential distribution, +/// with rate parameter `λ = 1`. This is equivalent to `Exp::new(1.0)` or +/// sampling with `-rng.gen::<f64>().ln()`, but faster. +/// +/// See `Exp` for the general exponential distribution. +/// +/// Implemented via the ZIGNOR variant[^1] of the Ziggurat method. The exact +/// description in the paper was adjusted to use tables for the exponential +/// distribution rather than normal. +/// +/// [^1]: Jurgen A. Doornik (2005). [*An Improved Ziggurat Method to +/// Generate Normal Random Samples*]( +/// https://www.doornik.com/research/ziggurat.pdf). +/// Nuffield College, Oxford +/// +/// # Example +/// ``` +/// use rand::prelude::*; +/// use rand_distr::Exp1; +/// +/// let val: f64 = thread_rng().sample(Exp1); +/// println!("{}", val); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Exp1; + +impl Distribution<f32> for Exp1 { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f32 { + // TODO: use optimal 32-bit implementation + let x: f64 = self.sample(rng); + x as f32 + } +} + +// This could be done via `-rng.gen::<f64>().ln()` but that is slower. +impl Distribution<f64> for Exp1 { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 { + #[inline] + fn pdf(x: f64) -> f64 { + (-x).exp() + } + #[inline] + fn zero_case<R: Rng + ?Sized>(rng: &mut R, _u: f64) -> f64 { + ziggurat_tables::ZIG_EXP_R - rng.gen::<f64>().ln() + } + + ziggurat(rng, false, + &ziggurat_tables::ZIG_EXP_X, + &ziggurat_tables::ZIG_EXP_F, + pdf, zero_case) + } +} + +/// The exponential distribution `Exp(lambda)`. +/// +/// This distribution has density function: `f(x) = lambda * exp(-lambda * x)` +/// for `x > 0`. +/// +/// Note that [`Exp1`](crate::Exp1) is an optimised implementation for `lambda = 1`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Exp, Distribution}; +/// +/// let exp = Exp::new(2.0).unwrap(); +/// let v = exp.sample(&mut rand::thread_rng()); +/// println!("{} is from a Exp(2) distribution", v); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Exp<N> { + /// `lambda` stored as `1/lambda`, since this is what we scale by. + lambda_inverse: N +} + +/// Error type returned from `Exp::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `lambda <= 0` or `nan`. + LambdaTooSmall, +} + +impl<N: Float> Exp<N> +where Exp1: Distribution<N> +{ + /// Construct a new `Exp` with the given shape parameter + /// `lambda`. + #[inline] + pub fn new(lambda: N) -> Result<Exp<N>, Error> { + if !(lambda > N::from(0.0)) { + return Err(Error::LambdaTooSmall); + } + Ok(Exp { lambda_inverse: N::from(1.0) / lambda }) + } +} + +impl<N: Float> Distribution<N> for Exp<N> +where Exp1: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + rng.sample(Exp1) * self.lambda_inverse + } +} + +#[cfg(test)] +mod test { + use crate::Distribution; + use super::Exp; + + #[test] + fn test_exp() { + let exp = Exp::new(10.0).unwrap(); + let mut rng = crate::test::rng(221); + for _ in 0..1000 { + assert!(exp.sample(&mut rng) >= 0.0); + } + } + #[test] + #[should_panic] + fn test_exp_invalid_lambda_zero() { + Exp::new(0.0).unwrap(); + } + #[test] + #[should_panic] + fn test_exp_invalid_lambda_neg() { + Exp::new(-10.0).unwrap(); + } +} diff --git a/rand/rand_distr/src/gamma.rs b/rand/rand_distr/src/gamma.rs new file mode 100644 index 0000000..4018361 --- /dev/null +++ b/rand/rand_distr/src/gamma.rs @@ -0,0 +1,485 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2013 The Rust Project Developers. +// +// 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. + +//! The Gamma and derived distributions. + +use self::GammaRepr::*; +use self::ChiSquaredRepr::*; + +use rand::Rng; +use crate::normal::StandardNormal; +use crate::{Distribution, Exp1, Exp, Open01}; +use crate::utils::Float; + +/// The Gamma distribution `Gamma(shape, scale)` distribution. +/// +/// The density function of this distribution is +/// +/// ```text +/// f(x) = x^(k - 1) * exp(-x / θ) / (Γ(k) * θ^k) +/// ``` +/// +/// where `Γ` is the Gamma function, `k` is the shape and `θ` is the +/// scale and both `k` and `θ` are strictly positive. +/// +/// The algorithm used is that described by Marsaglia & Tsang 2000[^1], +/// falling back to directly sampling from an Exponential for `shape +/// == 1`, and using the boosting technique described in that paper for +/// `shape < 1`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Distribution, Gamma}; +/// +/// let gamma = Gamma::new(2.0, 5.0).unwrap(); +/// let v = gamma.sample(&mut rand::thread_rng()); +/// println!("{} is from a Gamma(2, 5) distribution", v); +/// ``` +/// +/// [^1]: George Marsaglia and Wai Wan Tsang. 2000. "A Simple Method for +/// Generating Gamma Variables" *ACM Trans. Math. Softw.* 26, 3 +/// (September 2000), 363-372. +/// DOI:[10.1145/358407.358414](https://doi.acm.org/10.1145/358407.358414) +#[derive(Clone, Copy, Debug)] +pub struct Gamma<N> { + repr: GammaRepr<N>, +} + +/// Error type returned from `Gamma::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `shape <= 0` or `nan`. + ShapeTooSmall, + /// `scale <= 0` or `nan`. + ScaleTooSmall, + /// `1 / scale == 0`. + ScaleTooLarge, +} + +#[derive(Clone, Copy, Debug)] +enum GammaRepr<N> { + Large(GammaLargeShape<N>), + One(Exp<N>), + Small(GammaSmallShape<N>) +} + +// These two helpers could be made public, but saving the +// match-on-Gamma-enum branch from using them directly (e.g. if one +// knows that the shape is always > 1) doesn't appear to be much +// faster. + +/// Gamma distribution where the shape parameter is less than 1. +/// +/// Note, samples from this require a compulsory floating-point `pow` +/// call, which makes it significantly slower than sampling from a +/// gamma distribution where the shape parameter is greater than or +/// equal to 1. +/// +/// See `Gamma` for sampling from a Gamma distribution with general +/// shape parameters. +#[derive(Clone, Copy, Debug)] +struct GammaSmallShape<N> { + inv_shape: N, + large_shape: GammaLargeShape<N> +} + +/// Gamma distribution where the shape parameter is larger than 1. +/// +/// See `Gamma` for sampling from a Gamma distribution with general +/// shape parameters. +#[derive(Clone, Copy, Debug)] +struct GammaLargeShape<N> { + scale: N, + c: N, + d: N +} + +impl<N: Float> Gamma<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Construct an object representing the `Gamma(shape, scale)` + /// distribution. + #[inline] + pub fn new(shape: N, scale: N) -> Result<Gamma<N>, Error> { + if !(shape > N::from(0.0)) { + return Err(Error::ShapeTooSmall); + } + if !(scale > N::from(0.0)) { + return Err(Error::ScaleTooSmall); + } + + let repr = if shape == N::from(1.0) { + One(Exp::new(N::from(1.0) / scale).map_err(|_| Error::ScaleTooLarge)?) + } else if shape < N::from(1.0) { + Small(GammaSmallShape::new_raw(shape, scale)) + } else { + Large(GammaLargeShape::new_raw(shape, scale)) + }; + Ok(Gamma { repr }) + } +} + +impl<N: Float> GammaSmallShape<N> +where StandardNormal: Distribution<N>, Open01: Distribution<N> +{ + fn new_raw(shape: N, scale: N) -> GammaSmallShape<N> { + GammaSmallShape { + inv_shape: N::from(1.0) / shape, + large_shape: GammaLargeShape::new_raw(shape + N::from(1.0), scale) + } + } +} + +impl<N: Float> GammaLargeShape<N> +where StandardNormal: Distribution<N>, Open01: Distribution<N> +{ + fn new_raw(shape: N, scale: N) -> GammaLargeShape<N> { + let d = shape - N::from(1. / 3.); + GammaLargeShape { + scale, + c: N::from(1.0) / (N::from(9.) * d).sqrt(), + d + } + } +} + +impl<N: Float> Distribution<N> for Gamma<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + match self.repr { + Small(ref g) => g.sample(rng), + One(ref g) => g.sample(rng), + Large(ref g) => g.sample(rng), + } + } +} +impl<N: Float> Distribution<N> for GammaSmallShape<N> +where StandardNormal: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let u: N = rng.sample(Open01); + + self.large_shape.sample(rng) * u.powf(self.inv_shape) + } +} +impl<N: Float> Distribution<N> for GammaLargeShape<N> +where StandardNormal: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + // Marsaglia & Tsang method, 2000 + loop { + let x: N = rng.sample(StandardNormal); + let v_cbrt = N::from(1.0) + self.c * x; + if v_cbrt <= N::from(0.0) { // a^3 <= 0 iff a <= 0 + continue + } + + let v = v_cbrt * v_cbrt * v_cbrt; + let u: N = rng.sample(Open01); + + let x_sqr = x * x; + if u < N::from(1.0) - N::from(0.0331) * x_sqr * x_sqr || + u.ln() < N::from(0.5) * x_sqr + self.d * (N::from(1.0) - v + v.ln()) + { + return self.d * v * self.scale + } + } + } +} + +/// The chi-squared distribution `χ²(k)`, where `k` is the degrees of +/// freedom. +/// +/// For `k > 0` integral, this distribution is the sum of the squares +/// of `k` independent standard normal random variables. For other +/// `k`, this uses the equivalent characterisation +/// `χ²(k) = Gamma(k/2, 2)`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{ChiSquared, Distribution}; +/// +/// let chi = ChiSquared::new(11.0).unwrap(); +/// let v = chi.sample(&mut rand::thread_rng()); +/// println!("{} is from a χ²(11) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct ChiSquared<N> { + repr: ChiSquaredRepr<N>, +} + +/// Error type returned from `ChiSquared::new` and `StudentT::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum ChiSquaredError { + /// `0.5 * k <= 0` or `nan`. + DoFTooSmall, +} + +#[derive(Clone, Copy, Debug)] +enum ChiSquaredRepr<N> { + // k == 1, Gamma(alpha, ..) is particularly slow for alpha < 1, + // e.g. when alpha = 1/2 as it would be for this case, so special- + // casing and using the definition of N(0,1)^2 is faster. + DoFExactlyOne, + DoFAnythingElse(Gamma<N>), +} + +impl<N: Float> ChiSquared<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Create a new chi-squared distribution with degrees-of-freedom + /// `k`. + pub fn new(k: N) -> Result<ChiSquared<N>, ChiSquaredError> { + let repr = if k == N::from(1.0) { + DoFExactlyOne + } else { + if !(N::from(0.5) * k > N::from(0.0)) { + return Err(ChiSquaredError::DoFTooSmall); + } + DoFAnythingElse(Gamma::new(N::from(0.5) * k, N::from(2.0)).unwrap()) + }; + Ok(ChiSquared { repr }) + } +} +impl<N: Float> Distribution<N> for ChiSquared<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + match self.repr { + DoFExactlyOne => { + // k == 1 => N(0,1)^2 + let norm: N = rng.sample(StandardNormal); + norm * norm + } + DoFAnythingElse(ref g) => g.sample(rng) + } + } +} + +/// The Fisher F distribution `F(m, n)`. +/// +/// This distribution is equivalent to the ratio of two normalised +/// chi-squared distributions, that is, `F(m,n) = (χ²(m)/m) / +/// (χ²(n)/n)`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{FisherF, Distribution}; +/// +/// let f = FisherF::new(2.0, 32.0).unwrap(); +/// let v = f.sample(&mut rand::thread_rng()); +/// println!("{} is from an F(2, 32) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct FisherF<N> { + numer: ChiSquared<N>, + denom: ChiSquared<N>, + // denom_dof / numer_dof so that this can just be a straight + // multiplication, rather than a division. + dof_ratio: N, +} + +/// Error type returned from `FisherF::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum FisherFError { + /// `m <= 0` or `nan`. + MTooSmall, + /// `n <= 0` or `nan`. + NTooSmall, +} + +impl<N: Float> FisherF<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Create a new `FisherF` distribution, with the given parameter. + pub fn new(m: N, n: N) -> Result<FisherF<N>, FisherFError> { + if !(m > N::from(0.0)) { + return Err(FisherFError::MTooSmall); + } + if !(n > N::from(0.0)) { + return Err(FisherFError::NTooSmall); + } + + Ok(FisherF { + numer: ChiSquared::new(m).unwrap(), + denom: ChiSquared::new(n).unwrap(), + dof_ratio: n / m + }) + } +} +impl<N: Float> Distribution<N> for FisherF<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + self.numer.sample(rng) / self.denom.sample(rng) * self.dof_ratio + } +} + +/// The Student t distribution, `t(nu)`, where `nu` is the degrees of +/// freedom. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{StudentT, Distribution}; +/// +/// let t = StudentT::new(11.0).unwrap(); +/// let v = t.sample(&mut rand::thread_rng()); +/// println!("{} is from a t(11) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct StudentT<N> { + chi: ChiSquared<N>, + dof: N +} + +impl<N: Float> StudentT<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Create a new Student t distribution with `n` degrees of + /// freedom. + pub fn new(n: N) -> Result<StudentT<N>, ChiSquaredError> { + Ok(StudentT { + chi: ChiSquared::new(n)?, + dof: n + }) + } +} +impl<N: Float> Distribution<N> for StudentT<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let norm: N = rng.sample(StandardNormal); + norm * (self.dof / self.chi.sample(rng)).sqrt() + } +} + +/// The Beta distribution with shape parameters `alpha` and `beta`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Distribution, Beta}; +/// +/// let beta = Beta::new(2.0, 5.0).unwrap(); +/// let v = beta.sample(&mut rand::thread_rng()); +/// println!("{} is from a Beta(2, 5) distribution", v); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Beta<N> { + gamma_a: Gamma<N>, + gamma_b: Gamma<N>, +} + +/// Error type returned from `Beta::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum BetaError { + /// `alpha <= 0` or `nan`. + AlphaTooSmall, + /// `beta <= 0` or `nan`. + BetaTooSmall, +} + +impl<N: Float> Beta<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Construct an object representing the `Beta(alpha, beta)` + /// distribution. + pub fn new(alpha: N, beta: N) -> Result<Beta<N>, BetaError> { + Ok(Beta { + gamma_a: Gamma::new(alpha, N::from(1.)) + .map_err(|_| BetaError::AlphaTooSmall)?, + gamma_b: Gamma::new(beta, N::from(1.)) + .map_err(|_| BetaError::BetaTooSmall)?, + }) + } +} + +impl<N: Float> Distribution<N> for Beta<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let x = self.gamma_a.sample(rng); + let y = self.gamma_b.sample(rng); + x / (x + y) + } +} + +#[cfg(test)] +mod test { + use crate::Distribution; + use super::{Beta, ChiSquared, StudentT, FisherF}; + + #[test] + fn test_chi_squared_one() { + let chi = ChiSquared::new(1.0).unwrap(); + let mut rng = crate::test::rng(201); + for _ in 0..1000 { + chi.sample(&mut rng); + } + } + #[test] + fn test_chi_squared_small() { + let chi = ChiSquared::new(0.5).unwrap(); + let mut rng = crate::test::rng(202); + for _ in 0..1000 { + chi.sample(&mut rng); + } + } + #[test] + fn test_chi_squared_large() { + let chi = ChiSquared::new(30.0).unwrap(); + let mut rng = crate::test::rng(203); + for _ in 0..1000 { + chi.sample(&mut rng); + } + } + #[test] + #[should_panic] + fn test_chi_squared_invalid_dof() { + ChiSquared::new(-1.0).unwrap(); + } + + #[test] + fn test_f() { + let f = FisherF::new(2.0, 32.0).unwrap(); + let mut rng = crate::test::rng(204); + for _ in 0..1000 { + f.sample(&mut rng); + } + } + + #[test] + fn test_t() { + let t = StudentT::new(11.0).unwrap(); + let mut rng = crate::test::rng(205); + for _ in 0..1000 { + t.sample(&mut rng); + } + } + + #[test] + fn test_beta() { + let beta = Beta::new(1.0, 2.0).unwrap(); + let mut rng = crate::test::rng(201); + for _ in 0..1000 { + beta.sample(&mut rng); + } + } + + #[test] + #[should_panic] + fn test_beta_invalid_dof() { + Beta::new(0., 0.).unwrap(); + } +} diff --git a/rand/rand_distr/src/lib.rs b/rand/rand_distr/src/lib.rs new file mode 100644 index 0000000..baf65ed --- /dev/null +++ b/rand/rand_distr/src/lib.rs @@ -0,0 +1,134 @@ +// Copyright 2019 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. + +#![doc(html_logo_url = "https://www.rust-lang.org/logos/rust-logo-128x128-blk.png", + html_favicon_url = "https://www.rust-lang.org/favicon.ico", + html_root_url = "https://rust-random.github.io/rand/")] + +#![deny(missing_docs)] +#![deny(missing_debug_implementations)] + +#![allow(clippy::excessive_precision, clippy::float_cmp, clippy::unreadable_literal)] +#![allow(clippy::neg_cmp_op_on_partial_ord)] // suggested fix too verbose + +//! Generating random samples from probability distributions. +//! +//! ## Re-exports +//! +//! This crate is a super-set of the [`rand::distributions`] module. See the +//! [`rand::distributions`] module documentation for an overview of the core +//! [`Distribution`] trait and implementations. +//! +//! The following are re-exported: +//! +//! - The [`Distribution`] trait and [`DistIter`] helper type +//! - The [`Standard`], [`Alphanumeric`], [`Uniform`], [`OpenClosed01`], [`Open01`] and [`Bernoulli`] distributions +//! - The [`weighted`] sub-module +//! +//! ## Distributions +//! +//! This crate provides the following probability distributions: +//! +//! - Related to real-valued quantities that grow linearly +//! (e.g. errors, offsets): +//! - [`Normal`] distribution, and [`StandardNormal`] as a primitive +//! - [`Cauchy`] distribution +//! - Related to Bernoulli trials (yes/no events, with a given probability): +//! - [`Binomial`] distribution +//! - Related to positive real-valued quantities that grow exponentially +//! (e.g. prices, incomes, populations): +//! - [`LogNormal`] distribution +//! - Related to the occurrence of independent events at a given rate: +//! - [`Pareto`] distribution +//! - [`Poisson`] distribution +//! - [`Exp`]onential distribution, and [`Exp1`] as a primitive +//! - [`Weibull`] distribution +//! - Gamma and derived distributions: +//! - [`Gamma`] distribution +//! - [`ChiSquared`] distribution +//! - [`StudentT`] distribution +//! - [`FisherF`] distribution +//! - Triangular distribution: +//! - [`Beta`] distribution +//! - [`Triangular`] distribution +//! - Multivariate probability distributions +//! - [`Dirichlet`] distribution +//! - [`UnitSphere`] distribution +//! - [`UnitBall`] distribution +//! - [`UnitCircle`] distribution +//! - [`UnitDisc`] distribution + +pub use rand::distributions::{Distribution, DistIter, Standard, + Alphanumeric, Uniform, OpenClosed01, Open01, Bernoulli, uniform, weighted}; + +pub use self::unit_sphere::UnitSphere; +pub use self::unit_ball::UnitBall; +pub use self::unit_circle::UnitCircle; +pub use self::unit_disc::UnitDisc; +pub use self::gamma::{Gamma, Error as GammaError, ChiSquared, ChiSquaredError, + FisherF, FisherFError, StudentT, Beta, BetaError}; +pub use self::normal::{Normal, Error as NormalError, LogNormal, StandardNormal}; +pub use self::exponential::{Exp, Error as ExpError, Exp1}; +pub use self::pareto::{Pareto, Error as ParetoError}; +pub use self::pert::{Pert, PertError}; +pub use self::poisson::{Poisson, Error as PoissonError}; +pub use self::binomial::{Binomial, Error as BinomialError}; +pub use self::cauchy::{Cauchy, Error as CauchyError}; +pub use self::dirichlet::{Dirichlet, Error as DirichletError}; +pub use self::triangular::{Triangular, TriangularError}; +pub use self::weibull::{Weibull, Error as WeibullError}; +pub use self::utils::Float; + +mod unit_sphere; +mod unit_ball; +mod unit_circle; +mod unit_disc; +mod gamma; +mod normal; +mod exponential; +mod pareto; +mod pert; +mod poisson; +mod binomial; +mod cauchy; +mod dirichlet; +mod triangular; +mod weibull; +mod utils; +mod ziggurat_tables; + +#[cfg(test)] +mod test { + // Notes on testing + // + // Testing random number distributions correctly is hard. The following + // testing is desired: + // + // - Construction: test initialisation with a few valid parameter sets. + // - Erroneous usage: test that incorrect usage generates an error. + // - Vector: test that usage with fixed inputs (including RNG) generates a + // fixed output sequence on all platforms. + // - Correctness at fixed points (optional): using a specific mock RNG, + // check that specific values are sampled (e.g. end-points and median of + // distribution). + // - Correctness of PDF (extra): generate a histogram of samples within a + // certain range, and check this approximates the PDF. These tests are + // expected to be expensive, and should be behind a feature-gate. + // + // TODO: Vector and correctness tests are largely absent so far. + // NOTE: Some distributions have tests checking only that samples can be + // generated. This is redundant with vector and correctness tests. + + /// Construct a deterministic RNG with the given seed + pub fn rng(seed: u64) -> impl rand::RngCore { + // For tests, we want a statistically good, fast, reproducible RNG. + // PCG32 will do fine, and will be easy to embed if we ever need to. + const INC: u64 = 11634580027462260723; + rand_pcg::Pcg32::new(seed, INC) + } +} diff --git a/rand/rand_distr/src/normal.rs b/rand/rand_distr/src/normal.rs new file mode 100644 index 0000000..882754f --- /dev/null +++ b/rand/rand_distr/src/normal.rs @@ -0,0 +1,219 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2013 The Rust Project Developers. +// +// 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. + +//! The normal and derived distributions. + +use rand::Rng; +use crate::{ziggurat_tables, Distribution, Open01}; +use crate::utils::{ziggurat, Float}; + +/// Samples floating-point numbers according to the normal distribution +/// `N(0, 1)` (a.k.a. a standard normal, or Gaussian). This is equivalent to +/// `Normal::new(0.0, 1.0)` but faster. +/// +/// See `Normal` for the general normal distribution. +/// +/// Implemented via the ZIGNOR variant[^1] of the Ziggurat method. +/// +/// [^1]: Jurgen A. Doornik (2005). [*An Improved Ziggurat Method to +/// Generate Normal Random Samples*]( +/// https://www.doornik.com/research/ziggurat.pdf). +/// Nuffield College, Oxford +/// +/// # Example +/// ``` +/// use rand::prelude::*; +/// use rand_distr::StandardNormal; +/// +/// let val: f64 = thread_rng().sample(StandardNormal); +/// println!("{}", val); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct StandardNormal; + +impl Distribution<f32> for StandardNormal { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f32 { + // TODO: use optimal 32-bit implementation + let x: f64 = self.sample(rng); + x as f32 + } +} + +impl Distribution<f64> for StandardNormal { + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 { + #[inline] + fn pdf(x: f64) -> f64 { + (-x*x/2.0).exp() + } + #[inline] + fn zero_case<R: Rng + ?Sized>(rng: &mut R, u: f64) -> f64 { + // compute a random number in the tail by hand + + // strange initial conditions, because the loop is not + // do-while, so the condition should be true on the first + // run, they get overwritten anyway (0 < 1, so these are + // good). + let mut x = 1.0f64; + let mut y = 0.0f64; + + while -2.0 * y < x * x { + let x_: f64 = rng.sample(Open01); + let y_: f64 = rng.sample(Open01); + + x = x_.ln() / ziggurat_tables::ZIG_NORM_R; + y = y_.ln(); + } + + if u < 0.0 { x - ziggurat_tables::ZIG_NORM_R } else { ziggurat_tables::ZIG_NORM_R - x } + } + + ziggurat(rng, true, // this is symmetric + &ziggurat_tables::ZIG_NORM_X, + &ziggurat_tables::ZIG_NORM_F, + pdf, zero_case) + } +} + +/// The normal distribution `N(mean, std_dev**2)`. +/// +/// This uses the ZIGNOR variant of the Ziggurat method, see [`StandardNormal`] +/// for more details. +/// +/// Note that [`StandardNormal`] is an optimised implementation for mean 0, and +/// standard deviation 1. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Normal, Distribution}; +/// +/// // mean 2, standard deviation 3 +/// let normal = Normal::new(2.0, 3.0).unwrap(); +/// let v = normal.sample(&mut rand::thread_rng()); +/// println!("{} is from a N(2, 9) distribution", v) +/// ``` +/// +/// [`StandardNormal`]: crate::StandardNormal +#[derive(Clone, Copy, Debug)] +pub struct Normal<N> { + mean: N, + std_dev: N, +} + +/// Error type returned from `Normal::new` and `LogNormal::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `std_dev < 0` or `nan`. + StdDevTooSmall, +} + +impl<N: Float> Normal<N> +where StandardNormal: Distribution<N> +{ + /// Construct a new `Normal` distribution with the given mean and + /// standard deviation. + #[inline] + pub fn new(mean: N, std_dev: N) -> Result<Normal<N>, Error> { + if !(std_dev >= N::from(0.0)) { + return Err(Error::StdDevTooSmall); + } + Ok(Normal { + mean, + std_dev + }) + } +} + +impl<N: Float> Distribution<N> for Normal<N> +where StandardNormal: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let n: N = rng.sample(StandardNormal); + self.mean + self.std_dev * n + } +} + + +/// The log-normal distribution `ln N(mean, std_dev**2)`. +/// +/// If `X` is log-normal distributed, then `ln(X)` is `N(mean, std_dev**2)` +/// distributed. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{LogNormal, Distribution}; +/// +/// // mean 2, standard deviation 3 +/// let log_normal = LogNormal::new(2.0, 3.0).unwrap(); +/// let v = log_normal.sample(&mut rand::thread_rng()); +/// println!("{} is from an ln N(2, 9) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct LogNormal<N> { + norm: Normal<N> +} + +impl<N: Float> LogNormal<N> +where StandardNormal: Distribution<N> +{ + /// Construct a new `LogNormal` distribution with the given mean + /// and standard deviation of the logarithm of the distribution. + #[inline] + pub fn new(mean: N, std_dev: N) -> Result<LogNormal<N>, Error> { + if !(std_dev >= N::from(0.0)) { + return Err(Error::StdDevTooSmall); + } + Ok(LogNormal { norm: Normal::new(mean, std_dev).unwrap() }) + } +} + +impl<N: Float> Distribution<N> for LogNormal<N> +where StandardNormal: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + self.norm.sample(rng).exp() + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::{Normal, LogNormal}; + + #[test] + fn test_normal() { + let norm = Normal::new(10.0, 10.0).unwrap(); + let mut rng = crate::test::rng(210); + for _ in 0..1000 { + norm.sample(&mut rng); + } + } + #[test] + #[should_panic] + fn test_normal_invalid_sd() { + Normal::new(10.0, -1.0).unwrap(); + } + + + #[test] + fn test_log_normal() { + let lnorm = LogNormal::new(10.0, 10.0).unwrap(); + let mut rng = crate::test::rng(211); + for _ in 0..1000 { + lnorm.sample(&mut rng); + } + } + #[test] + #[should_panic] + fn test_log_normal_invalid_sd() { + LogNormal::new(10.0, -1.0).unwrap(); + } +} diff --git a/rand/rand_distr/src/pareto.rs b/rand/rand_distr/src/pareto.rs new file mode 100644 index 0000000..33ea382 --- /dev/null +++ b/rand/rand_distr/src/pareto.rs @@ -0,0 +1,89 @@ +// 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. + +//! The Pareto distribution. + +use rand::Rng; +use crate::{Distribution, OpenClosed01}; +use crate::utils::Float; + +/// Samples floating-point numbers according to the Pareto distribution +/// +/// # Example +/// ``` +/// use rand::prelude::*; +/// use rand_distr::Pareto; +/// +/// let val: f64 = thread_rng().sample(Pareto::new(1., 2.).unwrap()); +/// println!("{}", val); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Pareto<N> { + scale: N, + inv_neg_shape: N, +} + +/// Error type returned from `Pareto::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `scale <= 0` or `nan`. + ScaleTooSmall, + /// `shape <= 0` or `nan`. + ShapeTooSmall, +} + +impl<N: Float> Pareto<N> +where OpenClosed01: Distribution<N> +{ + /// Construct a new Pareto distribution with given `scale` and `shape`. + /// + /// In the literature, `scale` is commonly written as x<sub>m</sub> or k and + /// `shape` is often written as α. + pub fn new(scale: N, shape: N) -> Result<Pareto<N>, Error> { + if !(scale > N::from(0.0)) { + return Err(Error::ScaleTooSmall); + } + if !(shape > N::from(0.0)) { + return Err(Error::ShapeTooSmall); + } + Ok(Pareto { scale, inv_neg_shape: N::from(-1.0) / shape }) + } +} + +impl<N: Float> Distribution<N> for Pareto<N> +where OpenClosed01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let u: N = OpenClosed01.sample(rng); + self.scale * u.powf(self.inv_neg_shape) + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::Pareto; + + #[test] + #[should_panic] + fn invalid() { + Pareto::new(0., 0.).unwrap(); + } + + #[test] + fn sample() { + let scale = 1.0; + let shape = 2.0; + let d = Pareto::new(scale, shape).unwrap(); + let mut rng = crate::test::rng(1); + for _ in 0..1000 { + let r = d.sample(&mut rng); + assert!(r >= scale); + } + } +} diff --git a/rand/rand_distr/src/pert.rs b/rand/rand_distr/src/pert.rs new file mode 100644 index 0000000..040cd05 --- /dev/null +++ b/rand/rand_distr/src/pert.rs @@ -0,0 +1,132 @@ +// 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. +//! The PERT distribution. + +use rand::Rng; +use crate::{Distribution, Beta, StandardNormal, Exp1, Open01}; +use crate::utils::Float; + +/// The PERT distribution. +/// +/// Similar to the [`Triangular`] distribution, the PERT distribution is +/// parameterised by a range and a mode within that range. Unlike the +/// [`Triangular`] distribution, the probability density function of the PERT +/// distribution is smooth, with a configurable weighting around the mode. +/// +/// # Example +/// +/// ```rust +/// use rand_distr::{Pert, Distribution}; +/// +/// let d = Pert::new(0., 5., 2.5).unwrap(); +/// let v = d.sample(&mut rand::thread_rng()); +/// println!("{} is from a PERT distribution", v); +/// ``` +/// +/// [`Triangular`]: crate::Triangular +#[derive(Clone, Copy, Debug)] +pub struct Pert<N> { + min: N, + range: N, + beta: Beta<N>, +} + +/// Error type returned from [`Pert`] constructors. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum PertError { + /// `max < min` or `min` or `max` is NaN. + RangeTooSmall, + /// `mode < min` or `mode > max` or `mode` is NaN. + ModeRange, + /// `shape < 0` or `shape` is NaN + ShapeTooSmall, +} + +impl<N: Float> Pert<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + /// Set up the PERT distribution with defined `min`, `max` and `mode`. + /// + /// This is equivalent to calling `Pert::new_shape` with `shape == 4.0`. + #[inline] + pub fn new(min: N, max: N, mode: N) -> Result<Pert<N>, PertError> { + Pert::new_with_shape(min, max, mode, N::from(4.)) + } + + /// Set up the PERT distribution with defined `min`, `max`, `mode` and + /// `shape`. + pub fn new_with_shape(min: N, max: N, mode: N, shape: N) -> Result<Pert<N>, PertError> { + if !(max > min) { + return Err(PertError::RangeTooSmall); + } + if !(mode >= min && max >= mode) { + return Err(PertError::ModeRange); + } + if !(shape >= N::from(0.)) { + return Err(PertError::ShapeTooSmall); + } + + let range = max - min; + let mu = (min + max + shape * mode) / (shape + N::from(2.)); + let v = if mu == mode { + shape * N::from(0.5) + N::from(1.) + } else { + (mu - min) * (N::from(2.) * mode - min - max) + / ((mode - mu) * (max - min)) + }; + let w = v * (max - mu) / (mu - min); + let beta = Beta::new(v, w).map_err(|_| PertError::RangeTooSmall)?; + Ok(Pert{ min, range, beta }) + } +} + +impl<N: Float> Distribution<N> for Pert<N> +where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N> +{ + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + self.beta.sample(rng) * self.range + self.min + } +} + +#[cfg(test)] +mod test { + use std::f64; + use super::*; + + #[test] + fn test_pert() { + for &(min, max, mode) in &[ + (-1., 1., 0.), + (1., 2., 1.), + (5., 25., 25.), + ] { + let _distr = Pert::new(min, max, mode).unwrap(); + // TODO: test correctness + } + + for &(min, max, mode) in &[ + (-1., 1., 2.), + (-1., 1., -2.), + (2., 1., 1.), + ] { + assert!(Pert::new(min, max, mode).is_err()); + } + } + + #[test] + fn value_stability() { + let rng = crate::test::rng(860); + let distr = Pert::new(2., 10., 3.).unwrap(); // mean = 4, var = 12/7 + let seq = distr.sample_iter(rng).take(5).collect::<Vec<f64>>(); + println!("seq: {:?}", seq); + let expected = vec![4.631484136029422, 3.307201472321789, + 3.29995019556348, 3.66835483991721, 3.514246139933899]; + assert!(seq == expected); + } +} diff --git a/rand/rand_distr/src/poisson.rs b/rand/rand_distr/src/poisson.rs new file mode 100644 index 0000000..4f4a0b7 --- /dev/null +++ b/rand/rand_distr/src/poisson.rs @@ -0,0 +1,233 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2016-2017 The Rust Project Developers. +// +// 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. + +//! The Poisson distribution. + +use rand::Rng; +use crate::{Distribution, Cauchy, Standard}; +use crate::utils::Float; + +/// The Poisson distribution `Poisson(lambda)`. +/// +/// This distribution has a density function: +/// `f(k) = lambda^k * exp(-lambda) / k!` for `k >= 0`. +/// +/// # Example +/// +/// ``` +/// use rand_distr::{Poisson, Distribution}; +/// +/// let poi = Poisson::new(2.0).unwrap(); +/// let v: u64 = poi.sample(&mut rand::thread_rng()); +/// println!("{} is from a Poisson(2) distribution", v); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Poisson<N> { + lambda: N, + // precalculated values + exp_lambda: N, + log_lambda: N, + sqrt_2lambda: N, + magic_val: N, +} + +/// Error type returned from `Poisson::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `lambda <= 0` or `nan`. + ShapeTooSmall, +} + +impl<N: Float> Poisson<N> +where Standard: Distribution<N> +{ + /// Construct a new `Poisson` with the given shape parameter + /// `lambda`. + pub fn new(lambda: N) -> Result<Poisson<N>, Error> { + if !(lambda > N::from(0.0)) { + return Err(Error::ShapeTooSmall); + } + let log_lambda = lambda.ln(); + Ok(Poisson { + lambda, + exp_lambda: (-lambda).exp(), + log_lambda, + sqrt_2lambda: (N::from(2.0) * lambda).sqrt(), + magic_val: lambda * log_lambda - (N::from(1.0) + lambda).log_gamma(), + }) + } +} + +impl<N: Float> Distribution<N> for Poisson<N> +where Standard: Distribution<N> +{ + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + // using the algorithm from Numerical Recipes in C + + // for low expected values use the Knuth method + if self.lambda < N::from(12.0) { + let mut result = N::from(0.); + let mut p = N::from(1.0); + while p > self.exp_lambda { + p *= rng.gen::<N>(); + result += N::from(1.); + } + result - N::from(1.) + } + // high expected values - rejection method + else { + // we use the Cauchy distribution as the comparison distribution + // f(x) ~ 1/(1+x^2) + let cauchy = Cauchy::new(N::from(0.0), N::from(1.0)).unwrap(); + let mut result; + + loop { + let mut comp_dev; + + loop { + // draw from the Cauchy distribution + comp_dev = rng.sample(cauchy); + // shift the peak of the comparison ditribution + result = self.sqrt_2lambda * comp_dev + self.lambda; + // repeat the drawing until we are in the range of possible values + if result >= N::from(0.0) { + break; + } + } + // now the result is a random variable greater than 0 with Cauchy distribution + // the result should be an integer value + result = result.floor(); + + // this is the ratio of the Poisson distribution to the comparison distribution + // the magic value scales the distribution function to a range of approximately 0-1 + // since it is not exact, we multiply the ratio by 0.9 to avoid ratios greater than 1 + // this doesn't change the resulting distribution, only increases the rate of failed drawings + let check = N::from(0.9) * (N::from(1.0) + comp_dev * comp_dev) + * (result * self.log_lambda - (N::from(1.0) + result).log_gamma() - self.magic_val).exp(); + + // check with uniform random value - if below the threshold, we are within the target distribution + if rng.gen::<N>() <= check { + break; + } + } + result + } + } +} + +impl<N: Float> Distribution<u64> for Poisson<N> +where Standard: Distribution<N> +{ + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 { + let result: N = self.sample(rng); + result.to_u64().unwrap() + } +} + +#[cfg(test)] +mod test { + use crate::Distribution; + use super::Poisson; + + #[test] + fn test_poisson_10() { + let poisson = Poisson::new(10.0).unwrap(); + let mut rng = crate::test::rng(123); + let mut sum_u64 = 0; + let mut sum_f64 = 0.; + for _ in 0..1000 { + let s_u64: u64 = poisson.sample(&mut rng); + let s_f64: f64 = poisson.sample(&mut rng); + sum_u64 += s_u64; + sum_f64 += s_f64; + } + let avg_u64 = (sum_u64 as f64) / 1000.0; + let avg_f64 = sum_f64 / 1000.0; + println!("Poisson averages: {} (u64) {} (f64)", avg_u64, avg_f64); + for &avg in &[avg_u64, avg_f64] { + assert!((avg - 10.0).abs() < 0.5); // not 100% certain, but probable enough + } + } + + #[test] + fn test_poisson_15() { + // Take the 'high expected values' path + let poisson = Poisson::new(15.0).unwrap(); + let mut rng = crate::test::rng(123); + let mut sum_u64 = 0; + let mut sum_f64 = 0.; + for _ in 0..1000 { + let s_u64: u64 = poisson.sample(&mut rng); + let s_f64: f64 = poisson.sample(&mut rng); + sum_u64 += s_u64; + sum_f64 += s_f64; + } + let avg_u64 = (sum_u64 as f64) / 1000.0; + let avg_f64 = sum_f64 / 1000.0; + println!("Poisson average: {} (u64) {} (f64)", avg_u64, avg_f64); + for &avg in &[avg_u64, avg_f64] { + assert!((avg - 15.0).abs() < 0.5); // not 100% certain, but probable enough + } + } + + #[test] + fn test_poisson_10_f32() { + let poisson = Poisson::new(10.0f32).unwrap(); + let mut rng = crate::test::rng(123); + let mut sum_u64 = 0; + let mut sum_f32 = 0.; + for _ in 0..1000 { + let s_u64: u64 = poisson.sample(&mut rng); + let s_f32: f32 = poisson.sample(&mut rng); + sum_u64 += s_u64; + sum_f32 += s_f32; + } + let avg_u64 = (sum_u64 as f32) / 1000.0; + let avg_f32 = sum_f32 / 1000.0; + println!("Poisson averages: {} (u64) {} (f32)", avg_u64, avg_f32); + for &avg in &[avg_u64, avg_f32] { + assert!((avg - 10.0).abs() < 0.5); // not 100% certain, but probable enough + } + } + + #[test] + fn test_poisson_15_f32() { + // Take the 'high expected values' path + let poisson = Poisson::new(15.0f32).unwrap(); + let mut rng = crate::test::rng(123); + let mut sum_u64 = 0; + let mut sum_f32 = 0.; + for _ in 0..1000 { + let s_u64: u64 = poisson.sample(&mut rng); + let s_f32: f32 = poisson.sample(&mut rng); + sum_u64 += s_u64; + sum_f32 += s_f32; + } + let avg_u64 = (sum_u64 as f32) / 1000.0; + let avg_f32 = sum_f32 / 1000.0; + println!("Poisson average: {} (u64) {} (f32)", avg_u64, avg_f32); + for &avg in &[avg_u64, avg_f32] { + assert!((avg - 15.0).abs() < 0.5); // not 100% certain, but probable enough + } + } + + #[test] + #[should_panic] + fn test_poisson_invalid_lambda_zero() { + Poisson::new(0.0).unwrap(); + } + + #[test] + #[should_panic] + fn test_poisson_invalid_lambda_neg() { + Poisson::new(-10.0).unwrap(); + } +} diff --git a/rand/rand_distr/src/triangular.rs b/rand/rand_distr/src/triangular.rs new file mode 100644 index 0000000..dd0bbfb --- /dev/null +++ b/rand/rand_distr/src/triangular.rs @@ -0,0 +1,125 @@ +// 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. +//! The triangular distribution. + +use rand::Rng; +use crate::{Distribution, Standard}; +use crate::utils::Float; + +/// The triangular distribution. +/// +/// A continuous probability distribution parameterised by a range, and a mode +/// (most likely value) within that range. +/// +/// The probability density function is triangular. For a similar distribution +/// with a smooth PDF, see the [`Pert`] distribution. +/// +/// # Example +/// +/// ```rust +/// use rand_distr::{Triangular, Distribution}; +/// +/// let d = Triangular::new(0., 5., 2.5).unwrap(); +/// let v = d.sample(&mut rand::thread_rng()); +/// println!("{} is from a triangular distribution", v); +/// ``` +/// +/// [`Pert`]: crate::Pert +#[derive(Clone, Copy, Debug)] +pub struct Triangular<N> { + min: N, + max: N, + mode: N, +} + +/// Error type returned from [`Triangular::new`]. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum TriangularError { + /// `max < min` or `min` or `max` is NaN. + RangeTooSmall, + /// `mode < min` or `mode > max` or `mode` is NaN. + ModeRange, +} + +impl<N: Float> Triangular<N> +where Standard: Distribution<N> +{ + /// Set up the Triangular distribution with defined `min`, `max` and `mode`. + #[inline] + pub fn new(min: N, max: N, mode: N) -> Result<Triangular<N>, TriangularError> { + if !(max >= min) { + return Err(TriangularError::RangeTooSmall); + } + if !(mode >= min && max >= mode) { + return Err(TriangularError::ModeRange); + } + Ok(Triangular { min, max, mode }) + } +} + +impl<N: Float> Distribution<N> for Triangular<N> +where Standard: Distribution<N> +{ + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let f: N = rng.sample(Standard); + let diff_mode_min = self.mode - self.min; + let range = self.max - self.min; + let f_range = f * range; + if f_range < diff_mode_min { + self.min + (f_range * diff_mode_min).sqrt() + } else { + self.max - ((range - f_range) * (self.max - self.mode)).sqrt() + } + } +} + +#[cfg(test)] +mod test { + use std::f64; + use rand::{Rng, rngs::mock}; + use super::*; + + #[test] + fn test_triangular() { + let mut half_rng = mock::StepRng::new(0x8000_0000_0000_0000, 0); + assert_eq!(half_rng.gen::<f64>(), 0.5); + for &(min, max, mode, median) in &[ + (-1., 1., 0., 0.), + (1., 2., 1., 2. - 0.5f64.sqrt()), + (5., 25., 25., 5. + 200f64.sqrt()), + (1e-5, 1e5, 1e-3, 1e5 - 4999999949.5f64.sqrt()), + (0., 1., 0.9, 0.45f64.sqrt()), + (-4., -0.5, -2., -4.0 + 3.5f64.sqrt()), + ] { + println!("{} {} {} {}", min, max, mode, median); + let distr = Triangular::new(min, max, mode).unwrap(); + // Test correct value at median: + assert_eq!(distr.sample(&mut half_rng), median); + } + + for &(min, max, mode) in &[ + (-1., 1., 2.), + (-1., 1., -2.), + (2., 1., 1.), + ] { + assert!(Triangular::new(min, max, mode).is_err()); + } + } + + #[test] + fn value_stability() { + let rng = crate::test::rng(860); + let distr = Triangular::new(2., 10., 3.).unwrap(); + let seq = distr.sample_iter(rng).take(5).collect::<Vec<f64>>(); + println!("seq: {:?}", seq); + let expected = vec![5.74373257511361, 7.890059162791258, + 4.7256280652553455, 2.9474808121184077, 3.058301946314053]; + assert!(seq == expected); + } +} diff --git a/rand/rand_distr/src/unit_ball.rs b/rand/rand_distr/src/unit_ball.rs new file mode 100644 index 0000000..9d61627 --- /dev/null +++ b/rand/rand_distr/src/unit_ball.rs @@ -0,0 +1,69 @@ +// Copyright 2019 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. + +use rand::Rng; +use crate::{Distribution, Uniform, uniform::SampleUniform}; +use crate::utils::Float; + +/// Samples uniformly from the unit ball (surface and interior) in three +/// dimensions. +/// +/// Implemented via rejection sampling. +/// +/// +/// # Example +/// +/// ``` +/// use rand_distr::{UnitBall, Distribution}; +/// +/// let v: [f64; 3] = UnitBall.sample(&mut rand::thread_rng()); +/// println!("{:?} is from the unit ball.", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct UnitBall; + +impl<N: Float + SampleUniform> Distribution<[N; 3]> for UnitBall { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> [N; 3] { + let uniform = Uniform::new(N::from(-1.), N::from(1.)); + let mut x1; + let mut x2; + let mut x3; + loop { + x1 = uniform.sample(rng); + x2 = uniform.sample(rng); + x3 = uniform.sample(rng); + if x1*x1 + x2*x2 + x3*x3 <= N::from(1.) { + break; + } + } + [x1, x2, x3] + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::UnitBall; + + #[test] + fn value_stability() { + let mut rng = crate::test::rng(2); + let expected = [ + [0.018035709265959987, -0.4348771383120438, -0.07982762085055706], + [0.10588569388223945, -0.4734350111375454, -0.7392104908825501], + [0.11060237642041049, -0.16065642822852677, -0.8444043930440075] + ]; + let samples: [[f64; 3]; 3] = [ + UnitBall.sample(&mut rng), + UnitBall.sample(&mut rng), + UnitBall.sample(&mut rng), + ]; + assert_eq!(samples, expected); + } +} diff --git a/rand/rand_distr/src/unit_circle.rs b/rand/rand_distr/src/unit_circle.rs new file mode 100644 index 0000000..5863a1a --- /dev/null +++ b/rand/rand_distr/src/unit_circle.rs @@ -0,0 +1,99 @@ +// 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. + +use rand::Rng; +use crate::{Distribution, Uniform, uniform::SampleUniform}; +use crate::utils::Float; + +/// Samples uniformly from the edge of the unit circle in two dimensions. +/// +/// Implemented via a method by von Neumann[^1]. +/// +/// +/// # Example +/// +/// ``` +/// use rand_distr::{UnitCircle, Distribution}; +/// +/// let v: [f64; 2] = UnitCircle.sample(&mut rand::thread_rng()); +/// println!("{:?} is from the unit circle.", v) +/// ``` +/// +/// [^1]: von Neumann, J. (1951) [*Various Techniques Used in Connection with +/// Random Digits.*](https://mcnp.lanl.gov/pdf_files/nbs_vonneumann.pdf) +/// NBS Appl. Math. Ser., No. 12. Washington, DC: U.S. Government Printing +/// Office, pp. 36-38. +#[derive(Clone, Copy, Debug)] +pub struct UnitCircle; + +impl<N: Float + SampleUniform> Distribution<[N; 2]> for UnitCircle { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> [N; 2] { + let uniform = Uniform::new(N::from(-1.), N::from(1.)); + let mut x1; + let mut x2; + let mut sum; + loop { + x1 = uniform.sample(rng); + x2 = uniform.sample(rng); + sum = x1*x1 + x2*x2; + if sum < N::from(1.) { + break; + } + } + let diff = x1*x1 - x2*x2; + [diff / sum, N::from(2.)*x1*x2 / sum] + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::UnitCircle; + + /// Assert that two numbers are almost equal to each other. + /// + /// On panic, this macro will print the values of the expressions with their + /// debug representations. + macro_rules! assert_almost_eq { + ($a:expr, $b:expr, $prec:expr) => ( + let diff = ($a - $b).abs(); + if diff > $prec { + panic!(format!( + "assertion failed: `abs(left - right) = {:.1e} < {:e}`, \ + (left: `{}`, right: `{}`)", + diff, $prec, $a, $b)); + } + ); + } + + #[test] + fn norm() { + let mut rng = crate::test::rng(1); + for _ in 0..1000 { + let x: [f64; 2] = UnitCircle.sample(&mut rng); + assert_almost_eq!(x[0]*x[0] + x[1]*x[1], 1., 1e-15); + } + } + + #[test] + fn value_stability() { + let mut rng = crate::test::rng(2); + let expected = [ + [-0.9965658683520504, -0.08280380447614634], + [-0.9790853270389644, -0.20345004884984505], + [-0.8449189758898707, 0.5348943112253227], + ]; + let samples: [[f64; 2]; 3] = [ + UnitCircle.sample(&mut rng), + UnitCircle.sample(&mut rng), + UnitCircle.sample(&mut rng), + ]; + assert_eq!(samples, expected); + } +} diff --git a/rand/rand_distr/src/unit_disc.rs b/rand/rand_distr/src/unit_disc.rs new file mode 100644 index 0000000..97abc2f --- /dev/null +++ b/rand/rand_distr/src/unit_disc.rs @@ -0,0 +1,66 @@ +// Copyright 2019 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. + +use rand::Rng; +use crate::{Distribution, Uniform, uniform::SampleUniform}; +use crate::utils::Float; + +/// Samples uniformly from the unit disc in two dimensions. +/// +/// Implemented via rejection sampling. +/// +/// +/// # Example +/// +/// ``` +/// use rand_distr::{UnitDisc, Distribution}; +/// +/// let v: [f64; 2] = UnitDisc.sample(&mut rand::thread_rng()); +/// println!("{:?} is from the unit Disc.", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct UnitDisc; + +impl<N: Float + SampleUniform> Distribution<[N; 2]> for UnitDisc { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> [N; 2] { + let uniform = Uniform::new(N::from(-1.), N::from(1.)); + let mut x1; + let mut x2; + loop { + x1 = uniform.sample(rng); + x2 = uniform.sample(rng); + if x1*x1 + x2*x2 <= N::from(1.) { + break; + } + } + [x1, x2] + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::UnitDisc; + + #[test] + fn value_stability() { + let mut rng = crate::test::rng(2); + let expected = [ + [0.018035709265959987, -0.4348771383120438], + [-0.07982762085055706, 0.7765329819820659], + [0.21450745997299503, 0.7398636984333291] + ]; + let samples: [[f64; 2]; 3] = [ + UnitDisc.sample(&mut rng), + UnitDisc.sample(&mut rng), + UnitDisc.sample(&mut rng), + ]; + assert_eq!(samples, expected); + } +} diff --git a/rand/rand_distr/src/unit_sphere.rs b/rand/rand_distr/src/unit_sphere.rs new file mode 100644 index 0000000..8e0c361 --- /dev/null +++ b/rand/rand_distr/src/unit_sphere.rs @@ -0,0 +1,94 @@ +// Copyright 2018-2019 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. + +use rand::Rng; +use crate::{Distribution, Uniform, uniform::SampleUniform}; +use crate::utils::Float; + +/// Samples uniformly from the surface of the unit sphere in three dimensions. +/// +/// Implemented via a method by Marsaglia[^1]. +/// +/// +/// # Example +/// +/// ``` +/// use rand_distr::{UnitSphere, Distribution}; +/// +/// let v: [f64; 3] = UnitSphere.sample(&mut rand::thread_rng()); +/// println!("{:?} is from the unit sphere surface.", v) +/// ``` +/// +/// [^1]: Marsaglia, George (1972). [*Choosing a Point from the Surface of a +/// Sphere.*](https://doi.org/10.1214/aoms/1177692644) +/// Ann. Math. Statist. 43, no. 2, 645--646. +#[derive(Clone, Copy, Debug)] +pub struct UnitSphere; + +impl<N: Float + SampleUniform> Distribution<[N; 3]> for UnitSphere { + #[inline] + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> [N; 3] { + let uniform = Uniform::new(N::from(-1.), N::from(1.)); + loop { + let (x1, x2) = (uniform.sample(rng), uniform.sample(rng)); + let sum = x1*x1 + x2*x2; + if sum >= N::from(1.) { + continue; + } + let factor = N::from(2.) * (N::from(1.0) - sum).sqrt(); + return [x1 * factor, x2 * factor, N::from(1.) - N::from(2.)*sum]; + } + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::UnitSphere; + + /// Assert that two numbers are almost equal to each other. + /// + /// On panic, this macro will print the values of the expressions with their + /// debug representations. + macro_rules! assert_almost_eq { + ($a:expr, $b:expr, $prec:expr) => ( + let diff = ($a - $b).abs(); + if diff > $prec { + panic!(format!( + "assertion failed: `abs(left - right) = {:.1e} < {:e}`, \ + (left: `{}`, right: `{}`)", + diff, $prec, $a, $b)); + } + ); + } + + #[test] + fn norm() { + let mut rng = crate::test::rng(1); + for _ in 0..1000 { + let x: [f64; 3] = UnitSphere.sample(&mut rng); + assert_almost_eq!(x[0]*x[0] + x[1]*x[1] + x[2]*x[2], 1., 1e-15); + } + } + + #[test] + fn value_stability() { + let mut rng = crate::test::rng(2); + let expected = [ + [0.03247542860231647, -0.7830477442152738, 0.6211131755296027], + [-0.09978440840914075, 0.9706650829833128, -0.21875184231323952], + [0.2735582468624679, 0.9435374242279655, -0.1868234852870203], + ]; + let samples: [[f64; 3]; 3] = [ + UnitSphere.sample(&mut rng), + UnitSphere.sample(&mut rng), + UnitSphere.sample(&mut rng), + ]; + assert_eq!(samples, expected); + } +} diff --git a/rand/rand_distr/src/utils.rs b/rand/rand_distr/src/utils.rs new file mode 100644 index 0000000..75b3500 --- /dev/null +++ b/rand/rand_distr/src/utils.rs @@ -0,0 +1,234 @@ +// 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; + } + } +} diff --git a/rand/rand_distr/src/weibull.rs b/rand/rand_distr/src/weibull.rs new file mode 100644 index 0000000..ddde380 --- /dev/null +++ b/rand/rand_distr/src/weibull.rs @@ -0,0 +1,86 @@ +// 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. + +//! The Weibull distribution. + +use rand::Rng; +use crate::{Distribution, OpenClosed01}; +use crate::utils::Float; + +/// Samples floating-point numbers according to the Weibull distribution +/// +/// # Example +/// ``` +/// use rand::prelude::*; +/// use rand_distr::Weibull; +/// +/// let val: f64 = thread_rng().sample(Weibull::new(1., 10.).unwrap()); +/// println!("{}", val); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Weibull<N> { + inv_shape: N, + scale: N, +} + +/// Error type returned from `Weibull::new`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Error { + /// `scale <= 0` or `nan`. + ScaleTooSmall, + /// `shape <= 0` or `nan`. + ShapeTooSmall, +} + +impl<N: Float> Weibull<N> +where OpenClosed01: Distribution<N> +{ + /// Construct a new `Weibull` distribution with given `scale` and `shape`. + pub fn new(scale: N, shape: N) -> Result<Weibull<N>, Error> { + if !(scale > N::from(0.0)) { + return Err(Error::ScaleTooSmall); + } + if !(shape > N::from(0.0)) { + return Err(Error::ShapeTooSmall); + } + Ok(Weibull { inv_shape: N::from(1.)/shape, scale }) + } +} + +impl<N: Float> Distribution<N> for Weibull<N> +where OpenClosed01: Distribution<N> +{ + fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N { + let x: N = rng.sample(OpenClosed01); + self.scale * (-x.ln()).powf(self.inv_shape) + } +} + +#[cfg(test)] +mod tests { + use crate::Distribution; + use super::Weibull; + + #[test] + #[should_panic] + fn invalid() { + Weibull::new(0., 0.).unwrap(); + } + + #[test] + fn sample() { + let scale = 1.0; + let shape = 2.0; + let d = Weibull::new(scale, shape).unwrap(); + let mut rng = crate::test::rng(1); + for _ in 0..1000 { + let r = d.sample(&mut rng); + assert!(r >= 0.); + } + } +} diff --git a/rand/rand_distr/src/ziggurat_tables.rs b/rand/rand_distr/src/ziggurat_tables.rs new file mode 100644 index 0000000..ca1ce30 --- /dev/null +++ b/rand/rand_distr/src/ziggurat_tables.rs @@ -0,0 +1,279 @@ +// Copyright 2018 Developers of the Rand project. +// Copyright 2013 The Rust Project Developers. +// +// 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. + +// Tables for distributions which are sampled using the ziggurat +// algorithm. Autogenerated by `ziggurat_tables.py`. + +pub type ZigTable = &'static [f64; 257]; +pub const ZIG_NORM_R: f64 = 3.654152885361008796; +pub static ZIG_NORM_X: [f64; 257] = + [3.910757959537090045, 3.654152885361008796, 3.449278298560964462, 3.320244733839166074, + 3.224575052047029100, 3.147889289517149969, 3.083526132001233044, 3.027837791768635434, + 2.978603279880844834, 2.934366867207854224, 2.894121053612348060, 2.857138730872132548, + 2.822877396825325125, 2.790921174000785765, 2.760944005278822555, 2.732685359042827056, + 2.705933656121858100, 2.680514643284522158, 2.656283037575502437, 2.633116393630324570, + 2.610910518487548515, 2.589575986706995181, 2.569035452680536569, 2.549221550323460761, + 2.530075232158516929, 2.511544441625342294, 2.493583041269680667, 2.476149939669143318, + 2.459208374333311298, 2.442725318198956774, 2.426670984935725972, 2.411018413899685520, + 2.395743119780480601, 2.380822795170626005, 2.366237056715818632, 2.351967227377659952, + 2.337996148795031370, 2.324308018869623016, 2.310888250599850036, 2.297723348901329565, + 2.284800802722946056, 2.272108990226823888, 2.259637095172217780, 2.247375032945807760, + 2.235313384928327984, 2.223443340090905718, 2.211756642882544366, 2.200245546609647995, + 2.188902771624720689, 2.177721467738641614, 2.166695180352645966, 2.155817819875063268, + 2.145083634046203613, 2.134487182844320152, 2.124023315687815661, 2.113687150684933957, + 2.103474055713146829, 2.093379631137050279, 2.083399693996551783, 2.073530263516978778, + 2.063767547809956415, 2.054107931648864849, 2.044547965215732788, 2.035084353727808715, + 2.025713947862032960, 2.016433734904371722, 2.007240830558684852, 1.998132471356564244, + 1.989106007615571325, 1.980158896898598364, 1.971288697931769640, 1.962493064942461896, + 1.953769742382734043, 1.945116560006753925, 1.936531428273758904, 1.928012334050718257, + 1.919557336591228847, 1.911164563769282232, 1.902832208548446369, 1.894558525668710081, + 1.886341828534776388, 1.878180486290977669, 1.870072921069236838, 1.862017605397632281, + 1.854013059758148119, 1.846057850283119750, 1.838150586580728607, 1.830289919680666566, + 1.822474540091783224, 1.814703175964167636, 1.806974591348693426, 1.799287584547580199, + 1.791640986550010028, 1.784033659547276329, 1.776464495522344977, 1.768932414909077933, + 1.761436365316706665, 1.753975320315455111, 1.746548278279492994, 1.739154261283669012, + 1.731792314050707216, 1.724461502945775715, 1.717160915015540690, 1.709889657069006086, + 1.702646854797613907, 1.695431651932238548, 1.688243209434858727, 1.681080704722823338, + 1.673943330923760353, 1.666830296159286684, 1.659740822855789499, 1.652674147080648526, + 1.645629517902360339, 1.638606196773111146, 1.631603456932422036, 1.624620582830568427, + 1.617656869570534228, 1.610711622367333673, 1.603784156023583041, 1.596873794420261339, + 1.589979870021648534, 1.583101723393471438, 1.576238702733332886, 1.569390163412534456, + 1.562555467528439657, 1.555733983466554893, 1.548925085471535512, 1.542128153226347553, + 1.535342571438843118, 1.528567729435024614, 1.521803020758293101, 1.515047842773992404, + 1.508301596278571965, 1.501563685112706548, 1.494833515777718391, 1.488110497054654369, + 1.481394039625375747, 1.474683555695025516, 1.467978458615230908, 1.461278162507407830, + 1.454582081885523293, 1.447889631277669675, 1.441200224845798017, 1.434513276002946425, + 1.427828197027290358, 1.421144398672323117, 1.414461289772464658, 1.407778276843371534, + 1.401094763676202559, 1.394410150925071257, 1.387723835686884621, 1.381035211072741964, + 1.374343665770030531, 1.367648583594317957, 1.360949343030101844, 1.354245316759430606, + 1.347535871177359290, 1.340820365893152122, 1.334098153216083604, 1.327368577624624679, + 1.320630975217730096, 1.313884673146868964, 1.307128989027353860, 1.300363230327433728, + 1.293586693733517645, 1.286798664489786415, 1.279998415710333237, 1.273185207661843732, + 1.266358287014688333, 1.259516886060144225, 1.252660221891297887, 1.245787495544997903, + 1.238897891102027415, 1.231990574742445110, 1.225064693752808020, 1.218119375481726552, + 1.211153726239911244, 1.204166830140560140, 1.197157747875585931, 1.190125515422801650, + 1.183069142678760732, 1.175987612011489825, 1.168879876726833800, 1.161744859441574240, + 1.154581450355851802, 1.147388505416733873, 1.140164844363995789, 1.132909248648336975, + 1.125620459211294389, 1.118297174115062909, 1.110938046009249502, 1.103541679420268151, + 1.096106627847603487, 1.088631390649514197, 1.081114409698889389, 1.073554065787871714, + 1.065948674757506653, 1.058296483326006454, 1.050595664586207123, 1.042844313139370538, + 1.035040439828605274, 1.027181966030751292, 1.019266717460529215, 1.011292417434978441, + 1.003256679539591412, 0.995156999629943084, 0.986990747093846266, 0.978755155288937750, + 0.970447311058864615, 0.962064143217605250, 0.953602409875572654, 0.945058684462571130, + 0.936429340280896860, 0.927710533396234771, 0.918898183643734989, 0.909987953490768997, + 0.900975224455174528, 0.891855070726792376, 0.882622229578910122, 0.873271068082494550, + 0.863795545546826915, 0.854189171001560554, 0.844444954902423661, 0.834555354079518752, + 0.824512208745288633, 0.814306670128064347, 0.803929116982664893, 0.793369058833152785, + 0.782615023299588763, 0.771654424216739354, 0.760473406422083165, 0.749056662009581653, + 0.737387211425838629, 0.725446140901303549, 0.713212285182022732, 0.700661841097584448, + 0.687767892786257717, 0.674499822827436479, 0.660822574234205984, 0.646695714884388928, + 0.632072236375024632, 0.616896989996235545, 0.601104617743940417, 0.584616766093722262, + 0.567338257040473026, 0.549151702313026790, 0.529909720646495108, 0.509423329585933393, + 0.487443966121754335, 0.463634336771763245, 0.437518402186662658, 0.408389134588000746, + 0.375121332850465727, 0.335737519180459465, 0.286174591747260509, 0.215241895913273806, + 0.000000000000000000]; +pub static ZIG_NORM_F: [f64; 257] = + [0.000477467764586655, 0.001260285930498598, 0.002609072746106363, 0.004037972593371872, + 0.005522403299264754, 0.007050875471392110, 0.008616582769422917, 0.010214971439731100, + 0.011842757857943104, 0.013497450601780807, 0.015177088307982072, 0.016880083152595839, + 0.018605121275783350, 0.020351096230109354, 0.022117062707379922, 0.023902203305873237, + 0.025705804008632656, 0.027527235669693315, 0.029365939758230111, 0.031221417192023690, + 0.033093219458688698, 0.034980941461833073, 0.036884215688691151, 0.038802707404656918, + 0.040736110656078753, 0.042684144916619378, 0.044646552251446536, 0.046623094902089664, + 0.048613553216035145, 0.050617723861121788, 0.052635418276973649, 0.054666461325077916, + 0.056710690106399467, 0.058767952921137984, 0.060838108349751806, 0.062921024437977854, + 0.065016577971470438, 0.067124653828023989, 0.069245144397250269, 0.071377949059141965, + 0.073522973714240991, 0.075680130359194964, 0.077849336702372207, 0.080030515814947509, + 0.082223595813495684, 0.084428509570654661, 0.086645194450867782, 0.088873592068594229, + 0.091113648066700734, 0.093365311913026619, 0.095628536713353335, 0.097903279039215627, + 0.100189498769172020, 0.102487158942306270, 0.104796225622867056, 0.107116667775072880, + 0.109448457147210021, 0.111791568164245583, 0.114145977828255210, 0.116511665626037014, + 0.118888613443345698, 0.121276805485235437, 0.123676228202051403, 0.126086870220650349, + 0.128508722280473636, 0.130941777174128166, 0.133386029692162844, 0.135841476571757352, + 0.138308116449064322, 0.140785949814968309, 0.143274978974047118, 0.145775208006537926, + 0.148286642733128721, 0.150809290682410169, 0.153343161060837674, 0.155888264725064563, + 0.158444614156520225, 0.161012223438117663, 0.163591108232982951, 0.166181285765110071, + 0.168782774801850333, 0.171395595638155623, 0.174019770082499359, 0.176655321444406654, + 0.179302274523530397, 0.181960655600216487, 0.184630492427504539, 0.187311814224516926, + 0.190004651671193070, 0.192709036904328807, 0.195425003514885592, 0.198152586546538112, + 0.200891822495431333, 0.203642749311121501, 0.206405406398679298, 0.209179834621935651, + 0.211966076307852941, 0.214764175252008499, 0.217574176725178370, 0.220396127481011589, + 0.223230075764789593, 0.226076071323264877, 0.228934165415577484, 0.231804410825248525, + 0.234686861873252689, 0.237581574432173676, 0.240488605941449107, 0.243408015423711988, + 0.246339863502238771, 0.249284212419516704, 0.252241126056943765, 0.255210669955677150, + 0.258192911338648023, 0.261187919133763713, 0.264195763998317568, 0.267216518344631837, + 0.270250256366959984, 0.273297054069675804, 0.276356989296781264, 0.279430141762765316, + 0.282516593084849388, 0.285616426816658109, 0.288729728483353931, 0.291856585618280984, + 0.294997087801162572, 0.298151326697901342, 0.301319396102034120, 0.304501391977896274, + 0.307697412505553769, 0.310907558127563710, 0.314131931597630143, 0.317370638031222396, + 0.320623784958230129, 0.323891482377732021, 0.327173842814958593, 0.330470981380537099, + 0.333783015832108509, 0.337110066638412809, 0.340452257045945450, 0.343809713148291340, + 0.347182563958251478, 0.350570941482881204, 0.353974980801569250, 0.357394820147290515, + 0.360830600991175754, 0.364282468130549597, 0.367750569780596226, 0.371235057669821344, + 0.374736087139491414, 0.378253817247238111, 0.381788410875031348, 0.385340034841733958, + 0.388908860020464597, 0.392495061461010764, 0.396098818517547080, 0.399720314981931668, + 0.403359739222868885, 0.407017284331247953, 0.410693148271983222, 0.414387534042706784, + 0.418100649839684591, 0.421832709231353298, 0.425583931339900579, 0.429354541031341519, + 0.433144769114574058, 0.436954852549929273, 0.440785034667769915, 0.444635565397727750, + 0.448506701509214067, 0.452398706863882505, 0.456311852680773566, 0.460246417814923481, + 0.464202689050278838, 0.468180961407822172, 0.472181538469883255, 0.476204732721683788, + 0.480250865911249714, 0.484320269428911598, 0.488413284707712059, 0.492530263646148658, + 0.496671569054796314, 0.500837575128482149, 0.505028667945828791, 0.509245245998136142, + 0.513487720749743026, 0.517756517232200619, 0.522052074674794864, 0.526374847174186700, + 0.530725304406193921, 0.535103932383019565, 0.539511234259544614, 0.543947731192649941, + 0.548413963257921133, 0.552910490428519918, 0.557437893621486324, 0.561996775817277916, + 0.566587763258951771, 0.571211506738074970, 0.575868682975210544, 0.580559996103683473, + 0.585286179266300333, 0.590047996335791969, 0.594846243770991268, 0.599681752622167719, + 0.604555390700549533, 0.609468064928895381, 0.614420723892076803, 0.619414360609039205, + 0.624450015550274240, 0.629528779928128279, 0.634651799290960050, 0.639820277456438991, + 0.645035480824251883, 0.650298743114294586, 0.655611470583224665, 0.660975147780241357, + 0.666391343912380640, 0.671861719900766374, 0.677388036222513090, 0.682972161648791376, + 0.688616083008527058, 0.694321916130032579, 0.700091918140490099, 0.705928501336797409, + 0.711834248882358467, 0.717811932634901395, 0.723864533472881599, 0.729995264565802437, + 0.736207598131266683, 0.742505296344636245, 0.748892447223726720, 0.755373506511754500, + 0.761953346841546475, 0.768637315803334831, 0.775431304986138326, 0.782341832659861902, + 0.789376143571198563, 0.796542330428254619, 0.803849483176389490, 0.811307874318219935, + 0.818929191609414797, 0.826726833952094231, 0.834716292992930375, 0.842915653118441077, + 0.851346258465123684, 0.860033621203008636, 0.869008688043793165, 0.878309655816146839, + 0.887984660763399880, 0.898095921906304051, 0.908726440060562912, 0.919991505048360247, + 0.932060075968990209, 0.945198953453078028, 0.959879091812415930, 0.977101701282731328, + 1.000000000000000000]; +pub const ZIG_EXP_R: f64 = 7.697117470131050077; +pub static ZIG_EXP_X: [f64; 257] = + [8.697117470131052741, 7.697117470131050077, 6.941033629377212577, 6.478378493832569696, + 6.144164665772472667, 5.882144315795399869, 5.666410167454033697, 5.482890627526062488, + 5.323090505754398016, 5.181487281301500047, 5.054288489981304089, 4.938777085901250530, + 4.832939741025112035, 4.735242996601741083, 4.644491885420085175, 4.559737061707351380, + 4.480211746528421912, 4.405287693473573185, 4.334443680317273007, 4.267242480277365857, + 4.203313713735184365, 4.142340865664051464, 4.084051310408297830, 4.028208544647936762, + 3.974606066673788796, 3.923062500135489739, 3.873417670399509127, 3.825529418522336744, + 3.779270992411667862, 3.734528894039797375, 3.691201090237418825, 3.649195515760853770, + 3.608428813128909507, 3.568825265648337020, 3.530315889129343354, 3.492837654774059608, + 3.456332821132760191, 3.420748357251119920, 3.386035442460300970, 3.352149030900109405, + 3.319047470970748037, 3.286692171599068679, 3.255047308570449882, 3.224079565286264160, + 3.193757903212240290, 3.164053358025972873, 3.134938858084440394, 3.106389062339824481, + 3.078380215254090224, 3.050890016615455114, 3.023897504455676621, 2.997382949516130601, + 2.971327759921089662, 2.945714394895045718, 2.920526286512740821, 2.895747768600141825, + 2.871364012015536371, 2.847360965635188812, 2.823725302450035279, 2.800444370250737780, + 2.777506146439756574, 2.754899196562344610, 2.732612636194700073, 2.710636095867928752, + 2.688959688741803689, 2.667573980773266573, 2.646469963151809157, 2.625639026797788489, + 2.605072938740835564, 2.584763820214140750, 2.564704126316905253, 2.544886627111869970, + 2.525304390037828028, 2.505950763528594027, 2.486819361740209455, 2.467904050297364815, + 2.449198932978249754, 2.430698339264419694, 2.412396812688870629, 2.394289099921457886, + 2.376370140536140596, 2.358635057409337321, 2.341079147703034380, 2.323697874390196372, + 2.306486858283579799, 2.289441870532269441, 2.272558825553154804, 2.255833774367219213, + 2.239262898312909034, 2.222842503111036816, 2.206569013257663858, 2.190438966723220027, + 2.174449009937774679, 2.158595893043885994, 2.142876465399842001, 2.127287671317368289, + 2.111826546019042183, 2.096490211801715020, 2.081275874393225145, 2.066180819490575526, + 2.051202409468584786, 2.036338080248769611, 2.021585338318926173, 2.006941757894518563, + 1.992404978213576650, 1.977972700957360441, 1.963642687789548313, 1.949412758007184943, + 1.935280786297051359, 1.921244700591528076, 1.907302480018387536, 1.893452152939308242, + 1.879691795072211180, 1.866019527692827973, 1.852433515911175554, 1.838931967018879954, + 1.825513128903519799, 1.812175288526390649, 1.798916770460290859, 1.785735935484126014, + 1.772631179231305643, 1.759600930889074766, 1.746643651946074405, 1.733757834985571566, + 1.720942002521935299, 1.708194705878057773, 1.695514524101537912, 1.682900062917553896, + 1.670349953716452118, 1.657862852574172763, 1.645437439303723659, 1.633072416535991334, + 1.620766508828257901, 1.608518461798858379, 1.596327041286483395, 1.584191032532688892, + 1.572109239386229707, 1.560080483527888084, 1.548103603714513499, 1.536177455041032092, + 1.524300908219226258, 1.512472848872117082, 1.500692176842816750, 1.488957805516746058, + 1.477268661156133867, 1.465623682245745352, 1.454021818848793446, 1.442462031972012504, + 1.430943292938879674, 1.419464582769983219, 1.408024891569535697, 1.396623217917042137, + 1.385258568263121992, 1.373929956328490576, 1.362636402505086775, 1.351376933258335189, + 1.340150580529504643, 1.328956381137116560, 1.317793376176324749, 1.306660610415174117, + 1.295557131686601027, 1.284481990275012642, 1.273434238296241139, 1.262412929069615330, + 1.251417116480852521, 1.240445854334406572, 1.229498195693849105, 1.218573192208790124, + 1.207669893426761121, 1.196787346088403092, 1.185924593404202199, 1.175080674310911677, + 1.164254622705678921, 1.153445466655774743, 1.142652227581672841, 1.131873919411078511, + 1.121109547701330200, 1.110358108727411031, 1.099618588532597308, 1.088889961938546813, + 1.078171191511372307, 1.067461226479967662, 1.056759001602551429, 1.046063435977044209, + 1.035373431790528542, 1.024687873002617211, 1.014005623957096480, 1.003325527915696735, + 0.992646405507275897, 0.981967053085062602, 0.971286240983903260, 0.960602711668666509, + 0.949915177764075969, 0.939222319955262286, 0.928522784747210395, 0.917815182070044311, + 0.907098082715690257, 0.896370015589889935, 0.885629464761751528, 0.874874866291025066, + 0.864104604811004484, 0.853317009842373353, 0.842510351810368485, 0.831682837734273206, + 0.820832606554411814, 0.809957724057418282, 0.799056177355487174, 0.788125868869492430, + 0.777164609759129710, 0.766170112735434672, 0.755139984181982249, 0.744071715500508102, + 0.732962673584365398, 0.721810090308756203, 0.710611050909655040, 0.699362481103231959, + 0.688061132773747808, 0.676703568029522584, 0.665286141392677943, 0.653804979847664947, + 0.642255960424536365, 0.630634684933490286, 0.618936451394876075, 0.607156221620300030, + 0.595288584291502887, 0.583327712748769489, 0.571267316532588332, 0.559100585511540626, + 0.546820125163310577, 0.534417881237165604, 0.521885051592135052, 0.509211982443654398, + 0.496388045518671162, 0.483401491653461857, 0.470239275082169006, 0.456886840931420235, + 0.443327866073552401, 0.429543940225410703, 0.415514169600356364, 0.401214678896277765, + 0.386617977941119573, 0.371692145329917234, 0.356399760258393816, 0.340696481064849122, + 0.324529117016909452, 0.307832954674932158, 0.290527955491230394, 0.272513185478464703, + 0.253658363385912022, 0.233790483059674731, 0.212671510630966620, 0.189958689622431842, + 0.165127622564187282, 0.137304980940012589, 0.104838507565818778, 0.063852163815001570, + 0.000000000000000000]; +pub static ZIG_EXP_F: [f64; 257] = + [0.000167066692307963, 0.000454134353841497, 0.000967269282327174, 0.001536299780301573, + 0.002145967743718907, 0.002788798793574076, 0.003460264777836904, 0.004157295120833797, + 0.004877655983542396, 0.005619642207205489, 0.006381905937319183, 0.007163353183634991, + 0.007963077438017043, 0.008780314985808977, 0.009614413642502212, 0.010464810181029981, + 0.011331013597834600, 0.012212592426255378, 0.013109164931254991, 0.014020391403181943, + 0.014945968011691148, 0.015885621839973156, 0.016839106826039941, 0.017806200410911355, + 0.018786700744696024, 0.019780424338009740, 0.020787204072578114, 0.021806887504283581, + 0.022839335406385240, 0.023884420511558174, 0.024942026419731787, 0.026012046645134221, + 0.027094383780955803, 0.028188948763978646, 0.029295660224637411, 0.030414443910466622, + 0.031545232172893622, 0.032687963508959555, 0.033842582150874358, 0.035009037697397431, + 0.036187284781931443, 0.037377282772959382, 0.038578995503074871, 0.039792391023374139, + 0.041017441380414840, 0.042254122413316254, 0.043502413568888197, 0.044762297732943289, + 0.046033761076175184, 0.047316792913181561, 0.048611385573379504, 0.049917534282706379, + 0.051235237055126281, 0.052564494593071685, 0.053905310196046080, 0.055257689676697030, + 0.056621641283742870, 0.057997175631200659, 0.059384305633420280, 0.060783046445479660, + 0.062193415408541036, 0.063615431999807376, 0.065049117786753805, 0.066494496385339816, + 0.067951593421936643, 0.069420436498728783, 0.070901055162371843, 0.072393480875708752, + 0.073897746992364746, 0.075413888734058410, 0.076941943170480517, 0.078481949201606435, + 0.080033947542319905, 0.081597980709237419, 0.083174093009632397, 0.084762330532368146, + 0.086362741140756927, 0.087975374467270231, 0.089600281910032886, 0.091237516631040197, + 0.092887133556043569, 0.094549189376055873, 0.096223742550432825, 0.097910853311492213, + 0.099610583670637132, 0.101322997425953631, 0.103048160171257702, 0.104786139306570145, + 0.106537004050001632, 0.108300825451033755, 0.110077676405185357, 0.111867631670056283, + 0.113670767882744286, 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