From 986ad2f782cf944990e4eda8bf88ea1821233302 Mon Sep 17 00:00:00 2001 From: Robin Krahl Date: Tue, 11 Dec 2018 23:50:45 +0100 Subject: Add nitrokey as a dependency to nitrocli The nitrokey crate provides a simple interface to the Nitrokey Storage and the Nitrokey Pro based on the libnitrokey library developed by Nitrokey UG. The low-level bindings to this library are available in the nitrokey-sys crate. This patch adds version v0.2.1 of the nitrokey crate as a dependency for nitrocli. It includes the indirect dependencies nitrokey-sys (version 3.4.1) and rand (version 0.4.3). Import subrepo nitrokey/:nitrokey at 2eccc96ceec2282b868891befe9cda7f941fbe7b Import subrepo nitrokey-sys/:nitrokey-sys at f1a11ebf72610fb9cf80ac7f9f147b4ba1a5336f Import subrepo rand/:rand at d7d5da49daf7ceb3e5940072940d495cced3a1b3 --- rand/src/distributions/exponential.rs | 124 +++++++++ rand/src/distributions/gamma.rs | 386 ++++++++++++++++++++++++++++ rand/src/distributions/mod.rs | 409 ++++++++++++++++++++++++++++++ rand/src/distributions/normal.rs | 201 +++++++++++++++ rand/src/distributions/range.rs | 241 ++++++++++++++++++ rand/src/distributions/ziggurat_tables.rs | 280 ++++++++++++++++++++ 6 files changed, 1641 insertions(+) create mode 100644 rand/src/distributions/exponential.rs create mode 100644 rand/src/distributions/gamma.rs create mode 100644 rand/src/distributions/mod.rs create mode 100644 rand/src/distributions/normal.rs create mode 100644 rand/src/distributions/range.rs create mode 100644 rand/src/distributions/ziggurat_tables.rs (limited to 'rand/src/distributions') diff --git a/rand/src/distributions/exponential.rs b/rand/src/distributions/exponential.rs new file mode 100644 index 0000000..c3c924c --- /dev/null +++ b/rand/src/distributions/exponential.rs @@ -0,0 +1,124 @@ +// Copyright 2013 The Rust Project Developers. See the COPYRIGHT +// file at the top-level directory of this distribution and at +// http://rust-lang.org/COPYRIGHT. +// +// Licensed under the Apache License, Version 2.0 or the MIT license +// , at your +// option. This file may not be copied, modified, or distributed +// except according to those terms. + +//! The exponential distribution. + +use {Rng, Rand}; +use distributions::{ziggurat, ziggurat_tables, Sample, IndependentSample}; + +/// A wrapper around an `f64` to generate Exp(1) random numbers. +/// +/// 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*](http://www.doornik.com/research/ziggurat.pdf). Nuffield +/// College, Oxford +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::exponential::Exp1; +/// +/// let Exp1(x) = rand::random(); +/// println!("{}", x); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Exp1(pub f64); + +// This could be done via `-rng.gen::().ln()` but that is slower. +impl Rand for Exp1 { + #[inline] + fn rand(rng: &mut R) -> Exp1 { + #[inline] + fn pdf(x: f64) -> f64 { + (-x).exp() + } + #[inline] + fn zero_case(rng: &mut R, _u: f64) -> f64 { + ziggurat_tables::ZIG_EXP_R - rng.gen::().ln() + } + + Exp1(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`. +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::{Exp, IndependentSample}; +/// +/// let exp = Exp::new(2.0); +/// let v = exp.ind_sample(&mut rand::thread_rng()); +/// println!("{} is from a Exp(2) distribution", v); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Exp { + /// `lambda` stored as `1/lambda`, since this is what we scale by. + lambda_inverse: f64 +} + +impl Exp { + /// Construct a new `Exp` with the given shape parameter + /// `lambda`. Panics if `lambda <= 0`. + #[inline] + pub fn new(lambda: f64) -> Exp { + assert!(lambda > 0.0, "Exp::new called with `lambda` <= 0"); + Exp { lambda_inverse: 1.0 / lambda } + } +} + +impl Sample for Exp { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl IndependentSample for Exp { + fn ind_sample(&self, rng: &mut R) -> f64 { + let Exp1(n) = rng.gen::(); + n * self.lambda_inverse + } +} + +#[cfg(test)] +mod test { + use distributions::{Sample, IndependentSample}; + use super::Exp; + + #[test] + fn test_exp() { + let mut exp = Exp::new(10.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + assert!(exp.sample(&mut rng) >= 0.0); + assert!(exp.ind_sample(&mut rng) >= 0.0); + } + } + #[test] + #[should_panic] + fn test_exp_invalid_lambda_zero() { + Exp::new(0.0); + } + #[test] + #[should_panic] + fn test_exp_invalid_lambda_neg() { + Exp::new(-10.0); + } +} diff --git a/rand/src/distributions/gamma.rs b/rand/src/distributions/gamma.rs new file mode 100644 index 0000000..2806495 --- /dev/null +++ b/rand/src/distributions/gamma.rs @@ -0,0 +1,386 @@ +// Copyright 2013 The Rust Project Developers. See the COPYRIGHT +// file at the top-level directory of this distribution and at +// http://rust-lang.org/COPYRIGHT. +// +// Licensed under the Apache License, Version 2.0 or the MIT license +// , at your +// option. This file may not be copied, modified, or distributed +// except according to those terms. +// +// ignore-lexer-test FIXME #15679 + +//! The Gamma and derived distributions. + +use self::GammaRepr::*; +use self::ChiSquaredRepr::*; + +use {Rng, Open01}; +use super::normal::StandardNormal; +use super::{IndependentSample, Sample, Exp}; + +/// 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 [1] for +/// `shape < 1`. +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::{IndependentSample, Gamma}; +/// +/// let gamma = Gamma::new(2.0, 5.0); +/// let v = gamma.ind_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](http://doi.acm.org/10.1145/358407.358414) +#[derive(Clone, Copy, Debug)] +pub struct Gamma { + repr: GammaRepr, +} + +#[derive(Clone, Copy, Debug)] +enum GammaRepr { + Large(GammaLargeShape), + One(Exp), + Small(GammaSmallShape) +} + +// 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 { + inv_shape: f64, + large_shape: GammaLargeShape +} + +/// 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 { + scale: f64, + c: f64, + d: f64 +} + +impl Gamma { + /// Construct an object representing the `Gamma(shape, scale)` + /// distribution. + /// + /// Panics if `shape <= 0` or `scale <= 0`. + #[inline] + pub fn new(shape: f64, scale: f64) -> Gamma { + assert!(shape > 0.0, "Gamma::new called with shape <= 0"); + assert!(scale > 0.0, "Gamma::new called with scale <= 0"); + + let repr = if shape == 1.0 { + One(Exp::new(1.0 / scale)) + } else if shape < 1.0 { + Small(GammaSmallShape::new_raw(shape, scale)) + } else { + Large(GammaLargeShape::new_raw(shape, scale)) + }; + Gamma { repr: repr } + } +} + +impl GammaSmallShape { + fn new_raw(shape: f64, scale: f64) -> GammaSmallShape { + GammaSmallShape { + inv_shape: 1. / shape, + large_shape: GammaLargeShape::new_raw(shape + 1.0, scale) + } + } +} + +impl GammaLargeShape { + fn new_raw(shape: f64, scale: f64) -> GammaLargeShape { + let d = shape - 1. / 3.; + GammaLargeShape { + scale: scale, + c: 1. / (9. * d).sqrt(), + d: d + } + } +} + +impl Sample for Gamma { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl Sample for GammaSmallShape { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl Sample for GammaLargeShape { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} + +impl IndependentSample for Gamma { + fn ind_sample(&self, rng: &mut R) -> f64 { + match self.repr { + Small(ref g) => g.ind_sample(rng), + One(ref g) => g.ind_sample(rng), + Large(ref g) => g.ind_sample(rng), + } + } +} +impl IndependentSample for GammaSmallShape { + fn ind_sample(&self, rng: &mut R) -> f64 { + let Open01(u) = rng.gen::>(); + + self.large_shape.ind_sample(rng) * u.powf(self.inv_shape) + } +} +impl IndependentSample for GammaLargeShape { + fn ind_sample(&self, rng: &mut R) -> f64 { + loop { + let StandardNormal(x) = rng.gen::(); + let v_cbrt = 1.0 + self.c * x; + if v_cbrt <= 0.0 { // a^3 <= 0 iff a <= 0 + continue + } + + let v = v_cbrt * v_cbrt * v_cbrt; + let Open01(u) = rng.gen::>(); + + let x_sqr = x * x; + if u < 1.0 - 0.0331 * x_sqr * x_sqr || + u.ln() < 0.5 * x_sqr + self.d * (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 +/// +/// ```rust +/// use rand::distributions::{ChiSquared, IndependentSample}; +/// +/// let chi = ChiSquared::new(11.0); +/// let v = chi.ind_sample(&mut rand::thread_rng()); +/// println!("{} is from a χ²(11) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct ChiSquared { + repr: ChiSquaredRepr, +} + +#[derive(Clone, Copy, Debug)] +enum ChiSquaredRepr { + // 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), +} + +impl ChiSquared { + /// Create a new chi-squared distribution with degrees-of-freedom + /// `k`. Panics if `k < 0`. + pub fn new(k: f64) -> ChiSquared { + let repr = if k == 1.0 { + DoFExactlyOne + } else { + assert!(k > 0.0, "ChiSquared::new called with `k` < 0"); + DoFAnythingElse(Gamma::new(0.5 * k, 2.0)) + }; + ChiSquared { repr: repr } + } +} +impl Sample for ChiSquared { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl IndependentSample for ChiSquared { + fn ind_sample(&self, rng: &mut R) -> f64 { + match self.repr { + DoFExactlyOne => { + // k == 1 => N(0,1)^2 + let StandardNormal(norm) = rng.gen::(); + norm * norm + } + DoFAnythingElse(ref g) => g.ind_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 +/// +/// ```rust +/// use rand::distributions::{FisherF, IndependentSample}; +/// +/// let f = FisherF::new(2.0, 32.0); +/// let v = f.ind_sample(&mut rand::thread_rng()); +/// println!("{} is from an F(2, 32) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct FisherF { + numer: ChiSquared, + denom: ChiSquared, + // denom_dof / numer_dof so that this can just be a straight + // multiplication, rather than a division. + dof_ratio: f64, +} + +impl FisherF { + /// Create a new `FisherF` distribution, with the given + /// parameter. Panics if either `m` or `n` are not positive. + pub fn new(m: f64, n: f64) -> FisherF { + assert!(m > 0.0, "FisherF::new called with `m < 0`"); + assert!(n > 0.0, "FisherF::new called with `n < 0`"); + + FisherF { + numer: ChiSquared::new(m), + denom: ChiSquared::new(n), + dof_ratio: n / m + } + } +} +impl Sample for FisherF { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl IndependentSample for FisherF { + fn ind_sample(&self, rng: &mut R) -> f64 { + self.numer.ind_sample(rng) / self.denom.ind_sample(rng) * self.dof_ratio + } +} + +/// The Student t distribution, `t(nu)`, where `nu` is the degrees of +/// freedom. +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::{StudentT, IndependentSample}; +/// +/// let t = StudentT::new(11.0); +/// let v = t.ind_sample(&mut rand::thread_rng()); +/// println!("{} is from a t(11) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct StudentT { + chi: ChiSquared, + dof: f64 +} + +impl StudentT { + /// Create a new Student t distribution with `n` degrees of + /// freedom. Panics if `n <= 0`. + pub fn new(n: f64) -> StudentT { + assert!(n > 0.0, "StudentT::new called with `n <= 0`"); + StudentT { + chi: ChiSquared::new(n), + dof: n + } + } +} +impl Sample for StudentT { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl IndependentSample for StudentT { + fn ind_sample(&self, rng: &mut R) -> f64 { + let StandardNormal(norm) = rng.gen::(); + norm * (self.dof / self.chi.ind_sample(rng)).sqrt() + } +} + +#[cfg(test)] +mod test { + use distributions::{Sample, IndependentSample}; + use super::{ChiSquared, StudentT, FisherF}; + + #[test] + fn test_chi_squared_one() { + let mut chi = ChiSquared::new(1.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + chi.sample(&mut rng); + chi.ind_sample(&mut rng); + } + } + #[test] + fn test_chi_squared_small() { + let mut chi = ChiSquared::new(0.5); + let mut rng = ::test::rng(); + for _ in 0..1000 { + chi.sample(&mut rng); + chi.ind_sample(&mut rng); + } + } + #[test] + fn test_chi_squared_large() { + let mut chi = ChiSquared::new(30.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + chi.sample(&mut rng); + chi.ind_sample(&mut rng); + } + } + #[test] + #[should_panic] + fn test_chi_squared_invalid_dof() { + ChiSquared::new(-1.0); + } + + #[test] + fn test_f() { + let mut f = FisherF::new(2.0, 32.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + f.sample(&mut rng); + f.ind_sample(&mut rng); + } + } + + #[test] + fn test_t() { + let mut t = StudentT::new(11.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + t.sample(&mut rng); + t.ind_sample(&mut rng); + } + } +} diff --git a/rand/src/distributions/mod.rs b/rand/src/distributions/mod.rs new file mode 100644 index 0000000..5de8efb --- /dev/null +++ b/rand/src/distributions/mod.rs @@ -0,0 +1,409 @@ +// Copyright 2013 The Rust Project Developers. See the COPYRIGHT +// file at the top-level directory of this distribution and at +// http://rust-lang.org/COPYRIGHT. +// +// Licensed under the Apache License, Version 2.0 or the MIT license +// , at your +// option. This file may not be copied, modified, or distributed +// except according to those terms. + +//! Sampling from random distributions. +//! +//! This is a generalization of `Rand` to allow parameters to control the +//! exact properties of the generated values, e.g. the mean and standard +//! deviation of a normal distribution. The `Sample` trait is the most +//! general, and allows for generating values that change some state +//! internally. The `IndependentSample` trait is for generating values +//! that do not need to record state. + +use core::marker; + +use {Rng, Rand}; + +pub use self::range::Range; +#[cfg(feature="std")] +pub use self::gamma::{Gamma, ChiSquared, FisherF, StudentT}; +#[cfg(feature="std")] +pub use self::normal::{Normal, LogNormal}; +#[cfg(feature="std")] +pub use self::exponential::Exp; + +pub mod range; +#[cfg(feature="std")] +pub mod gamma; +#[cfg(feature="std")] +pub mod normal; +#[cfg(feature="std")] +pub mod exponential; + +#[cfg(feature="std")] +mod ziggurat_tables; + +/// Types that can be used to create a random instance of `Support`. +pub trait Sample { + /// Generate a random value of `Support`, using `rng` as the + /// source of randomness. + fn sample(&mut self, rng: &mut R) -> Support; +} + +/// `Sample`s that do not require keeping track of state. +/// +/// Since no state is recorded, each sample is (statistically) +/// independent of all others, assuming the `Rng` used has this +/// property. +// FIXME maybe having this separate is overkill (the only reason is to +// take &self rather than &mut self)? or maybe this should be the +// trait called `Sample` and the other should be `DependentSample`. +pub trait IndependentSample: Sample { + /// Generate a random value. + fn ind_sample(&self, &mut R) -> Support; +} + +/// A wrapper for generating types that implement `Rand` via the +/// `Sample` & `IndependentSample` traits. +#[derive(Debug)] +pub struct RandSample { + _marker: marker::PhantomData Sup>, +} + +impl Copy for RandSample {} +impl Clone for RandSample { + fn clone(&self) -> Self { *self } +} + +impl Sample for RandSample { + fn sample(&mut self, rng: &mut R) -> Sup { self.ind_sample(rng) } +} + +impl IndependentSample for RandSample { + fn ind_sample(&self, rng: &mut R) -> Sup { + rng.gen() + } +} + +impl RandSample { + pub fn new() -> RandSample { + RandSample { _marker: marker::PhantomData } + } +} + +/// A value with a particular weight for use with `WeightedChoice`. +#[derive(Copy, Clone, Debug)] +pub struct Weighted { + /// The numerical weight of this item + pub weight: u32, + /// The actual item which is being weighted + pub item: T, +} + +/// A distribution that selects from a finite collection of weighted items. +/// +/// Each item has an associated weight that influences how likely it +/// is to be chosen: higher weight is more likely. +/// +/// The `Clone` restriction is a limitation of the `Sample` and +/// `IndependentSample` traits. Note that `&T` is (cheaply) `Clone` for +/// all `T`, as is `u32`, so one can store references or indices into +/// another vector. +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::{Weighted, WeightedChoice, IndependentSample}; +/// +/// let mut items = vec!(Weighted { weight: 2, item: 'a' }, +/// Weighted { weight: 4, item: 'b' }, +/// Weighted { weight: 1, item: 'c' }); +/// let wc = WeightedChoice::new(&mut items); +/// let mut rng = rand::thread_rng(); +/// for _ in 0..16 { +/// // on average prints 'a' 4 times, 'b' 8 and 'c' twice. +/// println!("{}", wc.ind_sample(&mut rng)); +/// } +/// ``` +#[derive(Debug)] +pub struct WeightedChoice<'a, T:'a> { + items: &'a mut [Weighted], + weight_range: Range +} + +impl<'a, T: Clone> WeightedChoice<'a, T> { + /// Create a new `WeightedChoice`. + /// + /// Panics if: + /// + /// - `items` is empty + /// - the total weight is 0 + /// - the total weight is larger than a `u32` can contain. + pub fn new(items: &'a mut [Weighted]) -> WeightedChoice<'a, T> { + // strictly speaking, this is subsumed by the total weight == 0 case + assert!(!items.is_empty(), "WeightedChoice::new called with no items"); + + let mut running_total: u32 = 0; + + // we convert the list from individual weights to cumulative + // weights so we can binary search. This *could* drop elements + // with weight == 0 as an optimisation. + for item in items.iter_mut() { + running_total = match running_total.checked_add(item.weight) { + Some(n) => n, + None => panic!("WeightedChoice::new called with a total weight \ + larger than a u32 can contain") + }; + + item.weight = running_total; + } + assert!(running_total != 0, "WeightedChoice::new called with a total weight of 0"); + + WeightedChoice { + items: items, + // we're likely to be generating numbers in this range + // relatively often, so might as well cache it + weight_range: Range::new(0, running_total) + } + } +} + +impl<'a, T: Clone> Sample for WeightedChoice<'a, T> { + fn sample(&mut self, rng: &mut R) -> T { self.ind_sample(rng) } +} + +impl<'a, T: Clone> IndependentSample for WeightedChoice<'a, T> { + fn ind_sample(&self, rng: &mut R) -> T { + // we want to find the first element that has cumulative + // weight > sample_weight, which we do by binary since the + // cumulative weights of self.items are sorted. + + // choose a weight in [0, total_weight) + let sample_weight = self.weight_range.ind_sample(rng); + + // short circuit when it's the first item + if sample_weight < self.items[0].weight { + return self.items[0].item.clone(); + } + + let mut idx = 0; + let mut modifier = self.items.len(); + + // now we know that every possibility has an element to the + // left, so we can just search for the last element that has + // cumulative weight <= sample_weight, then the next one will + // be "it". (Note that this greatest element will never be the + // last element of the vector, since sample_weight is chosen + // in [0, total_weight) and the cumulative weight of the last + // one is exactly the total weight.) + while modifier > 1 { + let i = idx + modifier / 2; + if self.items[i].weight <= sample_weight { + // we're small, so look to the right, but allow this + // exact element still. + idx = i; + // we need the `/ 2` to round up otherwise we'll drop + // the trailing elements when `modifier` is odd. + modifier += 1; + } else { + // otherwise we're too big, so go left. (i.e. do + // nothing) + } + modifier /= 2; + } + return self.items[idx + 1].item.clone(); + } +} + +/// 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. +#[cfg(feature="std")] +#[inline(always)] +fn ziggurat( + rng: &mut R, + symmetric: bool, + x_tab: ziggurat_tables::ZigTable, + f_tab: ziggurat_tables::ZigTable, + mut pdf: P, + mut zero_case: Z) + -> f64 where P: FnMut(f64) -> f64, Z: FnMut(&mut R, f64) -> f64 { + const SCALE: f64 = (1u64 << 53) as f64; + loop { + // reimplement the f64 generation as an optimisation suggested + // by the Doornik paper: we have a lot of precision-space + // (i.e. there are 11 bits of the 64 of a u64 to use after + // creating a f64), so we might as well reuse some to save + // generating a whole extra random number. (Seems to be 15% + // faster.) + // + // This unfortunately misses out on the benefits of direct + // floating point generation if an RNG like dSMFT is + // used. (That is, such RNGs create floats directly, highly + // efficiently and overload next_f32/f64, so by not calling it + // this may be slower than it would be otherwise.) + // FIXME: investigate/optimise for the above. + let bits: u64 = rng.gen(); + let i = (bits & 0xff) as usize; + let f = (bits >> 11) as f64 / SCALE; + + // u is either U(-1, 1) or U(0, 1) depending on if this is a + // symmetric distribution or not. + let u = if symmetric {2.0 * f - 1.0} else {f}; + let x = u * x_tab[i]; + + let test_x = if symmetric { x.abs() } else {x}; + + // algebraically equivalent to |u| < x_tab[i+1]/x_tab[i] (or u < x_tab[i+1]/x_tab[i]) + if test_x < x_tab[i + 1] { + return x; + } + if i == 0 { + return zero_case(rng, u); + } + // algebraically equivalent to f1 + DRanU()*(f0 - f1) < 1 + if f_tab[i + 1] + (f_tab[i] - f_tab[i + 1]) * rng.gen::() < pdf(x) { + return x; + } + } +} + +#[cfg(test)] +mod tests { + + use {Rng, Rand}; + use super::{RandSample, WeightedChoice, Weighted, Sample, IndependentSample}; + + #[derive(PartialEq, Debug)] + struct ConstRand(usize); + impl Rand for ConstRand { + fn rand(_: &mut R) -> ConstRand { + ConstRand(0) + } + } + + // 0, 1, 2, 3, ... + struct CountingRng { i: u32 } + impl Rng for CountingRng { + fn next_u32(&mut self) -> u32 { + self.i += 1; + self.i - 1 + } + fn next_u64(&mut self) -> u64 { + self.next_u32() as u64 + } + } + + #[test] + fn test_rand_sample() { + let mut rand_sample = RandSample::::new(); + + assert_eq!(rand_sample.sample(&mut ::test::rng()), ConstRand(0)); + assert_eq!(rand_sample.ind_sample(&mut ::test::rng()), ConstRand(0)); + } + #[test] + fn test_weighted_choice() { + // this makes assumptions about the internal implementation of + // WeightedChoice, specifically: it doesn't reorder the items, + // it doesn't do weird things to the RNG (so 0 maps to 0, 1 to + // 1, internally; modulo a modulo operation). + + macro_rules! t { + ($items:expr, $expected:expr) => {{ + let mut items = $items; + let wc = WeightedChoice::new(&mut items); + let expected = $expected; + + let mut rng = CountingRng { i: 0 }; + + for &val in expected.iter() { + assert_eq!(wc.ind_sample(&mut rng), val) + } + }} + } + + t!(vec!(Weighted { weight: 1, item: 10}), [10]); + + // skip some + t!(vec!(Weighted { weight: 0, item: 20}, + Weighted { weight: 2, item: 21}, + Weighted { weight: 0, item: 22}, + Weighted { weight: 1, item: 23}), + [21,21, 23]); + + // different weights + t!(vec!(Weighted { weight: 4, item: 30}, + Weighted { weight: 3, item: 31}), + [30,30,30,30, 31,31,31]); + + // check that we're binary searching + // correctly with some vectors of odd + // length. + t!(vec!(Weighted { weight: 1, item: 40}, + Weighted { weight: 1, item: 41}, + Weighted { weight: 1, item: 42}, + Weighted { weight: 1, item: 43}, + Weighted { weight: 1, item: 44}), + [40, 41, 42, 43, 44]); + t!(vec!(Weighted { weight: 1, item: 50}, + Weighted { weight: 1, item: 51}, + Weighted { weight: 1, item: 52}, + Weighted { weight: 1, item: 53}, + Weighted { weight: 1, item: 54}, + Weighted { weight: 1, item: 55}, + Weighted { weight: 1, item: 56}), + [50, 51, 52, 53, 54, 55, 56]); + } + + #[test] + fn test_weighted_clone_initialization() { + let initial : Weighted = Weighted {weight: 1, item: 1}; + let clone = initial.clone(); + assert_eq!(initial.weight, clone.weight); + assert_eq!(initial.item, clone.item); + } + + #[test] #[should_panic] + fn test_weighted_clone_change_weight() { + let initial : Weighted = Weighted {weight: 1, item: 1}; + let mut clone = initial.clone(); + clone.weight = 5; + assert_eq!(initial.weight, clone.weight); + } + + #[test] #[should_panic] + fn test_weighted_clone_change_item() { + let initial : Weighted = Weighted {weight: 1, item: 1}; + let mut clone = initial.clone(); + clone.item = 5; + assert_eq!(initial.item, clone.item); + + } + + #[test] #[should_panic] + fn test_weighted_choice_no_items() { + WeightedChoice::::new(&mut []); + } + #[test] #[should_panic] + fn test_weighted_choice_zero_weight() { + WeightedChoice::new(&mut [Weighted { weight: 0, item: 0}, + Weighted { weight: 0, item: 1}]); + } + #[test] #[should_panic] + fn test_weighted_choice_weight_overflows() { + let x = ::std::u32::MAX / 2; // x + x + 2 is the overflow + WeightedChoice::new(&mut [Weighted { weight: x, item: 0 }, + Weighted { weight: 1, item: 1 }, + Weighted { weight: x, item: 2 }, + Weighted { weight: 1, item: 3 }]); + } +} diff --git a/rand/src/distributions/normal.rs b/rand/src/distributions/normal.rs new file mode 100644 index 0000000..280613d --- /dev/null +++ b/rand/src/distributions/normal.rs @@ -0,0 +1,201 @@ +// Copyright 2013 The Rust Project Developers. See the COPYRIGHT +// file at the top-level directory of this distribution and at +// http://rust-lang.org/COPYRIGHT. +// +// Licensed under the Apache License, Version 2.0 or the MIT license +// , at your +// option. This file may not be copied, modified, or distributed +// except according to those terms. + +//! The normal and derived distributions. + +use {Rng, Rand, Open01}; +use distributions::{ziggurat, ziggurat_tables, Sample, IndependentSample}; + +/// A wrapper around an `f64` to generate N(0, 1) random numbers +/// (a.k.a. a standard normal, or Gaussian). +/// +/// 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*](http://www.doornik.com/research/ziggurat.pdf). Nuffield +/// College, Oxford +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::normal::StandardNormal; +/// +/// let StandardNormal(x) = rand::random(); +/// println!("{}", x); +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct StandardNormal(pub f64); + +impl Rand for StandardNormal { + fn rand(rng: &mut R) -> StandardNormal { + #[inline] + fn pdf(x: f64) -> f64 { + (-x*x/2.0).exp() + } + #[inline] + fn zero_case(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 Open01(x_) = rng.gen::>(); + let Open01(y_) = rng.gen::>(); + + 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 } + } + + StandardNormal(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. +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::{Normal, IndependentSample}; +/// +/// // mean 2, standard deviation 3 +/// let normal = Normal::new(2.0, 3.0); +/// let v = normal.ind_sample(&mut rand::thread_rng()); +/// println!("{} is from a N(2, 9) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Normal { + mean: f64, + std_dev: f64, +} + +impl Normal { + /// Construct a new `Normal` distribution with the given mean and + /// standard deviation. + /// + /// # Panics + /// + /// Panics if `std_dev < 0`. + #[inline] + pub fn new(mean: f64, std_dev: f64) -> Normal { + assert!(std_dev >= 0.0, "Normal::new called with `std_dev` < 0"); + Normal { + mean: mean, + std_dev: std_dev + } + } +} +impl Sample for Normal { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl IndependentSample for Normal { + fn ind_sample(&self, rng: &mut R) -> f64 { + let StandardNormal(n) = rng.gen::(); + 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 +/// +/// ```rust +/// use rand::distributions::{LogNormal, IndependentSample}; +/// +/// // mean 2, standard deviation 3 +/// let log_normal = LogNormal::new(2.0, 3.0); +/// let v = log_normal.ind_sample(&mut rand::thread_rng()); +/// println!("{} is from an ln N(2, 9) distribution", v) +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct LogNormal { + norm: Normal +} + +impl LogNormal { + /// Construct a new `LogNormal` distribution with the given mean + /// and standard deviation. + /// + /// # Panics + /// + /// Panics if `std_dev < 0`. + #[inline] + pub fn new(mean: f64, std_dev: f64) -> LogNormal { + assert!(std_dev >= 0.0, "LogNormal::new called with `std_dev` < 0"); + LogNormal { norm: Normal::new(mean, std_dev) } + } +} +impl Sample for LogNormal { + fn sample(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) } +} +impl IndependentSample for LogNormal { + fn ind_sample(&self, rng: &mut R) -> f64 { + self.norm.ind_sample(rng).exp() + } +} + +#[cfg(test)] +mod tests { + use distributions::{Sample, IndependentSample}; + use super::{Normal, LogNormal}; + + #[test] + fn test_normal() { + let mut norm = Normal::new(10.0, 10.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + norm.sample(&mut rng); + norm.ind_sample(&mut rng); + } + } + #[test] + #[should_panic] + fn test_normal_invalid_sd() { + Normal::new(10.0, -1.0); + } + + + #[test] + fn test_log_normal() { + let mut lnorm = LogNormal::new(10.0, 10.0); + let mut rng = ::test::rng(); + for _ in 0..1000 { + lnorm.sample(&mut rng); + lnorm.ind_sample(&mut rng); + } + } + #[test] + #[should_panic] + fn test_log_normal_invalid_sd() { + LogNormal::new(10.0, -1.0); + } +} diff --git a/rand/src/distributions/range.rs b/rand/src/distributions/range.rs new file mode 100644 index 0000000..935a00a --- /dev/null +++ b/rand/src/distributions/range.rs @@ -0,0 +1,241 @@ +// Copyright 2013 The Rust Project Developers. See the COPYRIGHT +// file at the top-level directory of this distribution and at +// http://rust-lang.org/COPYRIGHT. +// +// Licensed under the Apache License, Version 2.0 or the MIT license +// , at your +// option. This file may not be copied, modified, or distributed +// except according to those terms. + +//! Generating numbers between two others. + +// this is surprisingly complicated to be both generic & correct + +use core::num::Wrapping as w; + +use Rng; +use distributions::{Sample, IndependentSample}; + +/// Sample values uniformly between two bounds. +/// +/// This gives a uniform distribution (assuming the RNG used to sample +/// it is itself uniform & the `SampleRange` implementation for the +/// given type is correct), even for edge cases like `low = 0u8`, +/// `high = 170u8`, for which a naive modulo operation would return +/// numbers less than 85 with double the probability to those greater +/// than 85. +/// +/// Types should attempt to sample in `[low, high)`, i.e., not +/// including `high`, but this may be very difficult. All the +/// primitive integer types satisfy this property, and the float types +/// normally satisfy it, but rounding may mean `high` can occur. +/// +/// # Example +/// +/// ```rust +/// use rand::distributions::{IndependentSample, Range}; +/// +/// fn main() { +/// let between = Range::new(10, 10000); +/// let mut rng = rand::thread_rng(); +/// let mut sum = 0; +/// for _ in 0..1000 { +/// sum += between.ind_sample(&mut rng); +/// } +/// println!("{}", sum); +/// } +/// ``` +#[derive(Clone, Copy, Debug)] +pub struct Range { + low: X, + range: X, + accept_zone: X +} + +impl Range { + /// Create a new `Range` instance that samples uniformly from + /// `[low, high)`. Panics if `low >= high`. + pub fn new(low: X, high: X) -> Range { + assert!(low < high, "Range::new called with `low >= high`"); + SampleRange::construct_range(low, high) + } +} + +impl Sample for Range { + #[inline] + fn sample(&mut self, rng: &mut R) -> Sup { self.ind_sample(rng) } +} +impl IndependentSample for Range { + fn ind_sample(&self, rng: &mut R) -> Sup { + SampleRange::sample_range(self, rng) + } +} + +/// The helper trait for types that have a sensible way to sample +/// uniformly between two values. This should not be used directly, +/// and is only to facilitate `Range`. +pub trait SampleRange : Sized { + /// Construct the `Range` object that `sample_range` + /// requires. This should not ever be called directly, only via + /// `Range::new`, which will check that `low < high`, so this + /// function doesn't have to repeat the check. + fn construct_range(low: Self, high: Self) -> Range; + + /// Sample a value from the given `Range` with the given `Rng` as + /// a source of randomness. + fn sample_range(r: &Range, rng: &mut R) -> Self; +} + +macro_rules! integer_impl { + ($ty:ty, $unsigned:ident) => { + impl SampleRange for $ty { + // we play free and fast with unsigned vs signed here + // (when $ty is signed), but that's fine, since the + // contract of this macro is for $ty and $unsigned to be + // "bit-equal", so casting between them is a no-op & a + // bijection. + + #[inline] + fn construct_range(low: $ty, high: $ty) -> Range<$ty> { + let range = (w(high as $unsigned) - w(low as $unsigned)).0; + let unsigned_max: $unsigned = ::core::$unsigned::MAX; + + // this is the largest number that fits into $unsigned + // that `range` divides evenly, so, if we've sampled + // `n` uniformly from this region, then `n % range` is + // uniform in [0, range) + let zone = unsigned_max - unsigned_max % range; + + Range { + low: low, + range: range as $ty, + accept_zone: zone as $ty + } + } + + #[inline] + fn sample_range(r: &Range<$ty>, rng: &mut R) -> $ty { + loop { + // rejection sample + let v = rng.gen::<$unsigned>(); + // until we find something that fits into the + // region which r.range evenly divides (this will + // be uniformly distributed) + if v < r.accept_zone as $unsigned { + // and return it, with some adjustments + return (w(r.low) + w((v % r.range as $unsigned) as $ty)).0; + } + } + } + } + } +} + +integer_impl! { i8, u8 } +integer_impl! { i16, u16 } +integer_impl! { i32, u32 } +integer_impl! { i64, u64 } +#[cfg(feature = "i128_support")] +integer_impl! { i128, u128 } +integer_impl! { isize, usize } +integer_impl! { u8, u8 } +integer_impl! { u16, u16 } +integer_impl! { u32, u32 } +integer_impl! { u64, u64 } +#[cfg(feature = "i128_support")] +integer_impl! { u128, u128 } +integer_impl! { usize, usize } + +macro_rules! float_impl { + ($ty:ty) => { + impl SampleRange for $ty { + fn construct_range(low: $ty, high: $ty) -> Range<$ty> { + Range { + low: low, + range: high - low, + accept_zone: 0.0 // unused + } + } + fn sample_range(r: &Range<$ty>, rng: &mut R) -> $ty { + r.low + r.range * rng.gen::<$ty>() + } + } + } +} + +float_impl! { f32 } +float_impl! { f64 } + +#[cfg(test)] +mod tests { + use distributions::{Sample, IndependentSample}; + use super::Range as Range; + + #[should_panic] + #[test] + fn test_range_bad_limits_equal() { + Range::new(10, 10); + } + #[should_panic] + #[test] + fn test_range_bad_limits_flipped() { + Range::new(10, 5); + } + + #[test] + fn test_integers() { + let mut rng = ::test::rng(); + macro_rules! t { + ($($ty:ident),*) => {{ + $( + let v: &[($ty, $ty)] = &[(0, 10), + (10, 127), + (::core::$ty::MIN, ::core::$ty::MAX)]; + for &(low, high) in v.iter() { + let mut sampler: Range<$ty> = Range::new(low, high); + for _ in 0..1000 { + let v = sampler.sample(&mut rng); + assert!(low <= v && v < high); + let v = sampler.ind_sample(&mut rng); + assert!(low <= v && v < high); + } + } + )* + }} + } + #[cfg(not(feature = "i128_support"))] + t!(i8, i16, i32, i64, isize, + u8, u16, u32, u64, usize); + #[cfg(feature = "i128_support")] + t!(i8, i16, i32, i64, i128, isize, + u8, u16, u32, u64, u128, usize); + } + + #[test] + fn test_floats() { + let mut rng = ::test::rng(); + macro_rules! t { + ($($ty:ty),*) => {{ + $( + let v: &[($ty, $ty)] = &[(0.0, 100.0), + (-1e35, -1e25), + (1e-35, 1e-25), + (-1e35, 1e35)]; + for &(low, high) in v.iter() { + let mut sampler: Range<$ty> = Range::new(low, high); + for _ in 0..1000 { + let v = sampler.sample(&mut rng); + assert!(low <= v && v < high); + let v = sampler.ind_sample(&mut rng); + assert!(low <= v && v < high); + } + } + )* + }} + } + + t!(f32, f64) + } + +} diff --git a/rand/src/distributions/ziggurat_tables.rs b/rand/src/distributions/ziggurat_tables.rs new file mode 100644 index 0000000..b6de4bf --- /dev/null +++ b/rand/src/distributions/ziggurat_tables.rs @@ -0,0 +1,280 @@ +// Copyright 2013 The Rust Project Developers. See the COPYRIGHT +// file at the top-level directory of this distribution and at +// http://rust-lang.org/COPYRIGHT. +// +// Licensed under the Apache License, Version 2.0 or the MIT license +// , at your +// option. This file may not be copied, modified, or distributed +// except according to those terms. + +// 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, + 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