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John Doty 2022-12-19 08:27:18 -08:00
parent 34d1830413
commit 9c435dc440
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124
vendor/rand/src/distributions/exponential.rs vendored Normal file
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// 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 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://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 {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::<f64>().ln()` but that is slower.
impl Rand for Exp1 {
#[inline]
fn rand<R:Rng>(rng: &mut R) -> Exp1 {
#[inline]
fn pdf(x: f64) -> f64 {
(-x).exp()
}
#[inline]
fn zero_case<R:Rng>(rng: &mut R, _u: f64) -> f64 {
ziggurat_tables::ZIG_EXP_R - rng.gen::<f64>().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<f64> for Exp {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for Exp {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
let Exp1(n) = rng.gen::<Exp1>();
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);
}
}

386
vendor/rand/src/distributions/gamma.rs vendored Normal file
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// 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 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, 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<f64> for Gamma {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl Sample<f64> for GammaSmallShape {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl Sample<f64> for GammaLargeShape {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for Gamma {
fn ind_sample<R: Rng>(&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<f64> for GammaSmallShape {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
let Open01(u) = rng.gen::<Open01<f64>>();
self.large_shape.ind_sample(rng) * u.powf(self.inv_shape)
}
}
impl IndependentSample<f64> for GammaLargeShape {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
loop {
let StandardNormal(x) = rng.gen::<StandardNormal>();
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::<Open01<f64>>();
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<f64> for ChiSquared {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for ChiSquared {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
match self.repr {
DoFExactlyOne => {
// k == 1 => N(0,1)^2
let StandardNormal(norm) = rng.gen::<StandardNormal>();
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<f64> for FisherF {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for FisherF {
fn ind_sample<R: Rng>(&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<f64> for StudentT {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for StudentT {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
let StandardNormal(norm) = rng.gen::<StandardNormal>();
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);
}
}
}

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// 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 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, 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<Support> {
/// Generate a random value of `Support`, using `rng` as the
/// source of randomness.
fn sample<R: Rng>(&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<Support>: Sample<Support> {
/// Generate a random value.
fn ind_sample<R: Rng>(&self, &mut R) -> Support;
}
/// A wrapper for generating types that implement `Rand` via the
/// `Sample` & `IndependentSample` traits.
#[derive(Debug)]
pub struct RandSample<Sup> {
_marker: marker::PhantomData<fn() -> Sup>,
}
impl<Sup> Copy for RandSample<Sup> {}
impl<Sup> Clone for RandSample<Sup> {
fn clone(&self) -> Self { *self }
}
impl<Sup: Rand> Sample<Sup> for RandSample<Sup> {
fn sample<R: Rng>(&mut self, rng: &mut R) -> Sup { self.ind_sample(rng) }
}
impl<Sup: Rand> IndependentSample<Sup> for RandSample<Sup> {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> Sup {
rng.gen()
}
}
impl<Sup> RandSample<Sup> {
pub fn new() -> RandSample<Sup> {
RandSample { _marker: marker::PhantomData }
}
}
/// A value with a particular weight for use with `WeightedChoice`.
#[derive(Copy, Clone, Debug)]
pub struct Weighted<T> {
/// 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<T>],
weight_range: Range<u32>
}
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<T>]) -> 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<T> for WeightedChoice<'a, T> {
fn sample<R: Rng>(&mut self, rng: &mut R) -> T { self.ind_sample(rng) }
}
impl<'a, T: Clone> IndependentSample<T> for WeightedChoice<'a, T> {
fn ind_sample<R: Rng>(&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<R: Rng, 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 {
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::<f64>() < 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<R: Rng>(_: &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::<ConstRand>::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<u32> = 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<u32> = 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<u32> = 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::<isize>::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 }]);
}
}

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// 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 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://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 {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<R:Rng>(rng: &mut R) -> StandardNormal {
#[inline]
fn pdf(x: f64) -> f64 {
(-x*x/2.0).exp()
}
#[inline]
fn zero_case<R:Rng>(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::<Open01<f64>>();
let Open01(y_) = rng.gen::<Open01<f64>>();
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<f64> for Normal {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for Normal {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
let StandardNormal(n) = rng.gen::<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
///
/// ```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<f64> for LogNormal {
fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
}
impl IndependentSample<f64> for LogNormal {
fn ind_sample<R: Rng>(&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);
}
}

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// 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 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, 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<X> {
low: X,
range: X,
accept_zone: X
}
impl<X: SampleRange + PartialOrd> Range<X> {
/// Create a new `Range` instance that samples uniformly from
/// `[low, high)`. Panics if `low >= high`.
pub fn new(low: X, high: X) -> Range<X> {
assert!(low < high, "Range::new called with `low >= high`");
SampleRange::construct_range(low, high)
}
}
impl<Sup: SampleRange> Sample<Sup> for Range<Sup> {
#[inline]
fn sample<R: Rng>(&mut self, rng: &mut R) -> Sup { self.ind_sample(rng) }
}
impl<Sup: SampleRange> IndependentSample<Sup> for Range<Sup> {
fn ind_sample<R: Rng>(&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<Self>;
/// Sample a value from the given `Range` with the given `Rng` as
/// a source of randomness.
fn sample_range<R: Rng>(r: &Range<Self>, 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 $ty
}
}
#[inline]
fn sample_range<R: Rng>(r: &Range<$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: Rng>(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)
}
}

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// 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 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://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,
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