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``````// Copyright 2018 Developers of the Rand project.
// Copyright 2016-2017 The Rust Project Developers.
//
// option. This file may not be copied, modified, or distributed
// except according to those terms.

//! The Cauchy distribution `Cauchy(x₀, γ)`.

use crate::{Distribution, Standard};
use core::fmt;
use num_traits::{Float, FloatConst};
use rand::Rng;

/// The [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution) `Cauchy(x₀, γ)`.
///
/// The Cauchy distribution is a continuous probability distribution with
/// parameters `x₀` (median) and `γ` (scale).
/// It describes the distribution of the ratio of two independent
/// normally distributed random variables with means `x₀` and scales `γ`.
/// In other words, if `X` and `Y` are independent normally distributed
/// random variables with means `x₀` and scales `γ`, respectively, then
/// `X / Y` is `Cauchy(x₀, γ)` distributed.
///
/// # Density function
///
/// `f(x) = 1 / (π * γ * (1 + ((x - x₀) / γ)²))`
///
/// # Plot
///
/// The plot illustrates the Cauchy distribution with various values of `x₀` and `γ`.
/// Note how the median parameter `x₀` shifts the distribution along the x-axis,
/// and how the scale `γ` changes the density around the median.
///
/// The standard Cauchy distribution is the special case with `x₀ = 0` and `γ = 1`,
/// which corresponds to the ratio of two [`StandardNormal`](crate::StandardNormal) distributions.
///
/// ![Cauchy distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/cauchy.svg)
///
/// # Example
///
/// ```
/// use rand_distr::{Cauchy, Distribution};
///
/// let cau = Cauchy::new(2.0, 5.0).unwrap();
/// let v = cau.sample(&mut rand::thread_rng());
/// println!("{} is from a Cauchy(2, 5) distribution", v);
/// ```
///
/// # Notes
///
/// Note that at least for `f32`, results are not fully portable due to minor
/// differences in the target system's *tan* implementation, `tanf`.
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
pub struct Cauchy<F>
where
F: Float + FloatConst,
Standard: Distribution<F>,
{
median: F,
scale: F,
}

/// Error type returned from [`Cauchy::new`].
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum Error {
/// `scale <= 0` or `nan`.
ScaleTooSmall,
}

impl fmt::Display for Error {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
Error::ScaleTooSmall => "scale is not positive in Cauchy distribution",
})
}
}

#[cfg(feature = "std")]
impl std::error::Error for Error {}

impl<F> Cauchy<F>
where
F: Float + FloatConst,
Standard: Distribution<F>,
{
/// Construct a new `Cauchy` with the given shape parameters
/// `median` the peak location and `scale` the scale factor.
pub fn new(median: F, scale: F) -> Result<Cauchy<F>, Error> {
if !(scale > F::zero()) {
return Err(Error::ScaleTooSmall);
}
Ok(Cauchy { median, scale })
}
}

impl<F> Distribution<F> for Cauchy<F>
where
F: Float + FloatConst,
Standard: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
// sample from [0, 1)
let x = Standard.sample(rng);
// get standard cauchy random number
// note that π/2 is not exactly representable, even if x=0.5 the result is finite
let comp_dev = (F::PI() * x).tan();
// shift and scale according to parameters
self.median + self.scale * comp_dev
}
}

#[cfg(test)]
mod test {
use super::*;

fn median(numbers: &mut [f64]) -> f64 {
sort(numbers);
let mid = numbers.len() / 2;
numbers[mid]
}

fn sort(numbers: &mut [f64]) {
numbers.sort_by(|a, b| a.partial_cmp(b).unwrap());
}

#[test]
fn test_cauchy_averages() {
// NOTE: given that the variance and mean are undefined,
// this test does not have any rigorous statistical meaning.
let cauchy = Cauchy::new(10.0, 5.0).unwrap();
let mut rng = crate::test::rng(123);
let mut numbers: [f64; 1000] = [0.0; 1000];
let mut sum = 0.0;
for number in &mut numbers[..] {
*number = cauchy.sample(&mut rng);
sum += *number;
}
let median = median(&mut numbers);
#[cfg(feature = "std")]
std::println!("Cauchy median: {}", median);
assert!((median - 10.0).abs() < 0.4); // not 100% certain, but probable enough
let mean = sum / 1000.0;
#[cfg(feature = "std")]
std::println!("Cauchy mean: {}", mean);
// for a Cauchy distribution the mean should not converge
assert!((mean - 10.0).abs() > 0.4); // not 100% certain, but probable enough
}

#[test]
#[should_panic]
fn test_cauchy_invalid_scale_zero() {
Cauchy::new(0.0, 0.0).unwrap();
}

#[test]
#[should_panic]
fn test_cauchy_invalid_scale_neg() {
Cauchy::new(0.0, -10.0).unwrap();
}

#[test]
fn value_stability() {
fn gen_samples<F: Float + FloatConst + fmt::Debug>(m: F, s: F, buf: &mut [F])
where
Standard: Distribution<F>,
{
let distr = Cauchy::new(m, s).unwrap();
let mut rng = crate::test::rng(353);
for x in buf {
*x = rng.sample(distr);
}
}

let mut buf = [0.0; 4];
gen_samples(100f64, 10.0, &mut buf);
assert_eq!(
&buf,
&[
77.93369152808678,
90.1606912098641,
125.31516221323625,
86.10217834773925
]
);

// Unfortunately this test is not fully portable due to reliance on the
// system's implementation of tanf (see doc on Cauchy struct).
let mut buf = [0.0; 4];
gen_samples(10f32, 7.0, &mut buf);
let expected = [15.023088, -5.446413, 3.7092876, 3.112482];
for (a, b) in buf.iter().zip(expected.iter()) {
assert_almost_eq!(*a, *b, 1e-5);
}
}

#[test]
fn cauchy_distributions_can_be_compared() {
assert_eq!(Cauchy::new(1.0, 2.0), Cauchy::new(1.0, 2.0));
}
}
``````