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fix: compute Cauchy::{cdf,sf} tails as atan(scale / d) / pi #476

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fix: compute Cauchy::{cdf,sf} tails as atan(scale / d) / pi #476

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43 changes: 39 additions & 4 deletions src/distribution/cauchy.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -133,7 +133,13 @@ impl ContinuousCDF<f64, f64> for Cauchy {
///
/// where `x_0` is the location and `γ` is the scale
fn cdf(&self, x: f64) -> f64 {
(1.0 / f64_consts::PI) * ((x - self.location) / self.scale).atan() + 0.5
let d = self.location - x;
// z < -1: 0.5 + atan(z) / pi cancels, and z = (x - loc) / scale can overflow
if d > self.scale {
(self.scale / d).atan() / f64_consts::PI
} else {
(1.0 / f64_consts::PI) * ((x - self.location) / self.scale).atan() + 0.5
}
}

/// Calculates the survival function for the
Expand All @@ -149,7 +155,12 @@ impl ContinuousCDF<f64, f64> for Cauchy {
/// note that this is identical to the cdf except for
/// the negative argument to the arctan function
fn sf(&self, x: f64) -> f64 {
(1.0 / f64_consts::PI) * ((self.location - x) / self.scale).atan() + 0.5
let d = x - self.location;
if d > self.scale {
(self.scale / d).atan() / f64_consts::PI
} else {
(1.0 / f64_consts::PI) * ((self.location - x) / self.scale).atan() + 0.5
}
}

/// Calculates the inverse cumulative distribution function for the
Expand Down Expand Up @@ -284,6 +295,7 @@ impl Continuous<f64, f64> for Cauchy {
mod tests {
use super::*;
use crate::distribution::internal::density_util;
use crate::prec;
crate::distribution::internal::testing_boiler!(location: f64, scale: f64; Cauchy; CauchyError);

#[test]
Expand Down Expand Up @@ -468,12 +480,12 @@ mod tests {
test_absolute(0.0, 0.1, 0.9682744825694465, 1e-16, sf(-1.0));
test_exact(0.0, 0.1, 0.5, sf(0.0));
test_absolute(0.0, 0.1, 0.03172551743055352, 1e-16, sf(1.0));
test_exact(0.0, 0.1, 0.006365349100972806, sf(5.0));
test_absolute(0.0, 0.1, 0.006365349100972796679298, 1e-16, sf(5.0));
test_absolute(0.0, 1.0, 0.9371670418109989, 1e-16, sf(-5.0));
test_exact(0.0, 1.0, 0.75, sf(-1.0));
test_exact(0.0, 1.0, 0.5, sf(0.0));
test_exact(0.0, 1.0, 0.25, sf(1.0));
test_exact(0.0, 1.0, 0.06283295818900114, sf(5.0));
test_absolute(0.0, 1.0, 0.06283295818900118381375, 1e-16, sf(5.0));
test_exact(0.0, 10.0, 0.6475836176504333, sf(-5.0));
test_exact(0.0, 10.0, 0.5317255174305535, sf(-1.0));
test_exact(0.0, 10.0, 0.5, sf(0.0));
Expand All @@ -496,6 +508,29 @@ mod tests {
test_exact(f64::INFINITY, 1.0, 1.0, sf(5.0));
}

#[test]
fn test_cdf_sf_tails() {
// cdf(-d) = sf(d) = atan2(scale, d) / pi, from mpmath at 50 digits on the exact f64 d
let cases = [
(100.0, 0.003182992764908255149119),
(1e12, 3.183098861837906715378e-13),
(1e16, 3.183098861837906715378e-17),
(1e300, 3.18309886183790654825e-301),
];

let dist = create_ok(0.0, 1.0);
for (d, expected) in cases {
prec::assert_relative_eq!(dist.cdf(-d), expected, epsilon = 0.0, max_relative = 1e-15);
prec::assert_relative_eq!(dist.sf(d), expected, epsilon = 0.0, max_relative = 1e-15);
}

// 1e10 / 1e-300 overflows and the result is subnormal, so the tolerance is absolute
let cdf = |arg: f64| move |x: Cauchy| x.cdf(arg);
let sf = |arg: f64| move |x: Cauchy| x.sf(arg);
test_absolute(0.0, 1e-300, 3.183098861837906795143e-311, 1e-322, cdf(-1e10));
test_absolute(0.0, 1e-300, 3.183098861837906795143e-311, 1e-322, sf(1e10));
}

#[test]
fn test_inverse_cdf() {
let func = |arg: f64| move |x: Cauchy| x.inverse_cdf(x.cdf(arg));
Expand Down
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