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use crate::betaln;
/// Regularized incomplete beta function <math><msub><mi>I</mi><mi>x</mi></msub><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math>.
///
/// Solves for:
/// - <math><msub><mi>I</mi><mi>x</mi></msub><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math> when `lower = true` (regularized lower incomplete beta)
/// - <math><mn>1</mn><mo>-</mo><msub><mi>I</mi><mi>x</mi></msub><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math> when `lower = false` (regularized upper incomplete beta)
///
/// # Domain
/// - `0 <= x <= 1`
/// - `z > 0`, `w > 0`
/// - Invalid inputs return `NaN`.
pub fn betainc(x: f64, z: f64, w: f64, lower: bool) -> f64 {
if x.is_nan() || z.is_nan() || w.is_nan() || x < 0.0 || x > 1.0 || z <= 0.0 || w <= 0.0 {
return f64::NAN;
}
if x == 0.0 {
return if lower { 0.0 } else { 1.0 };
}
if x == 1.0 { // Corrected from `x == 1.0` to `x == 1.0` (no change, just re-evaluating)
return if lower { 1.0 } else { 0.0 };
}
if z == w {
if z == 1.0 {
return x;
}
if x == 0.5 {
return 0.5;
}
}
// Use symmetry: I_x(a, b) = 1 - I_{1-x}(b, a)
// To ensure the continued fraction converges efficiently, we want x < (z+1)/(z+w+2)
if x > (z + 1.0) / (z + w + 2.0) {
return if lower {
1.0 - betainc_cf(1.0 - x, w, z)
} else {
betainc_cf(1.0 - x, w, z)
};
}
let val = betainc_cf(x, z, w);
if lower { val } else { 1.0 - val }
}
/// Evaluates the continued fraction for the regularized incomplete beta function.
/// Uses Lentz's method for stability.
fn betainc_cf(x: f64, z: f64, w: f64) -> f64 {
let ln_beta = betaln(z, w);
let front = (z * x.ln() + w * (1.0 - x).ln() - ln_beta).exp() / z;
let mut c = 1.0;
let mut d = 1.0 - (z + w) * x / (z + 1.0);
let tiny = 1e-30;
if d.abs() < tiny { d = tiny; }
d = 1.0 / d;
let mut h = d;
for m in 1..200 {
let m_f = m as f64;
let m2 = 2.0 * m_f;
// Even step (2m)
let aa = m_f * (w - m_f) * x / ((z + m2 - 1.0) * (z + m2));
d = 1.0 + aa * d;
if d.abs() < tiny { d = tiny; }
c = 1.0 + aa / c;
if c.abs() < tiny { c = tiny; }
d = 1.0 / d;
h *= d * c;
// Odd step (2m+1)
let aa = -(z + m_f) * (z + w + m_f) * x / ((z + m2) * (z + m2 + 1.0));
d = 1.0 + aa * d;
if d.abs() < tiny { d = tiny; }
c = 1.0 + aa / c;
if c.abs() < tiny { c = tiny; }
d = 1.0 / d;
let delta = d * c;
h *= delta;
if (delta - 1.0).abs() < 1e-16 {
break;
}
}
front * h
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_betainc_basic() {
// I_0.5(1, 1) = 0.5
assert!((betainc(0.5, 1.0, 1.0, true) - 0.5).abs() < 1e-15);
// I_0.5(2, 2) = 0.5
assert!((betainc(0.5, 2.0, 2.0, true) - 0.5).abs() < 1e-15);
// I_0.2(1, 3) = 1 - (1-0.2)^3 = 0.488
assert!((betainc(0.2, 1.0, 3.0, true) - 0.488).abs() < 1e-15);
}
#[test]
fn test_betainc_symmetry() {
let x = 0.3;
let z = 2.5;
let w = 1.5;
let lower = betainc(x, z, w, true);
let upper = betainc(x, z, w, false);
assert!((lower + upper - 1.0).abs() < 1e-15);
}
#[test]
fn test_betainc_boundaries() {
assert_eq!(betainc(0.0, 1.0, 1.0, true), 0.0);
assert_eq!(betainc(1.0, 1.0, 1.0, true), 1.0);
}
}