abax 0.1.21

A lightweight Rust library providing high-precision mathematical constants and special functions, including Bernoulli numbers, Riemann Zeta values, and robust incomplete gamma functions.
Documentation 
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use crate::erfc;
use std::f64::consts::SQRT_2;

/// Normal cumulative distribution function (CDF).
///
/// Given a value `x`, a mean `mu`, and a standard deviation `sigma`,
/// this function returns the probability that a normal random variable
/// is less than or equal to `x`.
///
/// # Mathematical Definition
/// For a normal distribution with mean <math><mi>μ</mi></math> and standard deviation <math><mi>σ</mi></math>:
/// <math display="block"><mi>Φ</mi><mo>(</mo><mi>x</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>σ</mi><mo>)</mo><mo>=</mo><msub><mi>Φ</mi><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>[</mo><mn>1</mn><mo>+</mo><mi>erf</mi><mo>(</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mi>μ</mi></mrow><mrow><mi>σ</mi><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>)</mo><mo>]</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>erfc</mi><mo>(</mo><mo>-</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mi>μ</mi></mrow><mrow><mi>σ</mi><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>)</mo></math>
///
/// # Domain
/// - `sigma >= 0`
/// - Returns `NaN` if `sigma` is negative or any input is `NaN`.
/// - If `sigma` is 0, the distribution is treated as a Dirac delta concentrated at `mu`.
///
/// # Examples
/// ```
/// use abax::normcdf;
///
/// // Standard normal median is 0.5
/// assert!((normcdf(0.0, 0.0, 1.0) - 0.5).abs() < 1e-15);
/// // One sigma upper bound
/// assert!((normcdf(1.0, 0.0, 1.0) - 0.8413447460685429).abs() < 1e-15);
/// ```
pub fn normcdf(x: f64, mu: f64, sigma: f64) -> f64 {
    if x.is_nan() || mu.is_nan() || sigma.is_nan() || sigma < 0.0 {
        return f64::NAN;
    }
 
    if sigma == 0.0 {
        return if x < mu { 0.0 } else { 1.0 };
    }
 
    let z = (x - mu) / sigma;
    0.5 * erfc(-z / SQRT_2)
}

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

    #[test]
    fn test_normcdf_standard() {
        let tol = 1e-14;
        assert!((normcdf(0.0, 0.0, 1.0) - 0.5).abs() < tol);
        assert!((normcdf(1.959963984540054, 0.0, 1.0) - 0.975).abs() < tol);
        assert!((normcdf(-1.959963984540054, 0.0, 1.0) - 0.025).abs() < tol);
    }
 
    #[test]
    fn test_normcdf_zero_sigma() {
        assert_eq!(normcdf(5.0, 5.0, 0.0), 1.0);
        assert_eq!(normcdf(4.99, 5.0, 0.0), 0.0);
        assert_eq!(normcdf(5.01, 5.0, 0.0), 1.0);
    }

    #[test]
    fn test_normcdf_invalid() {
        assert!(normcdf(0.5, 0.0, -1.0).is_nan());
    }
}