Code shared from the Rust Playground

playground.rs

use approx::assert_relative_eq;

fn gaussian_integral_approximation(start: f64, end: f64, steps: usize) -> f64 {
    let dx = (end - start) / steps as f64;
    let f = |x: f64| (-x.powi(2)).exp();

    let mut sum = (f(start) + f(end)) * 0.5;

    for i in 1..steps {
        let x = start + (i as f64 * dx);
        sum += f(x);
    }

    sum * dx
}

fn main() {
    let bounds = 10.0;
    let val = gaussian_integral_approximation(-bounds, bounds, 1_000_000);
    println!("Value: {}", val);
}

#[cfg(test)]
mod tests {
    use super::*;
    use std::f64::consts::PI;

    #[test]
    fn test_gaussian_full_range() {
        let theoretical = PI.sqrt();
        // Integration from -10 to 10 is sufficient for f64 precision
        let estimated = gaussian_integral_approximation(-10.0, 10.0, 1_000_000);
        
        assert_relative_eq!(theoretical, estimated, epsilon = 1e-10);
    }

    #[test]
    fn test_symmetry_half_integral() {
        let theoretical = PI.sqrt() / 2.0;
        // The Gaussian curve is symmetric; the integral from 0 to infinity is half
        let estimated = gaussian_integral_approximation(0.0, 10.0, 500_000);
        
        assert_relative_eq!(theoretical, estimated, epsilon = 1e-10);
    }

    #[test]
    #[ignore]
    fn test_convergence_improvement() {
        let theoretical = PI.sqrt();
        
        let coarse = gaussian_integral_approximation(-10.0, 10.0, 100);
        let fine = gaussian_integral_approximation(-10.0, 10.0, 10_000);
        
        let error_coarse = (theoretical - coarse).abs();
        let error_fine = (theoretical - fine).abs();

        // Higher step count should result in a smaller error
        assert!(error_fine < error_coarse);
    }

    #[test]
    fn test_zero_width_interval() {
        // Integrating from 'a' to 'a' should always be zero
        let result = gaussian_integral_approximation(5.0, 5.0, 100);
        assert_eq!(result, 0.0);
    }
}

https://play.rust-lang.org/?version=stable&mode=debug&edition=2024&gist=883e05211d6e1560bcfbde7fc85740bb