CDF cumulative density function F(x), Cook,

Stand-alone code for numerical computing

https://www.johndcook.com/blog/

https://www.johndcook.com/blog/stand_alone_code/

https://www.johndcook.com/blog/cpp_phi/

https://www.johndcook.com/erf_and_normal_cdf.pdf

#include <cmath>

double phi(double x)
{
    // constants
    double a1 =  0.254829592;
    double a2 = -0.284496736;
    double a3 =  1.421413741;
    double a4 = -1.453152027;
    double a5 =  1.061405429;
    double p  =  0.3275911;

    // Save the sign of x
    int sign = 1;
    if (x < 0)
        sign = -1;
    x = fabs(x)/sqrt(2.0);

    // A&S formula 7.1.26
    double t = 1.0/(1.0 + p*x);
    double y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t*exp(-x*x);

    return 0.5*(1.0 + sign*y);
}

void testPhi()
{
    // Select a few input values
    double x[] = 
    {
        -3, 
        -1, 
        0.0, 
        0.5, 
        2.1 
    };

    // Output computed by Mathematica
    // y = Phi[x]
    double y[] = 
    { 
        0.00134989803163, 
        0.158655253931, 
        0.5, 
        0.691462461274, 
        0.982135579437 
    };

        int numTests = sizeof(x)/sizeof(double);

    double maxError = 0.0;
    for (int i = 0; i < numTests; ++i)
    {
        double error = fabs(y[i] - phi(x[i]));
        if (error > maxError)
            maxError = error;
    }

        std::cout << "Maximum error: " << maxError << "\n";
}

C#

https://www.johndcook.com/blog/csharp_phi/

static double Phi(double x)
{
    // constants
    double a1 = 0.254829592;
    double a2 = -0.284496736;
    double a3 = 1.421413741;
    double a4 = -1.453152027;
    double a5 = 1.061405429;
    double p = 0.3275911;
        
    // Save the sign of x
    int sign = 1;
    if (x < 0)
        sign = -1;
    x = Math.Abs(x) / Math.Sqrt(2.0);
        
    // A&S formula 7.1.26
    double t = 1.0 / (1.0 + p*x);
    double y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t * Math.Exp(-x*x);
        
    return 0.5 * (1.0 + sign*y);
}

static void TestPhi()
{
    // Select a few input values
    double[] x = 
    {
        -3, 
        -1, 
        0.0, 
        0.5, 
        2.1 
    };

    // Output computed by Mathematica
    // y = Phi[x]
    double[] y = 
    { 
        0.00134989803163, 
        0.158655253931, 
        0.5, 
        0.691462461274, 
        0.982135579437 
    };

    double maxError = 0.0;
    for (int i = 0; i < x.Length; ++i)
    {
        double error = Math.Abs(y[i] - Phi(x[i]));
        if (error > maxError)
            maxError = error;
    }

        Console.WriteLine("Maximum error: {0}", maxError);
} 

Python

https://www.johndcook.com/blog/python_phi/

import math

def phi(x):
    # constants
    a1 =  0.254829592
    a2 = -0.284496736
    a3 =  1.421413741
    a4 = -1.453152027
    a5 =  1.061405429
    p  =  0.3275911

    # Save the sign of x
    sign = 1
    if x < 0:
        sign = -1
    x = abs(x)/math.sqrt(2.0)

    # A&S formula 7.1.26
    t = 1.0/(1.0 + p*x)
    y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t*math.exp(-x*x)

    return 0.5*(1.0 + sign*y)