The function Φ(x) is the cumulative density function (CDF) of a standard normal (Gaussian) random variable. It is closely related to the error function erf(x).
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); }
A&S refers to Handbook of Mathematical Functions by Abramowitz and Stegun. See Stand-alone error function for details of the algorithm.
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