/// <summary>
/// Normal (Gaussian) inverse cumulative distribution function.
/// </summary>
///
/// <remarks>
/// <para>
/// For small arguments <c>0 < y < exp(-2)</c>, the program computes <c>z =
/// sqrt( -2.0 * log(y) )</c>; then the approximation is <c>x = z - log(z)/z -
/// (1/z) P(1/z) / Q(1/z)</c>.</para>
/// <para>
/// There are two rational functions P/Q, one for <c>0 < y < exp(-32)</c> and
/// the other for <c>y</c> up to <c>exp(-2)</c>. For larger arguments, <c>w = y - 0.5</c>,
/// and <c>x/sqrt(2pi) = w + w^3 * R(w^2)/S(w^2))</c>.</para>
/// </remarks>
///
/// <returns>
/// Returns the value, <c>x</c>, for which the area under the Normal (Gaussian)
/// probability density function (integrated from minus infinity to <c>x</c>) is
/// equal to the argument <c>y</c> (assumes mean is zero, variance is one).
/// </returns>
///
public static double Inverse(double y0)
{
if (y0 <= 0.0)
{
if (y0 == 0)
{
return(Double.NegativeInfinity);
}
throw new ArgumentOutOfRangeException("y0");
}
if (y0 >= 1.0)
{
if (y0 == 1)
{
return(Double.PositiveInfinity);
}
throw new ArgumentOutOfRangeException("y0");
}
double s2pi = Math.Sqrt(2.0 * Math.PI);
int code = 1;
double y = y0;
double x;
if (y > 0.8646647167633873)
{
y = 1.0 - y;
code = 0;
}
if (y > 0.1353352832366127)
{
y -= 0.5;
double y2 = y * y;
x = y + y * ((y2 * Polynomial.Evaluate(y2, inverse_P0, 4)) / Polynomial.EvaluateSpecial(y2, inverse_Q0, 8));
x *= s2pi;
return(x);
}
x = Math.Sqrt(-2.0 * Math.Log(y));
double x0 = x - Math.Log(x) / x;
double z = 1.0 / x;
double x1;
if (x < 8.0)
{
x1 = (z * Polynomial.Evaluate(z, inverse_P1, 8)) / Polynomial.EvaluateSpecial(z, inverse_Q1, 8);
}
else
{
x1 = (z * Polynomial.Evaluate(z, inverse_P2, 8)) / Polynomial.EvaluateSpecial(z, inverse_Q2, 8);
}
x = x0 - x1;
if (code != 0)
{
x = -x;
}
return(x);
}