private static double inverse(double a, double y) { // bound the solution var x0 = double.MaxValue; double yl = 0; double x1 = 0; var yh = 1.0; var dithresh = 5.0 * Constants.DoubleEpsilon; // approximation to inverse function var d = 1.0 / (9.0 * a); var yy = 1.0 - d - Normal.Inverse(y) * Math.Sqrt(d); var x = a * yy * yy * yy; var lgm = Log(a); for (var i = 0; i < 10; i++) { if (x > x0 || x < x1) { goto ihalve; } yy = UpperIncomplete(a, x); if (yy < yl || yy > yh) { goto ihalve; } if (yy < y) { x0 = x; yl = yy; } else { x1 = x; yh = yy; } // compute the derivative of the function at this point d = (a - 1.0) * Math.Log(x) - x - lgm; if (d < -Constants.LogMax) { goto ihalve; } d = -Math.Exp(d); // compute the step to the next approximation of x d = (yy - y) / d; if (Math.Abs(d / x) < Constants.DoubleEpsilon) { return(x); } x = x - d; } // Resort to interval halving if Newton iteration did not converge. ihalve: d = 0.0625; if (x0 == double.MaxValue) { if (x <= 0.0) { x = 1.0; } while (x0 == double.MaxValue && !double.IsNaN(x)) { x = (1.0 + d) * x; yy = UpperIncomplete(a, x); if (yy < y) { x0 = x; yl = yy; break; } d = d + d; } } d = 0.5; double dir = 0; for (var i = 0; i < 400; i++) { var t = x1 + d * (x0 - x1); if (double.IsNaN(t)) { break; } x = t; yy = UpperIncomplete(a, x); lgm = (x0 - x1) / (x1 + x0); if (Math.Abs(lgm) < dithresh) { break; } lgm = (yy - y) / y; if (Math.Abs(lgm) < dithresh) { break; } if (x <= 0.0) { break; } if (yy >= y) { x1 = x; yh = yy; if (dir < 0) { dir = 0; d = 0.5; } else if (dir > 1) { d = 0.5 * d + 0.5; } else { d = (y - yl) / (yh - yl); } dir += 1; } else { x0 = x; yl = yy; if (dir > 0) { dir = 0; d = 0.5; } else if (dir < -1) { d = 0.5 * d; } else { d = (y - yl) / (yh - yl); } dir -= 1; } } if (x == 0.0 || double.IsNaN(x)) { throw new ArithmeticException(); } return(x); }