source: sasview/sansmodels/src/c_models/gamma_win.c @ acd0fd10

// Visit http://www.johndcook.com/stand_alone_code.html for the source of this code and more like it.
// Modified for the DANSE project.

#include <math.h>

#include "gamma_win.h"

// Note that the functions Gamma and LogGamma are mutually dependent.

double gamma(double x)
{
  // Split the function domain into three intervals:
  // (0, 0.001), [0.001, 12), and (12, infinity)

  ///////////////////////////////////////////////////////////////////////////
  // First interval: (0, 0.001)
  //
  // For small x, 1/Gamma(x) has power series x + gamma x^2  - ...
  // So in this range, 1/Gamma(x) = x + gamma x^2 with error on the order of x^3.
  // The relative error over this interval is less than 6e-7.

  const double gamma = 0.577215664901532860606512090; // Euler's gamma constant
  double y;
  int arg_was_less_than_one = 0;
  int n = 0;
  // numerator coefficients for approximation over the interval (1,2)
  static const double p[] =
  {
      -1.71618513886549492533811E+0,
       2.47656508055759199108314E+1,
      -3.79804256470945635097577E+2,
       6.29331155312818442661052E+2,
       8.66966202790413211295064E+2,
      -3.14512729688483675254357E+4,
      -3.61444134186911729807069E+4,
       6.64561438202405440627855E+4
  };
  // denominator coefficients for approximation over the interval (1,2)
  static const double q[] =
  {
      -3.08402300119738975254353E+1,
       3.15350626979604161529144E+2,
      -1.01515636749021914166146E+3,
      -3.10777167157231109440444E+3,
       2.25381184209801510330112E+4,
       4.75584627752788110767815E+3,
      -1.34659959864969306392456E+5,
      -1.15132259675553483497211E+5
  };
  double num = 0.0;
  double den = 1.0;
  int i;
  double z;
  double result;

  if (x < 0.001)
      return 1.0/(x*(1.0 + gamma*x));

  ///////////////////////////////////////////////////////////////////////////
  // Second interval: [0.001, 12)

  if (x < 12.0)
    {
      // The algorithm directly approximates gamma over (1,2) and uses
      // reduction identities to reduce other arguments to this interval.

      y = x;

      if (y < 1.0) arg_was_less_than_one = 1;

      // Add or subtract integers as necessary to bring y into (1,2)
      // Will correct for this below
      if (arg_was_less_than_one==1) {
          y += 1.0;
      } else {
          n = (int) (floor(y)) - 1;  // will use n later
          y -= n;
      }

      z = y - 1;
      for (i = 0; i < 8; i++) {
          num = (num + p[i])*z;
          den = den*z + q[i];
      }
      result = num/den + 1.0;

      // Apply correction if argument was not initially in (1,2)
      if (arg_was_less_than_one==1) {
          // Use identity gamma(z) = gamma(z+1)/z
          // The variable "result" now holds gamma of the original y + 1
          // Thus we use y-1 to get back the orginal y.
          result /= (y-1.0);
      } else {
          // Use the identity gamma(z+n) = z*(z+1)* ... *(z+n-1)*gamma(z)
          for (i = 0; i < n; i++)
              result *= y++;
      }

      return result;
    }

    return exp(lgamma(x));
}

double lgamma (double x)
{
  static const double c[8] =
  {
   1.0/12.0,
  -1.0/360.0,
  1.0/1260.0,
  -1.0/1680.0,
  1.0/1188.0,
  -691.0/360360.0,
  1.0/156.0,
  -3617.0/122400.0
  };
  double z = 1.0/(x*x);
  double sum = c[7];
  static const double halfLogTwoPi = 0.91893853320467274178032973640562;
  double series;
  double logGamma;
  int i;

  if (x < 12.0) {
     return log(fabs(gamma(x)));
  }

  // Abramowitz and Stegun 6.1.41
  // Asymptotic series should be good to at least 11 or 12 figures
  // For error analysis, see Whittiker and Watson
  // A Course in Modern Analysis (1927), page 252

  for (i=6; i >= 0; i--) {
      sum *= z;
      sum += c[i];
  }
  series = sum/x;
  logGamma = (x - 0.5)*log(x) - x + halfLogTwoPi + series;
  return logGamma;
}