text: /* erf.c  - public domain implementation of error function erf(3m)
  
  reference - Haruhiko Okumura: C-gengo niyoru saishin algorithm jiten
              (New Algorithm handbook in C language) (Gijyutsu hyouron
              sha, Tokyo, 1991) p.227 [in Japanese]                 */
  #include <stdio.h>
  #include <math.h>
  
  //RHO
  int rhoRubyFPrintf(FILE *, const char *, ...);
  #ifndef USE_STD_PRINTF
  #define fprintf rhoRubyFPrintf
  #endif
  //RHO
  
  #ifdef _WIN32
  # include <float.h>
  # if !defined __MINGW32__ || defined __NO_ISOCEXT
  #  ifndef isnan
  #   define isnan(x) _isnan(x)
  #  endif
  #  ifndef isinf
  #   define isinf(x) (!_finite(x) && !_isnan(x))
  #  endif
  #  ifndef finite
  #   define finite(x) _finite(x)
  #  endif
  # endif
  #endif
  
  static double q_gamma(double, double, double);
  
  /* Incomplete gamma function
     1 / Gamma(a) * Int_0^x exp(-t) t^(a-1) dt  */
  static double p_gamma(double a, double x, double loggamma_a)
  {
      int k;
      double result, term, previous;
  
      if (x >= 1 + a) return 1 - q_gamma(a, x, loggamma_a);
      if (x == 0)     return 0;
      result = term = exp(a * log(x) - x - loggamma_a) / a;
      for (k = 1; k < 1000; k++) {
          term *= x / (a + k);
          previous = result;  result += term;
          if (result == previous) return result;
      }
      fprintf(stderr, "erf.c:%d:p_gamma() could not converge.", __LINE__);
      return result;
  }
  
  /* Incomplete gamma function
     1 / Gamma(a) * Int_x^inf exp(-t) t^(a-1) dt  */
  static double q_gamma(double a, double x, double loggamma_a)
  {
      int k;
      double result, w, temp, previous;
      double la = 1, lb = 1 + x - a;  /* Laguerre polynomial */
  
      if (x < 1 + a) return 1 - p_gamma(a, x, loggamma_a);
      w = exp(a * log(x) - x - loggamma_a);
      result = w / lb;
      for (k = 2; k < 1000; k++) {
          temp = ((k - 1 - a) * (lb - la) + (k + x) * lb) / k;
          la = lb;  lb = temp;
          w *= (k - 1 - a) / k;
          temp = w / (la * lb);
          previous = result;  result += temp;
          if (result == previous) return result;
      }
      fprintf(stderr, "erf.c:%d:q_gamma() could not converge.", __LINE__);
      return result;
  }
  
  #define LOG_PI_OVER_2 0.572364942924700087071713675675 /* log_e(PI)/2 */
  
  double erf(double x)
  {
      if (!finite(x)) {
          if (isnan(x)) return x;      /* erf(NaN)   = NaN   */
          return (x>0 ? 1.0 : -1.0);   /* erf(+-inf) = +-1.0 */
      }
      if (x >= 0) return   p_gamma(0.5, x * x, LOG_PI_OVER_2);
      else        return - p_gamma(0.5, x * x, LOG_PI_OVER_2);
  }
  
  double erfc(double x)
  {
      if (!finite(x)) {
          if (isnan(x)) return x;      /* erfc(NaN)   = NaN      */
          return (x>0 ? 0.0 : 2.0);    /* erfc(+-inf) = 0.0, 2.0 */
      }
      if (x >= 0) return  q_gamma(0.5, x * x, LOG_PI_OVER_2);
      else        return  1 + p_gamma(0.5, x * x, LOG_PI_OVER_2);
  }
  