erf.c

/* 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 "ruby/missing.h"
#include <stdio.h>
#include <math.h>

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);
}