src/lib/gauss.c
/* cdf/gauss.c
*
* Copyright (C) 2002, 2004 Jason H. Stover.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software Foundation,
* Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/*
* Computes the cumulative distribution function for the Gaussian
* distribution using a rational function approximation. The
* computation is for the standard Normal distribution, i.e., mean 0
* and standard deviation 1. If you want to compute Pr(X < t) for a
* Gaussian random variable X with non-zero mean m and standard
* deviation sd not equal to 1, find gsl_cdf_ugaussian ((t-m)/sd).
* This approximation is accurate to at least double precision. The
* accuracy was verified with a pari-gp script. The largest error
* found was about 1.4E-20. The coefficients were derived by Cody.
*
* References:
*
* W.J. Cody. "Rational Chebyshev Approximations for the Error
* Function," Mathematics of Computation, v23 n107 1969, 631-637.
*
* W. Fraser, J.F Hart. "On the Computation of Rational Approximations
* to Continuous Functions," Communications of the ACM, v5 1962.
*
* W.J. Kennedy Jr., J.E. Gentle. "Statistical Computing." Marcel Dekker. 1980.
*
*
*/
//#include <config.h>
#include <gauss.h>
#include <math.h>
#include <gsl_math.h>
#include <gsl_cdf.h>
#ifndef GSL_DBL_EPSILON
#define GSL_DBL_EPSILON 2.2204460492503131e-16
#endif
#ifndef M_1_SQRT2PI
#define M_1_SQRT2PI (M_2_SQRTPI * M_SQRT1_2 / 2.0)
#endif
#define SQRT32 (4.0 * M_SQRT2)
/*
* IEEE double precision dependent constants.
*
* GAUSS_EPSILON: Smallest positive value such that
* gsl_cdf_gaussian(x) > 0.5.
* GAUSS_XUPPER: Largest value x such that gsl_cdf_gaussian(x) < 1.0.
* GAUSS_XLOWER: Smallest value x such that gsl_cdf_gaussian(x) > 0.0.
*/
#define GAUSS_EPSILON (GSL_DBL_EPSILON / 2)
#define GAUSS_XUPPER (8.572)
#define GAUSS_XLOWER (-37.519)
#define GAUSS_SCALE (16.0)
static double
get_del (double x, double rational)
{
double xsq = 0.0;
double del = 0.0;
double result = 0.0;
xsq = floor (x * GAUSS_SCALE) / GAUSS_SCALE;
del = (x - xsq) * (x + xsq);
del *= 0.5;
result = exp (-0.5 * xsq * xsq) * exp (-1.0 * del) * rational;
return result;
}
/*
* Normal cdf for fabs(x) < 0.66291
*/
static double
gauss_small (const double x)
{
unsigned int i;
double result = 0.0;
double xsq;
double xnum;
double xden;
const double a[5] = {
2.2352520354606839287,
161.02823106855587881,
1067.6894854603709582,
18154.981253343561249,
0.065682337918207449113
};
const double b[4] = {
47.20258190468824187,
976.09855173777669322,
10260.932208618978205,
45507.789335026729956
};
xsq = x * x;
xnum = a[4] * xsq;
xden = xsq;
for (i = 0; i < 3; i++)
{
xnum = (xnum + a[i]) * xsq;
xden = (xden + b[i]) * xsq;
}
result = x * (xnum + a[3]) / (xden + b[3]);
return result;
}
/*
* Normal cdf for 0.66291 < fabs(x) < sqrt(32).
*/
static double
gauss_medium (const double x)
{
unsigned int i;
double temp = 0.0;
double result = 0.0;
double xnum;
double xden;
double absx;
const double c[9] = {
0.39894151208813466764,
8.8831497943883759412,
93.506656132177855979,
597.27027639480026226,
2494.5375852903726711,
6848.1904505362823326,
11602.651437647350124,
9842.7148383839780218,
1.0765576773720192317e-8
};
const double d[8] = {
22.266688044328115691,
235.38790178262499861,
1519.377599407554805,
6485.558298266760755,
18615.571640885098091,
34900.952721145977266,
38912.003286093271411,
19685.429676859990727
};
absx = fabs (x);
xnum = c[8] * absx;
xden = absx;
for (i = 0; i < 7; i++)
{
xnum = (xnum + c[i]) * absx;
xden = (xden + d[i]) * absx;
}
temp = (xnum + c[7]) / (xden + d[7]);
result = get_del (x, temp);
return result;
}
/*
* Normal cdf for
* {sqrt(32) < x < GAUSS_XUPPER} union { GAUSS_XLOWER < x < -sqrt(32) }.
*/
static double
gauss_large (const double x)
{
int i;
double result;
double xsq;
double temp;
double xnum;
double xden;
double absx;
const double p[6] = {
0.21589853405795699,
0.1274011611602473639,
0.022235277870649807,
0.001421619193227893466,
2.9112874951168792e-5,
0.02307344176494017303
};
const double q[5] = {
1.28426009614491121,
0.468238212480865118,
0.0659881378689285515,
0.00378239633202758244,
7.29751555083966205e-5
};
absx = fabs (x);
xsq = 1.0 / (x * x);
xnum = p[5] * xsq;
xden = xsq;
for (i = 0; i < 4; i++)
{
xnum = (xnum + p[i]) * xsq;
xden = (xden + q[i]) * xsq;
}
temp = xsq * (xnum + p[4]) / (xden + q[4]);
temp = (M_1_SQRT2PI - temp) / absx;
result = get_del (x, temp);
return result;
}
double
gsl_cdf_ugaussian_P (const double x)
{
double result;
double absx = fabs (x);
if (absx < GAUSS_EPSILON)
{
result = 0.5;
return result;
}
else if (absx < 0.66291)
{
result = 0.5 + gauss_small (x);
return result;
}
else if (absx < SQRT32)
{
result = gauss_medium (x);
if (x > 0.0)
{
result = 1.0 - result;
}
return result;
}
else if (x > GAUSS_XUPPER)
{
result = 1.0;
return result;
}
else if (x < GAUSS_XLOWER)
{
result = 0.0;
return result;
}
else
{
result = gauss_large (x);
if (x > 0.0)
{
result = 1.0 - result;
}
}
return result;
}
double
gsl_cdf_ugaussian_Q (const double x)
{
double result;
double absx = fabs (x);
if (absx < GAUSS_EPSILON)
{
result = 0.5;
return result;
}
else if (absx < 0.66291)
{
result = gauss_small (x);
if (x < 0.0)
{
result = fabs (result) + 0.5;
}
else
{
result = 0.5 - result;
}
return result;
}
else if (absx < SQRT32)
{
result = gauss_medium (x);
if (x < 0.0)
{
result = 1.0 - result;
}
return result;
}
else if (x > -(GAUSS_XLOWER))
{
result = 0.0;
return result;
}
else if (x < -(GAUSS_XUPPER))
{
result = 1.0;
return result;
}
else
{
result = gauss_large (x);
if (x < 0.0)
{
result = 1.0 - result;
}
}
return result;
}
double
gsl_cdf_gaussian_P (const double x, const double sigma)
{
return gsl_cdf_ugaussian_P (x / sigma);
}
double
gsl_cdf_gaussian_Q (const double x, const double sigma)
{
return gsl_cdf_ugaussian_Q (x / sigma);
}