/* specfunc/erfc.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003 Gerard Jungman
*
* 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.
*/
/* Author: J. Theiler (modifications by G. Jungman) */
/*
* See Hart et al, Computer Approximations, John Wiley and Sons, New York (1968)
* (This applies only to the erfc8 stuff, which is the part
* of the original code that survives. I have replaced much of
* the other stuff with Chebyshev fits. These are simpler and
* more precise than the original approximations. [GJ])
*/
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_erf.h>
#include "check.h"
#include "chebyshev.h"
#include "cheb_eval.c"
#define LogRootPi_ 0.57236494292470008706
static double erfc8_sum(double x)
{
/* estimates erfc(x) valid for 8 < x < 100 */
/* This is based on index 5725 in Hart et al */
static double P[] = {
2.97886562639399288862,
7.409740605964741794425,
6.1602098531096305440906,
5.019049726784267463450058,
1.275366644729965952479585264,
0.5641895835477550741253201704
};
static double Q[] = {
3.3690752069827527677,
9.608965327192787870698,
17.08144074746600431571095,
12.0489519278551290360340491,
9.396034016235054150430579648,
2.260528520767326969591866945,
1.0
};
double num=0.0, den=0.0;
int i;
num = P[5];
for (i=4; i>=0; --i) {
num = x*num + P[i];
}
den = Q[6];
for (i=5; i>=0; --i) {
den = x*den + Q[i];
}
return num/den;
}
inline
static double erfc8(double x)
{
double e;
e = erfc8_sum(x);
e *= exp(-x*x);
return e;
}
inline
static double log_erfc8(double x)
{
double e;
e = erfc8_sum(x);
e = log(e) - x*x;
return e;
}
#if 0
/* Abramowitz+Stegun, 7.2.14 */
static double erfcasympsum(double x)
{
int i;
double e = 1.;
double coef = 1.;
for (i=1; i<5; ++i) {
/* coef *= -(2*i-1)/(2*x*x); ??? [GJ] */
coef *= -(2*i+1)/(i*(4*x*x*x*x));
e += coef;
/*
if (fabs(coef) < 1.0e-15) break;
if (fabs(coef) > 1.0e10) break;
[GJ]: These tests are not useful. This function is only
used below. Took them out; they gum up the pipeline.
*/
}
return e;
}
#endif /* 0 */
/* Abramowitz+Stegun, 7.1.5 */
static int erfseries(double x, gsl_sf_result * result)
{
double coef = x;
double e = coef;
double del;
int k;
for (k=1; k<30; ++k) {
coef *= -x*x/k;
del = coef/(2.0*k+1.0);
e += del;
}
result->val = 2.0 / M_SQRTPI * e;
result->err = 2.0 / M_SQRTPI * (fabs(del) + GSL_DBL_EPSILON);
return GSL_SUCCESS;
}
/* Chebyshev fit for erfc((t+1)/2), -1 < t < 1
*/
static double erfc_xlt1_data[20] = {
1.06073416421769980345174155056,
-0.42582445804381043569204735291,
0.04955262679620434040357683080,
0.00449293488768382749558001242,
-0.00129194104658496953494224761,
-0.00001836389292149396270416979,
0.00002211114704099526291538556,
-5.23337485234257134673693179020e-7,
-2.78184788833537885382530989578e-7,
1.41158092748813114560316684249e-8,
2.72571296330561699984539141865e-9,
-2.06343904872070629406401492476e-10,
-2.14273991996785367924201401812e-11,
2.22990255539358204580285098119e-12,
1.36250074650698280575807934155e-13,
-1.95144010922293091898995913038e-14,
-6.85627169231704599442806370690e-16,
1.44506492869699938239521607493e-16,
2.45935306460536488037576200030e-18,
-9.29599561220523396007359328540e-19
};
static cheb_series erfc_xlt1_cs = {
erfc_xlt1_data,
19,
-1, 1,
12
};
/* Chebyshev fit for erfc(x) exp(x^2), 1 < x < 5, x = 2t + 3, -1 < t < 1
*/
static double erfc_x15_data[25] = {
0.44045832024338111077637466616,
-0.143958836762168335790826895326,
0.044786499817939267247056666937,
-0.013343124200271211203618353102,
0.003824682739750469767692372556,
-0.001058699227195126547306482530,
0.000283859419210073742736310108,
-0.000073906170662206760483959432,
0.000018725312521489179015872934,
-4.62530981164919445131297264430e-6,
1.11558657244432857487884006422e-6,
-2.63098662650834130067808832725e-7,
6.07462122724551777372119408710e-8,
-1.37460865539865444777251011793e-8,
3.05157051905475145520096717210e-9,
-6.65174789720310713757307724790e-10,
1.42483346273207784489792999706e-10,
-3.00141127395323902092018744545e-11,
6.22171792645348091472914001250e-12,
-1.26994639225668496876152836555e-12,
2.55385883033257575402681845385e-13,
-5.06258237507038698392265499770e-14,
9.89705409478327321641264227110e-15,
-1.90685978789192181051961024995e-15,
3.50826648032737849245113757340e-16
};
static cheb_series erfc_x15_cs = {
erfc_x15_data,
24,
-1, 1,
16
};
/* Chebyshev fit for erfc(x) x exp(x^2), 5 < x < 10, x = (5t + 15)/2, -1 < t < 1
*/
static double erfc_x510_data[20] = {
1.11684990123545698684297865808,
0.003736240359381998520654927536,
-0.000916623948045470238763619870,
0.000199094325044940833965078819,
-0.000040276384918650072591781859,
7.76515264697061049477127605790e-6,
-1.44464794206689070402099225301e-6,
2.61311930343463958393485241947e-7,
-4.61833026634844152345304095560e-8,
8.00253111512943601598732144340e-9,
-1.36291114862793031395712122089e-9,
2.28570483090160869607683087722e-10,
-3.78022521563251805044056974560e-11,
6.17253683874528285729910462130e-12,
-9.96019290955316888445830597430e-13,
1.58953143706980770269506726000e-13,
-2.51045971047162509999527428316e-14,
3.92607828989125810013581287560e-15,
-6.07970619384160374392535453420e-16,
9.12600607264794717315507477670e-17
};
static cheb_series erfc_x510_cs = {
erfc_x510_data,
19,
-1, 1,
12
};
#if 0
inline
static double
erfc_asymptotic(double x)
{
return exp(-x*x)/x * erfcasympsum(x) / M_SQRTPI;
}
inline
static double
log_erfc_asymptotic(double x)
{
return log(erfcasympsum(x)/x) - x*x - LogRootPi_;
}
#endif /* 0 */
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int gsl_sf_erfc_e(double x, gsl_sf_result * result)
{
const double ax = fabs(x);
double e_val, e_err;
/* CHECK_POINTER(result) */
if(ax <= 1.0) {
double t = 2.0*ax - 1.0;
gsl_sf_result c;
cheb_eval_e(&erfc_xlt1_cs, t, &c);
e_val = c.val;
e_err = c.err;
}
else if(ax <= 5.0) {
double ex2 = exp(-x*x);
double t = 0.5*(ax-3.0);
gsl_sf_result c;
cheb_eval_e(&erfc_x15_cs, t, &c);
e_val = ex2 * c.val;
e_err = ex2 * (c.err + 2.0*fabs(x)*GSL_DBL_EPSILON);
}
else if(ax < 10.0) {
double exterm = exp(-x*x) / ax;
double t = (2.0*ax - 15.0)/5.0;
gsl_sf_result c;
cheb_eval_e(&erfc_x510_cs, t, &c);
e_val = exterm * c.val;
e_err = exterm * (c.err + 2.0*fabs(x)*GSL_DBL_EPSILON + GSL_DBL_EPSILON);
}
else {
e_val = erfc8(ax);
e_err = (x*x + 1.0) * GSL_DBL_EPSILON * fabs(e_val);
}
if(x < 0.0) {
result->val = 2.0 - e_val;
result->err = e_err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
else {
result->val = e_val;
result->err = e_err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
return GSL_SUCCESS;
}
int gsl_sf_log_erfc_e(double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x*x < 10.0*GSL_ROOT6_DBL_EPSILON) {
const double y = x / M_SQRTPI;
/* series for -1/2 Log[Erfc[Sqrt[Pi] y]] */
const double c3 = (4.0 - M_PI)/3.0;
const double c4 = 2.0*(1.0 - M_PI/3.0);
const double c5 = -0.001829764677455021; /* (96.0 - 40.0*M_PI + 3.0*M_PI*M_PI)/30.0 */
const double c6 = 0.02629651521057465; /* 2.0*(120.0 - 60.0*M_PI + 7.0*M_PI*M_PI)/45.0 */
const double c7 = -0.01621575378835404;
const double c8 = 0.00125993961762116;
const double c9 = 0.00556964649138;
const double c10 = -0.0045563339802;
const double c11 = 0.0009461589032;
const double c12 = 0.0013200243174;
const double c13 = -0.00142906;
const double c14 = 0.00048204;
double series = c8 + y*(c9 + y*(c10 + y*(c11 + y*(c12 + y*(c13 + c14*y)))));
series = y*(1.0 + y*(1.0 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*(c7 + y*series)))))));
result->val = -2.0 * series;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/*
don't like use of log1p(); added above series stuff for small x instead, should be ok [GJ]
else if (fabs(x) < 1.0) {
gsl_sf_result result_erf;
gsl_sf_erf_e(x, &result_erf);
result->val = log1p(-result_erf.val);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
*/
else if(x > 8.0) {
result->val = log_erfc8(x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
gsl_sf_result result_erfc;
gsl_sf_erfc_e(x, &result_erfc);
result->val = log(result_erfc.val);
result->err = fabs(result_erfc.err / result_erfc.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int gsl_sf_erf_e(double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(fabs(x) < 1.0) {
return erfseries(x, result);
}
else {
gsl_sf_result result_erfc;
gsl_sf_erfc_e(x, &result_erfc);
result->val = 1.0 - result_erfc.val;
result->err = result_erfc.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int gsl_sf_erf_Z_e(double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
{
const double ex2 = exp(-x*x/2.0);
result->val = ex2 / (M_SQRT2 * M_SQRTPI);
result->err = fabs(x * result->val) * GSL_DBL_EPSILON;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_erf_Q_e(double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
{
gsl_sf_result result_erfc;
int stat = gsl_sf_erfc_e(x/M_SQRT2, &result_erfc);
result->val = 0.5 * result_erfc.val;
result->err = 0.5 * result_erfc.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat;
}
}
int gsl_sf_hazard_e(double x, gsl_sf_result * result)
{
if(x < 25.0)
{
gsl_sf_result result_ln_erfc;
const int stat_l = gsl_sf_log_erfc_e(x/M_SQRT2, &result_ln_erfc);
const double lnc = -0.22579135264472743236; /* ln(sqrt(2/pi)) */
const double arg = lnc - 0.5*x*x - result_ln_erfc.val;
const int stat_e = gsl_sf_exp_e(arg, result);
result->err += 3.0 * (1.0 + fabs(x)) * GSL_DBL_EPSILON * fabs(result->val);
result->err += fabs(result_ln_erfc.err * result->val);
return GSL_ERROR_SELECT_2(stat_l, stat_e);
}
else
{
const double ix2 = 1.0/(x*x);
const double corrB = 1.0 - 9.0*ix2 * (1.0 - 11.0*ix2);
const double corrM = 1.0 - 5.0*ix2 * (1.0 - 7.0*ix2 * corrB);
const double corrT = 1.0 - ix2 * (1.0 - 3.0*ix2*corrM);
result->val = x / corrT;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_erfc(double x)
{
EVAL_RESULT(gsl_sf_erfc_e(x, &result));
}
double gsl_sf_log_erfc(double x)
{
EVAL_RESULT(gsl_sf_log_erfc_e(x, &result));
}
double gsl_sf_erf(double x)
{
EVAL_RESULT(gsl_sf_erf_e(x, &result));
}
double gsl_sf_erf_Z(double x)
{
EVAL_RESULT(gsl_sf_erf_Z_e(x, &result));
}
double gsl_sf_erf_Q(double x)
{
EVAL_RESULT(gsl_sf_erf_Q_e(x, &result));
}
double gsl_sf_hazard(double x)
{
EVAL_RESULT(gsl_sf_hazard_e(x, &result));
}