/* specfunc/exp.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 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: G. Jungman */
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_exp.h>
#include "error.h"
/* Evaluate the continued fraction for exprel.
* [Abramowitz+Stegun, 4.2.41]
*/
static
int
exprel_n_CF(const double N, const double x, gsl_sf_result * result)
{
const double RECUR_BIG = GSL_SQRT_DBL_MAX;
const int maxiter = 5000;
int n = 1;
double Anm2 = 1.0;
double Bnm2 = 0.0;
double Anm1 = 0.0;
double Bnm1 = 1.0;
double a1 = 1.0;
double b1 = 1.0;
double a2 = -x;
double b2 = N+1;
double an, bn;
double fn;
double An = b1*Anm1 + a1*Anm2; /* A1 */
double Bn = b1*Bnm1 + a1*Bnm2; /* B1 */
/* One explicit step, before we get to the main pattern. */
n++;
Anm2 = Anm1;
Bnm2 = Bnm1;
Anm1 = An;
Bnm1 = Bn;
An = b2*Anm1 + a2*Anm2; /* A2 */
Bn = b2*Bnm1 + a2*Bnm2; /* B2 */
fn = An/Bn;
while(n < maxiter) {
double old_fn;
double del;
n++;
Anm2 = Anm1;
Bnm2 = Bnm1;
Anm1 = An;
Bnm1 = Bn;
an = ( GSL_IS_ODD(n) ? ((n-1)/2)*x : -(N+(n/2)-1)*x );
bn = N + n - 1;
An = bn*Anm1 + an*Anm2;
Bn = bn*Bnm1 + an*Bnm2;
if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {
An /= RECUR_BIG;
Bn /= RECUR_BIG;
Anm1 /= RECUR_BIG;
Bnm1 /= RECUR_BIG;
Anm2 /= RECUR_BIG;
Bnm2 /= RECUR_BIG;
}
old_fn = fn;
fn = An/Bn;
del = old_fn/fn;
if(fabs(del - 1.0) < 2.0*GSL_DBL_EPSILON) break;
}
result->val = fn;
result->err = 4.0*(n+1.0)*GSL_DBL_EPSILON*fabs(fn);
if(n == maxiter)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int gsl_sf_exp_e(const double x, gsl_sf_result * result)
{
if(x > GSL_LOG_DBL_MAX) {
OVERFLOW_ERROR(result);
}
else if(x < GSL_LOG_DBL_MIN) {
UNDERFLOW_ERROR(result);
}
else {
result->val = exp(x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int gsl_sf_exp_e10_e(const double x, gsl_sf_result_e10 * result)
{
if(x > INT_MAX-1) {
OVERFLOW_ERROR_E10(result);
}
else if(x < INT_MIN+1) {
UNDERFLOW_ERROR_E10(result);
}
else {
const int N = (x > GSL_LOG_DBL_MAX || x < GSL_LOG_DBL_MIN) ? (int) floor(x/M_LN10) : 0;
result->val = exp(x-N*M_LN10);
result->err = 2.0 * (fabs(x)+1.0) * GSL_DBL_EPSILON * fabs(result->val);
result->e10 = N;
return GSL_SUCCESS;
}
}
int gsl_sf_exp_mult_e(const double x, const double y, gsl_sf_result * result)
{
const double ay = fabs(y);
if(y == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if( ( x < 0.5*GSL_LOG_DBL_MAX && x > 0.5*GSL_LOG_DBL_MIN)
&& (ay < 0.8*GSL_SQRT_DBL_MAX && ay > 1.2*GSL_SQRT_DBL_MIN)
) {
const double ex = exp(x);
result->val = y * ex;
result->err = (2.0 + fabs(x)) * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
const double ly = log(ay);
const double lnr = x + ly;
if(lnr > GSL_LOG_DBL_MAX - 0.01) {
OVERFLOW_ERROR(result);
}
else if(lnr < GSL_LOG_DBL_MIN + 0.01) {
UNDERFLOW_ERROR(result);
}
else {
const double sy = GSL_SIGN(y);
const double M = floor(x);
const double N = floor(ly);
const double a = x - M;
const double b = ly - N;
const double berr = 2.0 * GSL_DBL_EPSILON * (fabs(ly) + fabs(N));
result->val = sy * exp(M+N) * exp(a+b);
result->err = berr * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * (M + N + 1.0) * fabs(result->val);
return GSL_SUCCESS;
}
}
}
int gsl_sf_exp_mult_e10_e(const double x, const double y, gsl_sf_result_e10 * result)
{
const double ay = fabs(y);
if(y == 0.0) {
result->val = 0.0;
result->err = 0.0;
result->e10 = 0;
return GSL_SUCCESS;
}
else if( ( x < 0.5*GSL_LOG_DBL_MAX && x > 0.5*GSL_LOG_DBL_MIN)
&& (ay < 0.8*GSL_SQRT_DBL_MAX && ay > 1.2*GSL_SQRT_DBL_MIN)
) {
const double ex = exp(x);
result->val = y * ex;
result->err = (2.0 + fabs(x)) * GSL_DBL_EPSILON * fabs(result->val);
result->e10 = 0;
return GSL_SUCCESS;
}
else {
const double ly = log(ay);
const double l10_val = (x + ly)/M_LN10;
if(l10_val > INT_MAX-1) {
OVERFLOW_ERROR_E10(result);
}
else if(l10_val < INT_MIN+1) {
UNDERFLOW_ERROR_E10(result);
}
else {
const double sy = GSL_SIGN(y);
const int N = (int) floor(l10_val);
const double arg_val = (l10_val - N) * M_LN10;
const double arg_err = 2.0 * GSL_DBL_EPSILON * (fabs(x) + fabs(ly) + M_LN10*fabs(N));
result->val = sy * exp(arg_val);
result->err = arg_err * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
result->e10 = N;
return GSL_SUCCESS;
}
}
}
int gsl_sf_exp_mult_err_e(const double x, const double dx,
const double y, const double dy,
gsl_sf_result * result)
{
const double ay = fabs(y);
if(y == 0.0) {
result->val = 0.0;
result->err = fabs(dy * exp(x));
return GSL_SUCCESS;
}
else if( ( x < 0.5*GSL_LOG_DBL_MAX && x > 0.5*GSL_LOG_DBL_MIN)
&& (ay < 0.8*GSL_SQRT_DBL_MAX && ay > 1.2*GSL_SQRT_DBL_MIN)
) {
double ex = exp(x);
result->val = y * ex;
result->err = ex * (fabs(dy) + fabs(y*dx));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
const double ly = log(ay);
const double lnr = x + ly;
if(lnr > GSL_LOG_DBL_MAX - 0.01) {
OVERFLOW_ERROR(result);
}
else if(lnr < GSL_LOG_DBL_MIN + 0.01) {
UNDERFLOW_ERROR(result);
}
else {
const double sy = GSL_SIGN(y);
const double M = floor(x);
const double N = floor(ly);
const double a = x - M;
const double b = ly - N;
const double eMN = exp(M+N);
const double eab = exp(a+b);
result->val = sy * eMN * eab;
result->err = eMN * eab * 2.0*GSL_DBL_EPSILON;
result->err += eMN * eab * fabs(dy/y);
result->err += eMN * eab * fabs(dx);
return GSL_SUCCESS;
}
}
}
int gsl_sf_exp_mult_err_e10_e(const double x, const double dx,
const double y, const double dy,
gsl_sf_result_e10 * result)
{
const double ay = fabs(y);
if(y == 0.0) {
result->val = 0.0;
result->err = fabs(dy * exp(x));
result->e10 = 0;
return GSL_SUCCESS;
}
else if( ( x < 0.5*GSL_LOG_DBL_MAX && x > 0.5*GSL_LOG_DBL_MIN)
&& (ay < 0.8*GSL_SQRT_DBL_MAX && ay > 1.2*GSL_SQRT_DBL_MIN)
) {
const double ex = exp(x);
result->val = y * ex;
result->err = ex * (fabs(dy) + fabs(y*dx));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
result->e10 = 0;
return GSL_SUCCESS;
}
else {
const double ly = log(ay);
const double l10_val = (x + ly)/M_LN10;
if(l10_val > INT_MAX-1) {
OVERFLOW_ERROR_E10(result);
}
else if(l10_val < INT_MIN+1) {
UNDERFLOW_ERROR_E10(result);
}
else {
const double sy = GSL_SIGN(y);
const int N = (int) floor(l10_val);
const double arg_val = (l10_val - N) * M_LN10;
const double arg_err = dy/fabs(y) + dx + 2.0*GSL_DBL_EPSILON*fabs(arg_val);
result->val = sy * exp(arg_val);
result->err = arg_err * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
result->e10 = N;
return GSL_SUCCESS;
}
}
}
int gsl_sf_expm1_e(const double x, gsl_sf_result * result)
{
const double cut = 0.002;
if(x < GSL_LOG_DBL_MIN) {
result->val = -1.0;
result->err = GSL_DBL_EPSILON;
return GSL_SUCCESS;
}
else if(x < -cut) {
result->val = exp(x) - 1.0;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < cut) {
result->val = x * (1.0 + 0.5*x*(1.0 + x/3.0*(1.0 + 0.25*x*(1.0 + 0.2*x))));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < GSL_LOG_DBL_MAX) {
result->val = exp(x) - 1.0;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
OVERFLOW_ERROR(result);
}
}
int gsl_sf_exprel_e(const double x, gsl_sf_result * result)
{
const double cut = 0.002;
if(x < GSL_LOG_DBL_MIN) {
result->val = -1.0/x;
result->err = GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < -cut) {
result->val = (exp(x) - 1.0)/x;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < cut) {
result->val = (1.0 + 0.5*x*(1.0 + x/3.0*(1.0 + 0.25*x*(1.0 + 0.2*x))));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < GSL_LOG_DBL_MAX) {
result->val = (exp(x) - 1.0)/x;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
OVERFLOW_ERROR(result);
}
}
int gsl_sf_exprel_2_e(double x, gsl_sf_result * result)
{
const double cut = 0.002;
if(x < GSL_LOG_DBL_MIN) {
result->val = -2.0/x*(1.0 + 1.0/x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < -cut) {
result->val = 2.0*(exp(x) - 1.0 - x)/(x*x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < cut) {
result->val = (1.0 + 1.0/3.0*x*(1.0 + 0.25*x*(1.0 + 0.2*x*(1.0 + 1.0/6.0*x))));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < GSL_LOG_DBL_MAX) {
result->val = 2.0*(exp(x) - 1.0 - x)/(x*x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
OVERFLOW_ERROR(result);
}
}
int
gsl_sf_exprel_n_CF_e(const double N, const double x, gsl_sf_result * result)
{
return exprel_n_CF(N, x, result);
}
int
gsl_sf_exprel_n_e(const int N, const double x, gsl_sf_result * result)
{
if(N < 0) {
DOMAIN_ERROR(result);
}
else if(x == 0.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(fabs(x) < GSL_ROOT3_DBL_EPSILON * N) {
result->val = 1.0 + x/(N+1) * (1.0 + x/(N+2));
result->err = 2.0 * GSL_DBL_EPSILON;
return GSL_SUCCESS;
}
else if(N == 0) {
return gsl_sf_exp_e(x, result);
}
else if(N == 1) {
return gsl_sf_exprel_e(x, result);
}
else if(N == 2) {
return gsl_sf_exprel_2_e(x, result);
}
else {
if(x > N && (-x + N*(1.0 + log(x/N)) < GSL_LOG_DBL_EPSILON)) {
/* x is much larger than n.
* Ignore polynomial part, so
* exprel_N(x) ~= e^x N!/x^N
*/
gsl_sf_result lnf_N;
double lnr_val;
double lnr_err;
double lnterm;
gsl_sf_lnfact_e(N, &lnf_N);
lnterm = N*log(x);
lnr_val = x + lnf_N.val - lnterm;
lnr_err = GSL_DBL_EPSILON * (fabs(x) + fabs(lnf_N.val) + fabs(lnterm));
lnr_err += lnf_N.err;
return gsl_sf_exp_err_e(lnr_val, lnr_err, result);
}
else if(x > N) {
/* Write the identity
* exprel_n(x) = e^x n! / x^n (1 - Gamma[n,x]/Gamma[n])
* then use the asymptotic expansion
* Gamma[n,x] ~ x^(n-1) e^(-x) (1 + (n-1)/x + (n-1)(n-2)/x^2 + ...)
*/
double ln_x = log(x);
gsl_sf_result lnf_N;
double lg_N;
double lnpre_val;
double lnpre_err;
gsl_sf_lnfact_e(N, &lnf_N); /* log(N!) */
lg_N = lnf_N.val - log(N); /* log(Gamma(N)) */
lnpre_val = x + lnf_N.val - N*ln_x;
lnpre_err = GSL_DBL_EPSILON * (fabs(x) + fabs(lnf_N.val) + fabs(N*ln_x));
lnpre_err += lnf_N.err;
if(lnpre_val < GSL_LOG_DBL_MAX - 5.0) {
int stat_eG;
gsl_sf_result bigG_ratio;
gsl_sf_result pre;
int stat_ex = gsl_sf_exp_err_e(lnpre_val, lnpre_err, &pre);
double ln_bigG_ratio_pre = -x + (N-1)*ln_x - lg_N;
double bigGsum = 1.0;
double term = 1.0;
int k;
for(k=1; k<N; k++) {
term *= (N-k)/x;
bigGsum += term;
}
stat_eG = gsl_sf_exp_mult_e(ln_bigG_ratio_pre, bigGsum, &bigG_ratio);
if(stat_eG == GSL_SUCCESS) {
result->val = pre.val * (1.0 - bigG_ratio.val);
result->err = pre.val * (2.0*GSL_DBL_EPSILON + bigG_ratio.err);
result->err += pre.err * fabs(1.0 - bigG_ratio.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_ex;
}
else {
result->val = 0.0;
result->err = 0.0;
return stat_eG;
}
}
else {
OVERFLOW_ERROR(result);
}
}
else if(x > -10.0*N) {
return exprel_n_CF(N, x, result);
}
else {
/* x -> -Inf asymptotic:
* exprel_n(x) ~ e^x n!/x^n - n/x (1 + (n-1)/x + (n-1)(n-2)/x + ...)
* ~ - n/x (1 + (n-1)/x + (n-1)(n-2)/x + ...)
*/
double sum = 1.0;
double term = 1.0;
int k;
for(k=1; k<N; k++) {
term *= (N-k)/x;
sum += term;
}
result->val = -N/x * sum;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
}
int
gsl_sf_exp_err_e(const double x, const double dx, gsl_sf_result * result)
{
const double adx = fabs(dx);
/* CHECK_POINTER(result) */
if(x + adx > GSL_LOG_DBL_MAX) {
OVERFLOW_ERROR(result);
}
else if(x - adx < GSL_LOG_DBL_MIN) {
UNDERFLOW_ERROR(result);
}
else {
const double ex = exp(x);
const double edx = exp(adx);
result->val = ex;
result->err = ex * GSL_MAX_DBL(GSL_DBL_EPSILON, edx - 1.0/edx);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int
gsl_sf_exp_err_e10_e(const double x, const double dx, gsl_sf_result_e10 * result)
{
const double adx = fabs(dx);
/* CHECK_POINTER(result) */
if(x + adx > INT_MAX - 1) {
OVERFLOW_ERROR_E10(result);
}
else if(x - adx < INT_MIN + 1) {
UNDERFLOW_ERROR_E10(result);
}
else {
const int N = (int)floor(x/M_LN10);
const double ex = exp(x-N*M_LN10);
result->val = ex;
result->err = ex * (2.0 * GSL_DBL_EPSILON * (fabs(x) + 1.0) + adx);
result->e10 = N;
return GSL_SUCCESS;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_exp(const double x)
{
EVAL_RESULT(gsl_sf_exp_e(x, &result));
}
double gsl_sf_exp_mult(const double x, const double y)
{
EVAL_RESULT(gsl_sf_exp_mult_e(x, y, &result));
}
double gsl_sf_expm1(const double x)
{
EVAL_RESULT(gsl_sf_expm1_e(x, &result));
}
double gsl_sf_exprel(const double x)
{
EVAL_RESULT(gsl_sf_exprel_e(x, &result));
}
double gsl_sf_exprel_2(const double x)
{
EVAL_RESULT(gsl_sf_exprel_2_e(x, &result));
}
double gsl_sf_exprel_n(const int n, const double x)
{
EVAL_RESULT(gsl_sf_exprel_n_e(n, x, &result));
}