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/* specfunc/lambert.c
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*
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* Copyright (C) 2007 Brian Gough
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* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 Gerard Jungman
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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/* Author: G. Jungman */
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#include <config.h>
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#include <math.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_errno.h>
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#include <gsl/gsl_sf_lambert.h>
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/* Started with code donated by K. Briggs; added
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* error estimates, GSL foo, and minor tweaks.
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* Some Lambert-ology from
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* [Corless, Gonnet, Hare, and Jeffrey, "On Lambert's W Function".]
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*/
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/* Halley iteration (eqn. 5.12, Corless et al) */
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static int
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halley_iteration(
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double x,
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double w_initial,
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unsigned int max_iters,
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gsl_sf_result * result
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)
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{
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double w = w_initial;
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unsigned int i;
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for(i=0; i
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double tol;
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const double e = exp(w);
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const double p = w + 1.0;
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double t = w*e - x;
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/* printf("FOO: %20.16g %20.16g\n", w, t); */
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if (w > 0) {
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t = (t/p)/e; /* Newton iteration */
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} else {
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t /= e*p - 0.5*(p + 1.0)*t/p; /* Halley iteration */
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};
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w -= t;
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tol = 10 * GSL_DBL_EPSILON * GSL_MAX_DBL(fabs(w), 1.0/(fabs(p)*e));
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if(fabs(t) < tol)
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{
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result->val = w;
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result->err = 2.0*tol;
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return GSL_SUCCESS;
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}
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}
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/* should never get here */
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result->val = w;
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result->err = fabs(w);
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return GSL_EMAXITER;
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}
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/* series which appears for q near zero;
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* only the argument is different for the different branches
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*/
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static double
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series_eval(double r)
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{
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static const double c[12] = {
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-1.0,
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2.331643981597124203363536062168,
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-1.812187885639363490240191647568,
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1.936631114492359755363277457668,
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-2.353551201881614516821543561516,
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3.066858901050631912893148922704,
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-4.175335600258177138854984177460,
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5.858023729874774148815053846119,
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-8.401032217523977370984161688514,
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12.250753501314460424,
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-18.100697012472442755,
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27.029044799010561650
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};
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const double t_8 = c[8] + r*(c[9] + r*(c[10] + r*c[11]));
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const double t_5 = c[5] + r*(c[6] + r*(c[7] + r*t_8));
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const double t_1 = c[1] + r*(c[2] + r*(c[3] + r*(c[4] + r*t_5)));
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return c[0] + r*t_1;
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}
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/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
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int
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gsl_sf_lambert_W0_e(double x, gsl_sf_result * result)
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{
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const double one_over_E = 1.0/M_E;
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const double q = x + one_over_E;
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if(x == 0.0) {
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result->val = 0.0;
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result->err = 0.0;
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return GSL_SUCCESS;
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}
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else if(q < 0.0) {
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/* Strictly speaking this is an error. But because of the
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* arithmetic operation connecting x and q, I am a little
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* lenient in case of some epsilon overshoot. The following
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* answer is quite accurate in that case. Anyway, we have
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* to return GSL_EDOM.
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*/
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result->val = -1.0;
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result->err = sqrt(-q);
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return GSL_EDOM;
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}
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else if(q == 0.0) {
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result->val = -1.0;
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result->err = GSL_DBL_EPSILON; /* cannot error is zero, maybe q == 0 by "accident" */
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return GSL_SUCCESS;
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}
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else if(q < 1.0e-03) {
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/* series near -1/E in sqrt(q) */
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const double r = sqrt(q);
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result->val = series_eval(r);
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result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
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return GSL_SUCCESS;
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}
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else {
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static const unsigned int MAX_ITERS = 10;
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double w;
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if (x < 1.0) {
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/* obtain initial approximation from series near x=0;
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* no need for extra care, since the Halley iteration
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* converges nicely on this branch
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*/
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const double p = sqrt(2.0 * M_E * q);
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w = -1.0 + p*(1.0 + p*(-1.0/3.0 + p*11.0/72.0));
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}
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else {
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/* obtain initial approximation from rough asymptotic */
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w = log(x);
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if(x > 3.0) w -= log(w);
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}
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return halley_iteration(x, w, MAX_ITERS, result);
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}
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}
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int
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gsl_sf_lambert_Wm1_e(double x, gsl_sf_result * result)
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{
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if(x > 0.0) {
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return gsl_sf_lambert_W0_e(x, result);
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}
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else if(x == 0.0) {
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result->val = 0.0;
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result->err = 0.0;
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return GSL_SUCCESS;
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}
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else {
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static const unsigned int MAX_ITERS = 32;
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const double one_over_E = 1.0/M_E;
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const double q = x + one_over_E;
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double w;
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if (q < 0.0) {
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/* As in the W0 branch above, return some reasonable answer anyway. */
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result->val = -1.0;
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result->err = sqrt(-q);
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return GSL_EDOM;
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}
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if(x < -1.0e-6) {
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/* Obtain initial approximation from series about q = 0,
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* as long as we're not very close to x = 0.
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* Use full series and try to bail out if q is too small,
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* since the Halley iteration has bad convergence properties
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* in finite arithmetic for q very small, because the
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* increment alternates and p is near zero.
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*/
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const double r = -sqrt(q);
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w = series_eval(r);
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if(q < 3.0e-3) {
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/* this approximation is good enough */
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result->val = w;
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result->err = 5.0 * GSL_DBL_EPSILON * fabs(w);
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return GSL_SUCCESS;
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}
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}
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else {
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/* Obtain initial approximation from asymptotic near zero. */
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const double L_1 = log(-x);
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const double L_2 = log(-L_1);
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w = L_1 - L_2 + L_2/L_1;
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}
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return halley_iteration(x, w, MAX_ITERS, result);
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}
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}
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/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
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#include "eval.h"
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double gsl_sf_lambert_W0(double x)
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{
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EVAL_RESULT(gsl_sf_lambert_W0_e(x, &result));
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}
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double gsl_sf_lambert_Wm1(double x)
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{
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EVAL_RESULT(gsl_sf_lambert_Wm1_e(x, &result));
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}
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