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/* mpfr_gamma -- gamma function
Copyright 2001 Free Software Foundation.
This file is part of the MPFR Library, and was contributed by Mathieu Dutour.
The MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.
The MPFR Library 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 Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "gmp.h"
#include "gmp-impl.h"
#include "mpfr.h"
#include "mpfr-impl.h"
int mpfr_gamma _PROTO ((mpfr_ptr, mpfr_srcptr, mp_rnd_t));
/* We use the reflection formula
Gamma(1+x)Gamma(1-x)=\pi x/(sin(\pi x))
in order to treat the case x<=1 */
#define CST 0.38 /* CST=ln(2)/(ln(2*pi)) */
#define zCST 0.26 /* zCST=1/(2*ln(2*pi)) */
int
#if __STDC__
mpfr_gamma (mpfr_ptr gamma, mpfr_srcptr x, mp_rnd_t rnd_mode)
#else
mpfr_gamma (gamma, x, rnd_mode)
mpfr_ptr gamma;
mpfr_srcptr x;
mp_rnd_t rnd_mode;
#endif
{
mpfr_t xp;
mpfr_t product;
mpfr_t driv;
mpfr_t xminusone;
mpfr_t the_pi_constant;
mpfr_t Csin, Ccos;
mpfr_t GammaTrial;
int reflex;
mpfr_t tmp, tmp2;
int Prec;
int Prec_nec;
int prec_gamma;
int prec_nec;
int good = 0;
double C;
long A, N;
int realprec;
int estimated_delta;
int compared;
int compar;
int factorial_case;
int k;
/* Trivial cases */
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(gamma);
return 1;
}
if (!MPFR_NOTZERO(x))
{
MPFR_SET_INF(gamma);
return 1;
}
if (MPFR_IS_INF(x))
{
MPFR_SET_INF(gamma);
return 1;
}
/* Set x_p=x if x> 1 else set x_p=2-x */
prec_gamma = MPFR_PREC(gamma);
compared = mpfr_cmp_ui(x, 1);
if (compared == 0)
{
mpfr_set_ui(gamma, 1, rnd_mode);
return 1;
}
if (compared == -1)
{
prec_nec = 2+prec_gamma;
reflex = 1;
return 1;
}
else
{
prec_nec = prec_gamma;
reflex = 0;
return 1;
}
realprec = prec_nec+10;
while (!good){
C = ((double) realprec)*CST-0.5;
A = (long) ceil(C-zCST*log(C));
N = A-1;
/* Compute the correct estimated_delta as a function of realprec */
Prec = realprec+estimated_delta;
mpfr_init2(xp, Prec);
if (compared == -1)
{
mpfr_ui_sub(xp, 2, x, GMP_RNDN);
}
else
{
mpfr_set(xp, x, GMP_RNDN);
}
/* Initialisation */
mpfr_init2(product, Prec);
mpfr_init2(driv, Prec);
mpfr_init2(tmp, Prec);
mpfr_set(driv, xp, GMP_RNDN);
mpfr_set_ui(product, 1, GMP_RNDN);
/* We use a naugthy algorithm to reduce to the domain 1<=x <2*/
while (1)
{
compar = mpfr_cmp_ui(driv, 2);
if (compar == 0) /* the factorial in fact */
{
mpfr_mul_ui(GammaTrial, product, 2, GMP_RNDN);
factorial_case = 1;
break;
}
if (compar == -1)
{
factorial_case = 0;
break;
}
mpfr_sub_ui(driv, driv, 1, GMP_RNDN);
mpfr_mul(product, product, driv, GMP_RNDN);
}
/* now we run into the trivial case */
if (factorial_case == 0 && reflex == 1)
{
mpfr_clear(product);
mpfr_clear(driv);
mpfr_clear(tmp);
MPFR_SET_INF(gamma);
return 1;
}
mpfr_init2(GammaTrial, Prec);
if (factorial_case == 0)
{
/* compute the Gamma for 1< driv < 2 */
mpfr_init2(tmp2, Prec);
mpfr_add_ui(tmp2, driv,6*N, GMP_RNDN);
mpfr_ui_div(tmp, 1, tmp2, GMP_RNDN);
for(k=6*N; k>=1; k--)
{
mpfr_mul_ui(tmp, tmp, N, GMP_RNDN);
mpfr_neg(tmp, tmp, GMP_RNDN);
mpfr_div_ui(tmp, tmp, k, GMP_RNDN);
mpfr_add_ui(tmp2, driv, k-1, GMP_RNDN);
mpfr_ui_div(tmp2, 1, tmp2, GMP_RNDN);
mpfr_add(tmp, tmp, tmp2, GMP_RNDN);
}
mpfr_set_ui(tmp2, N, GMP_RNDN);
mpfr_log(tmp2, tmp2, GMP_RNDN);
mpfr_mul(tmp2, tmp2, driv, GMP_RNDN);
mpfr_exp(tmp2, tmp2, GMP_RNDN);
mpfr_mul(tmp, tmp2, tmp, GMP_RNDN);
mpfr_mul(GammaTrial, tmp, product, GMP_RNDN);
}
if (reflex == 1)
{
mpfr_init2(xminusone, Prec);
mpfr_init2(Csin, Prec);
mpfr_init2(Ccos, Prec);
mpfr_sub_ui(xminusone, x, 1, GMP_RNDN);
mpfr_const_pi(the_pi_constant, Prec_nec);
mpfr_mul(tmp, the_pi_constant, xminusone, GMP_RNDN);
mpfr_sin(Csin, tmp, GMP_RNDN);
mpfr_cos(Ccos, tmp, GMP_RNDN);
mpfr_div(tmp, tmp, Csin, GMP_RNDN);
mpfr_neg(tmp, tmp, GMP_RNDN);
mpfr_div(GammaTrial, tmp, GammaTrial, GMP_RNDN);
}
if (mpfr_can_round(GammaTrial, realprec, GMP_RNDD, rnd_mode, MPFR_PREC(gamma)))
{
mpfr_set(gamma, GammaTrial, rnd_mode);
good = 1;
}
else
{
realprec += __gmpfr_ceil_log2 ((double) realprec);
#ifdef DEBUG
printf("RETRY\n");
#endif
}
mpfr_clear(tmp);
mpfr_clear(driv);
mpfr_clear(product);
mpfr_clear(GammaTrial);
if (reflex == 1)
{
mpfr_clear(xminusone);
mpfr_clear(Csin);
mpfr_clear(Ccos);
}
}
mpfr_clear(xp);
return 1; /* inexact result */
}
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