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/* mpfr_gamma -- gamma function
Copyright 2001, 2002, 2003 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. */
#ifdef DEBUG
#include <stdio.h>
#include <stdlib.h>
#endif
#include "gmp.h"
#include "gmp-impl.h"
#include "mpfr.h"
#include "mpfr-impl.h"
/* We use the reflection formula
Gamma(1+t) Gamma(1-t) = - Pi t / sin(Pi (1 + t))
in order to treat the case x <= 1,
i.e. if x = 1-t, then Gamma(x) = -Pi*(1-x)/sin(Pi*(2-x))/GAMMA(2-x)
*/
#define CST 0.38 /* CST=ln(2)/(ln(2*pi)) */
#define zCST 0.26 /* zCST=1/(2*ln(2*pi)) */
#define ecCST 1.84 /* {1+sup_{x\in [0,1]} x*ln((1-x)/x)}/ln(2) */
int
mpfr_gamma (mpfr_ptr gamma, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t xp;
mpfr_t product;
mpfr_t the_pi;
mpfr_t GammaTrial;
mpfr_t tmp, tmp2;
mp_prec_t Prec;
mp_prec_t prec_gamma;
mp_prec_t prec_nec;
int good = 0;
double C;
mp_prec_t A, N, estimated_cancel;
mp_prec_t realprec;
int compared;
int k;
int sign;
int inex;
/* Trivial cases */
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(gamma);
MPFR_RET_NAN;
}
if (MPFR_IS_INF(x))
{
if (MPFR_SIGN(x) < 0)
{
MPFR_SET_NAN(gamma);
MPFR_RET_NAN;
}
else
{
MPFR_CLEAR_NAN(gamma);
MPFR_SET_INF(gamma);
MPFR_SET_POS(gamma);
return 0; /* exact */
}
}
if (MPFR_IS_ZERO(x))
{
MPFR_CLEAR_NAN(gamma);
MPFR_SET_INF(gamma);
MPFR_SET_SAME_SIGN(gamma, x);
return 0; /* exact */
}
/* 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)
return mpfr_set_ui (gamma, 1, rnd_mode);
realprec = prec_gamma + 10;
mpfr_init2 (xp, 2);
while (!good)
{
/* Precision stuff */
prec_nec = compared < 0 ?
2 + realprec /* We will use the reflexion formula! */
: realprec;
C = (double)(((double) prec_nec)*CST-0.5);
A = (mp_prec_t) C;
N = A - 1;
#ifdef DEBUG
printf("C=%u", (int)C);
printf(" A=%u", (int)A);
printf(" N=%u", (int)N);
printf("\n");
#endif
/* estimated_cancel is the amount of bit that will be flushed */
estimated_cancel= (mp_prec_t) (ecCST * (double) A + 1.0);
Prec = prec_nec + estimated_cancel + 20;
mpfr_set_prec (xp, Prec);
if (compared < 0)
{
mpfr_ui_sub (xp, 1, x, GMP_RNDN);
}
else
{
mpfr_sub_ui (xp, x, 1, GMP_RNDN);
}
/* Initialisation */
mpfr_init2(tmp, Prec);
mpfr_init2(tmp2, Prec);
mpfr_init2(the_pi, Prec);
mpfr_init2(product, Prec);
mpfr_init2(GammaTrial, Prec);
mpfr_set_ui(GammaTrial, 0, GMP_RNDN);
sign = 1;
for (k = 1; k <= N; k++)
{
mpfr_set_ui(tmp, A-k, GMP_RNDN);
mpfr_exp(product, tmp, GMP_RNDN);
mpfr_ui_pow_ui(tmp, A-k, k-1, GMP_RNDN);
mpfr_mul(product, product, tmp, GMP_RNDN);
mpfr_sqrt_ui(tmp, A-k, GMP_RNDN);
mpfr_mul(product, product, tmp, GMP_RNDN);
mpfr_fac_ui(tmp, k-1, GMP_RNDN);
mpfr_div(product, product, tmp, GMP_RNDN);
mpfr_add_ui(tmp, xp, k, GMP_RNDN);
mpfr_div(product, product, tmp, GMP_RNDN);
sign = -sign;
if (sign == 1)
{
mpfr_neg(product, product, GMP_RNDN);
#ifdef DEBUG
/* printf(" k=%u", k);
printf("\n");*/
#endif
}
mpfr_add(GammaTrial, GammaTrial, product, GMP_RNDN);
}
#ifdef DEBUG
printf("GammaTrial =");
mpfr_out_str (stdout, 10, 0, GammaTrial, GMP_RNDD);
printf ("\n");
#endif
mpfr_const_pi(the_pi, GMP_RNDN);
mpfr_const_pi(tmp, GMP_RNDN);
mpfr_mul_2ui(tmp, tmp, 1, GMP_RNDN);
mpfr_sqrt(tmp, tmp, GMP_RNDN);
mpfr_add(GammaTrial, GammaTrial, tmp, GMP_RNDN);
mpfr_add_ui(tmp2, xp, A, GMP_RNDN);
mpfr_set_ui(tmp, 1, GMP_RNDN);
mpfr_div_2ui(tmp, tmp, 1, GMP_RNDN);
mpfr_add(tmp, tmp, xp, GMP_RNDN);
mpfr_pow(tmp, tmp2, tmp, GMP_RNDN);
mpfr_mul(GammaTrial, GammaTrial, tmp, GMP_RNDN);
mpfr_neg(tmp, tmp2, GMP_RNDN);
mpfr_exp(tmp, tmp, GMP_RNDN);
mpfr_mul(GammaTrial, GammaTrial, tmp, GMP_RNDN);
if (compared < 0)
{
mpfr_sub_ui (tmp, x, 1, GMP_RNDN);
mpfr_mul (tmp, the_pi, tmp, GMP_RNDN);
mpfr_div (GammaTrial, tmp, GammaTrial, GMP_RNDN);
mpfr_sin (tmp, tmp, GMP_RNDN);
mpfr_div (GammaTrial, GammaTrial, tmp, GMP_RNDN);
}
#ifdef DEBUG
printf("GammaTrial =");
mpfr_out_str (stdout, 10, 0, GammaTrial, GMP_RNDD);
printf ("\n");
#endif
if (mpfr_can_round (GammaTrial, realprec, GMP_RNDD, GMP_RNDZ,
MPFR_PREC(gamma) + (rnd_mode == GMP_RNDN)))
{
inex = 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(tmp2);
mpfr_clear(the_pi);
mpfr_clear(product);
mpfr_clear(GammaTrial);
}
mpfr_clear (xp);
return inex; /* inexact result */
}
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