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/* mpfr_gamma -- gamma function
Copyright 2001, 2002, 2003, 2004, 2005 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
#define MPFR_NEED_LONGLONG_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)
*/
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;
mp_prec_t A, N, estimated_cancel;
mp_prec_t realprec;
int compared;
unsigned long k;
int sign;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
("gamma[%#R]=%R inexact=%d", gamma, gamma, inex));
/* Trivial cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_NEG (x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
else
{
MPFR_SET_INF (gamma);
MPFR_SET_POS (gamma);
MPFR_RET (0); /* exact */
}
}
else
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
MPFR_SET_INF(gamma);
MPFR_SET_SAME_SIGN(gamma, x);
MPFR_RET (0); /* exact */
}
}
MPFR_CLEAR_FLAGS(gamma);
/* 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);
/* if x is an integer that fits into an unsigned long, use mpfr_fac_ui */
if (mpfr_integer_p (x) && mpfr_fits_ulong_p (x, GMP_RNDN))
{
unsigned long int u;
u = mpfr_get_ui (x, GMP_RNDN);
return mpfr_fac_ui (gamma, u - 1, rnd_mode);
}
MPFR_SAVE_EXPO_MARK (expo);
realprec = prec_gamma + MPFR_INT_CEIL_LOG2 (prec_gamma) + 10;
mpfr_init2 (xp, realprec + BITS_PER_MP_LIMB);
mpfr_init2 (tmp, realprec + BITS_PER_MP_LIMB);
mpfr_init2 (tmp2, realprec + BITS_PER_MP_LIMB);
mpfr_init2 (the_pi, realprec + BITS_PER_MP_LIMB);
mpfr_init2 (product, realprec + BITS_PER_MP_LIMB);
mpfr_init2 (GammaTrial, realprec + BITS_PER_MP_LIMB);
MPFR_ZIV_INIT (loop, realprec);
for (;;)
{
/* Precision stuff */
prec_nec = compared < 0 ?
2 + realprec /* We will use the reflexion formula! */
: realprec;
/* A = (prec_nec-0.5)*CST
CST = ln(2)/(ln(2*pi))) = 0.38
This strange formula is just to avoid any overflow */
A = (prec_nec/100)*38 + ((prec_nec%100)*38+100-38/2)/100 - 1;
N = A - 1;
#ifdef DEBUG
printf("A=%d N=%d\n", (int)A, (int)N);
#endif
/* Estimated_cancel is the amount of bit that will be flushed */
/* estimated_cancel = A + ecCST * A;
ecCST = {1+sup_{x\in [0,1]} x*ln((1-x)/x)}/ln(2) = 1.84
This strange formula is just to avoid any overflow */
estimated_cancel = A + (A + (A/100)*84 + ((A%100)*84)/100);
Prec = prec_nec + estimated_cancel + 16;
MPFR_ASSERTD (Prec > prec_nec);
MPFR_ASSERTD (Prec > estimated_cancel);
MPFR_ASSERTD (estimated_cancel > A);
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);
/* Set prec */
mpfr_set_prec (tmp, Prec);
mpfr_set_prec (tmp2, Prec);
mpfr_set_prec (the_pi, Prec);
mpfr_set_prec (product, Prec);
mpfr_set_prec (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)))
break;
MPFR_ZIV_NEXT (loop, realprec);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (gamma, GammaTrial, rnd_mode);
mpfr_clear(tmp);
mpfr_clear(tmp2);
mpfr_clear(the_pi);
mpfr_clear(product);
mpfr_clear(GammaTrial);
mpfr_clear (xp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range(gamma, inex, rnd_mode);
}
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