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219 lines
6.3 KiB
C
219 lines
6.3 KiB
C
/* Implementation of gamma function according to ISO C.
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Copyright (C) 1997-2016 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and
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Jakub Jelinek <jj@ultra.linux.cz, 1999.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but 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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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<http://www.gnu.org/licenses/>. */
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#include <math.h>
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#include <math_private.h>
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#include <float.h>
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/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
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approximation to gamma function. */
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static const long double gamma_coeff[] =
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{
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0x1.555555555555555555555555558p-4L,
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-0xb.60b60b60b60b60b60b60b60b6p-12L,
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0x3.4034034034034034034034034p-12L,
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-0x2.7027027027027027027027027p-12L,
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0x3.72a3c5631fe46ae1d4e700dca9p-12L,
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-0x7.daac36664f1f207daac36664f2p-12L,
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0x1.a41a41a41a41a41a41a41a41a4p-8L,
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-0x7.90a1b2c3d4e5f708192a3b4c5ep-8L,
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0x2.dfd2c703c0cfff430edfd2c704p-4L,
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-0x1.6476701181f39edbdb9ce625988p+0L,
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0xd.672219167002d3a7a9c886459cp+0L,
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-0x9.cd9292e6660d55b3f712eb9e08p+4L,
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0x8.911a740da740da740da740da74p+8L,
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};
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#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
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/* Return gamma (X), for positive X less than 191, in the form R *
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2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
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avoid overflow or underflow in intermediate calculations. */
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static long double
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gammal_positive (long double x, int *exp2_adj)
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{
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int local_signgam;
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if (x < 0.5L)
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{
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*exp2_adj = 0;
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return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
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}
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else if (x <= 1.5L)
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{
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*exp2_adj = 0;
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return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
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}
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else if (x < 11.5L)
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{
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/* Adjust into the range for using exp (lgamma). */
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*exp2_adj = 0;
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long double n = __ceill (x - 1.5L);
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long double x_adj = x - n;
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long double eps;
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long double prod = __gamma_productl (x_adj, 0, n, &eps);
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return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
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* prod * (1.0L + eps));
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}
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else
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{
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long double eps = 0;
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long double x_eps = 0;
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long double x_adj = x;
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long double prod = 1;
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if (x < 23.0L)
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{
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/* Adjust into the range for applying Stirling's
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approximation. */
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long double n = __ceill (23.0L - x);
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x_adj = x + n;
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x_eps = (x - (x_adj - n));
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prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
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}
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/* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
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Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
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starting by computing pow (X_ADJ, X_ADJ) with a power of 2
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factored out. */
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long double exp_adj = -eps;
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long double x_adj_int = __roundl (x_adj);
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long double x_adj_frac = x_adj - x_adj_int;
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int x_adj_log2;
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long double x_adj_mant = __frexpl (x_adj, &x_adj_log2);
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if (x_adj_mant < M_SQRT1_2l)
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{
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x_adj_log2--;
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x_adj_mant *= 2.0L;
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}
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*exp2_adj = x_adj_log2 * (int) x_adj_int;
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long double ret = (__ieee754_powl (x_adj_mant, x_adj)
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* __ieee754_exp2l (x_adj_log2 * x_adj_frac)
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* __ieee754_expl (-x_adj)
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* __ieee754_sqrtl (2 * M_PIl / x_adj)
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/ prod);
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exp_adj += x_eps * __ieee754_logl (x_adj);
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long double bsum = gamma_coeff[NCOEFF - 1];
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long double x_adj2 = x_adj * x_adj;
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for (size_t i = 1; i <= NCOEFF - 1; i++)
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bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
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exp_adj += bsum / x_adj;
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return ret + ret * __expm1l (exp_adj);
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}
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}
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long double
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__ieee754_gammal_r (long double x, int *signgamp)
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{
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int64_t hx;
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double xhi;
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long double ret;
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xhi = ldbl_high (x);
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EXTRACT_WORDS64 (hx, xhi);
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if ((hx & 0x7fffffffffffffffLL) == 0)
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{
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/* Return value for x == 0 is Inf with divide by zero exception. */
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*signgamp = 0;
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return 1.0 / x;
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}
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if (hx < 0 && (u_int64_t) hx < 0xfff0000000000000ULL && __rintl (x) == x)
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{
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/* Return value for integer x < 0 is NaN with invalid exception. */
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*signgamp = 0;
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return (x - x) / (x - x);
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}
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if (hx == 0xfff0000000000000ULL)
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{
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/* x == -Inf. According to ISO this is NaN. */
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*signgamp = 0;
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return x - x;
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}
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if ((hx & 0x7ff0000000000000ULL) == 0x7ff0000000000000ULL)
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{
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/* Positive infinity (return positive infinity) or NaN (return
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NaN). */
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*signgamp = 0;
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return x + x;
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}
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if (x >= 172.0L)
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{
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/* Overflow. */
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*signgamp = 0;
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return LDBL_MAX * LDBL_MAX;
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}
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else
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{
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SET_RESTORE_ROUNDL (FE_TONEAREST);
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if (x > 0.0L)
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{
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*signgamp = 0;
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int exp2_adj;
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ret = gammal_positive (x, &exp2_adj);
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ret = __scalbnl (ret, exp2_adj);
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}
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else if (x >= -0x1p-110L)
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{
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*signgamp = 0;
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ret = 1.0L / x;
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}
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else
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{
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long double tx = __truncl (x);
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*signgamp = (tx == 2.0L * __truncl (tx / 2.0L)) ? -1 : 1;
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if (x <= -191.0L)
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/* Underflow. */
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ret = LDBL_MIN * LDBL_MIN;
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else
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{
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long double frac = tx - x;
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if (frac > 0.5L)
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frac = 1.0L - frac;
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long double sinpix = (frac <= 0.25L
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? __sinl (M_PIl * frac)
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: __cosl (M_PIl * (0.5L - frac)));
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int exp2_adj;
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ret = M_PIl / (-x * sinpix
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* gammal_positive (-x, &exp2_adj));
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ret = __scalbnl (ret, -exp2_adj);
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math_check_force_underflow_nonneg (ret);
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}
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}
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}
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if (isinf (ret) && x != 0)
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{
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if (*signgamp < 0)
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return -(-__copysignl (LDBL_MAX, ret) * LDBL_MAX);
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else
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return __copysignl (LDBL_MAX, ret) * LDBL_MAX;
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}
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else if (ret == 0)
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{
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if (*signgamp < 0)
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return -(-__copysignl (LDBL_MIN, ret) * LDBL_MIN);
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else
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return __copysignl (LDBL_MIN, ret) * LDBL_MIN;
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}
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else
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return ret;
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}
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strong_alias (__ieee754_gammal_r, __gammal_r_finite)
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