223 lines
6.3 KiB
C
223 lines
6.3 KiB
C
/* Implementation of gamma function according to ISO C.
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Copyright (C) 1997-2018 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 "quadmath-imp.h"
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__float128
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tgammaq (__float128 x)
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{
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int sign;
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__float128 ret;
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ret = __quadmath_gammaq_r (x, &sign);
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return sign < 0 ? -ret : ret;
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}
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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 __float128 gamma_coeff[] =
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{
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0x1.5555555555555555555555555555p-4Q,
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-0xb.60b60b60b60b60b60b60b60b60b8p-12Q,
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0x3.4034034034034034034034034034p-12Q,
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-0x2.7027027027027027027027027028p-12Q,
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0x3.72a3c5631fe46ae1d4e700dca8f2p-12Q,
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-0x7.daac36664f1f207daac36664f1f4p-12Q,
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0x1.a41a41a41a41a41a41a41a41a41ap-8Q,
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-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8Q,
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0x2.dfd2c703c0cfff430edfd2c703cp-4Q,
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-0x1.6476701181f39edbdb9ce625987dp+0Q,
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0xd.672219167002d3a7a9c886459cp+0Q,
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-0x9.cd9292e6660d55b3f712eb9e07c8p+4Q,
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0x8.911a740da740da740da740da741p+8Q,
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-0x8.d0cc570e255bf59ff6eec24b49p+12Q,
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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 1775, 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 __float128
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gammal_positive (__float128 x, int *exp2_adj)
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{
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int local_signgam;
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if (x < 0.5Q)
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{
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*exp2_adj = 0;
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return expq (__quadmath_lgammaq_r (x + 1, &local_signgam)) / x;
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}
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else if (x <= 1.5Q)
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{
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*exp2_adj = 0;
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return expq (__quadmath_lgammaq_r (x, &local_signgam));
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}
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else if (x < 12.5Q)
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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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__float128 n = ceilq (x - 1.5Q);
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__float128 x_adj = x - n;
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__float128 eps;
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__float128 prod = __quadmath_gamma_productq (x_adj, 0, n, &eps);
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return (expq (__quadmath_lgammaq_r (x_adj, &local_signgam))
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* prod * (1 + eps));
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}
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else
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{
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__float128 eps = 0;
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__float128 x_eps = 0;
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__float128 x_adj = x;
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__float128 prod = 1;
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if (x < 24)
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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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__float128 n = ceilq (24 - x);
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x_adj = x + n;
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x_eps = (x - (x_adj - n));
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prod = __quadmath_gamma_productq (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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__float128 exp_adj = -eps;
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__float128 x_adj_int = roundq (x_adj);
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__float128 x_adj_frac = x_adj - x_adj_int;
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int x_adj_log2;
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__float128 x_adj_mant = frexpq (x_adj, &x_adj_log2);
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if (x_adj_mant < M_SQRT1_2q)
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{
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x_adj_log2--;
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x_adj_mant *= 2;
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}
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*exp2_adj = x_adj_log2 * (int) x_adj_int;
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__float128 ret = (powq (x_adj_mant, x_adj)
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* exp2q (x_adj_log2 * x_adj_frac)
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* expq (-x_adj)
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* sqrtq (2 * M_PIq / x_adj)
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/ prod);
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exp_adj += x_eps * logq (x_adj);
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__float128 bsum = gamma_coeff[NCOEFF - 1];
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__float128 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 * expm1q (exp_adj);
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}
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}
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__float128
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__quadmath_gammaq_r (__float128 x, int *signgamp)
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{
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int64_t hx;
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uint64_t lx;
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__float128 ret;
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GET_FLT128_WORDS64 (hx, lx, x);
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if (((hx & 0x7fffffffffffffffLL) | lx) == 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 && (uint64_t) hx < 0xffff000000000000ULL && rintq (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 == 0xffff000000000000ULL && lx == 0)
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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 & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
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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 >= 1756)
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{
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/* Overflow. */
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*signgamp = 0;
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return FLT128_MAX * FLT128_MAX;
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}
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else
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{
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SET_RESTORE_ROUNDF128 (FE_TONEAREST);
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if (x > 0)
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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 = scalbnq (ret, exp2_adj);
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}
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else if (x >= -FLT128_EPSILON / 4)
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{
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*signgamp = 0;
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ret = 1 / x;
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}
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else
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{
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__float128 tx = truncq (x);
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*signgamp = (tx == 2 * truncq (tx / 2)) ? -1 : 1;
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if (x <= -1775)
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/* Underflow. */
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ret = FLT128_MIN * FLT128_MIN;
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else
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{
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__float128 frac = tx - x;
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if (frac > 0.5Q)
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frac = 1 - frac;
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__float128 sinpix = (frac <= 0.25Q
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? sinq (M_PIq * frac)
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: cosq (M_PIq * (0.5Q - frac)));
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int exp2_adj;
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ret = M_PIq / (-x * sinpix
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* gammal_positive (-x, &exp2_adj));
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ret = scalbnq (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 (isinfq (ret) && x != 0)
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{
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if (*signgamp < 0)
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return -(-copysignq (FLT128_MAX, ret) * FLT128_MAX);
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else
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return copysignq (FLT128_MAX, ret) * FLT128_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 -(-copysignq (FLT128_MIN, ret) * FLT128_MIN);
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else
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return copysignq (FLT128_MIN, ret) * FLT128_MIN;
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}
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else
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return ret;
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}
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