/* Implementation of gamma function according to ISO C.
- Copyright (C) 1997, 1999 Free Software Foundation, Inc.
+ Copyright (C) 1997-2015 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and
- Jakub Jelinek <jj@ultra.linux.cz, 1999.
+ Jakub Jelinek <jj@ultra.linux.cz, 1999.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
- License along with the GNU C Library; if not, write to the Free
- Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
- 02111-1307 USA. */
+ License along with the GNU C Library; if not, see
+ <http://www.gnu.org/licenses/>. */
#include <math.h>
#include <math_private.h>
+#include <float.h>
+/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
+ approximation to gamma function. */
+
+static const long double gamma_coeff[] =
+ {
+ 0x1.5555555555555555555555555555p-4L,
+ -0xb.60b60b60b60b60b60b60b60b60b8p-12L,
+ 0x3.4034034034034034034034034034p-12L,
+ -0x2.7027027027027027027027027028p-12L,
+ 0x3.72a3c5631fe46ae1d4e700dca8f2p-12L,
+ -0x7.daac36664f1f207daac36664f1f4p-12L,
+ 0x1.a41a41a41a41a41a41a41a41a41ap-8L,
+ -0x7.90a1b2c3d4e5f708192a3b4c5d7p-8L,
+ 0x2.dfd2c703c0cfff430edfd2c703cp-4L,
+ -0x1.6476701181f39edbdb9ce625987dp+0L,
+ 0xd.672219167002d3a7a9c886459cp+0L,
+ -0x9.cd9292e6660d55b3f712eb9e07c8p+4L,
+ 0x8.911a740da740da740da740da741p+8L,
+ -0x8.d0cc570e255bf59ff6eec24b49p+12L,
+ };
+
+#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
+
+/* Return gamma (X), for positive X less than 1775, in the form R *
+ 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
+ avoid overflow or underflow in intermediate calculations. */
+
+static long double
+gammal_positive (long double x, int *exp2_adj)
+{
+ int local_signgam;
+ if (x < 0.5L)
+ {
+ *exp2_adj = 0;
+ return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
+ }
+ else if (x <= 1.5L)
+ {
+ *exp2_adj = 0;
+ return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
+ }
+ else if (x < 12.5L)
+ {
+ /* Adjust into the range for using exp (lgamma). */
+ *exp2_adj = 0;
+ long double n = __ceill (x - 1.5L);
+ long double x_adj = x - n;
+ long double eps;
+ long double prod = __gamma_productl (x_adj, 0, n, &eps);
+ return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
+ * prod * (1.0L + eps));
+ }
+ else
+ {
+ long double eps = 0;
+ long double x_eps = 0;
+ long double x_adj = x;
+ long double prod = 1;
+ if (x < 24.0L)
+ {
+ /* Adjust into the range for applying Stirling's
+ approximation. */
+ long double n = __ceill (24.0L - x);
+ x_adj = x + n;
+ x_eps = (x - (x_adj - n));
+ prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
+ }
+ /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
+ Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
+ starting by computing pow (X_ADJ, X_ADJ) with a power of 2
+ factored out. */
+ long double exp_adj = -eps;
+ long double x_adj_int = __roundl (x_adj);
+ long double x_adj_frac = x_adj - x_adj_int;
+ int x_adj_log2;
+ long double x_adj_mant = __frexpl (x_adj, &x_adj_log2);
+ if (x_adj_mant < M_SQRT1_2l)
+ {
+ x_adj_log2--;
+ x_adj_mant *= 2.0L;
+ }
+ *exp2_adj = x_adj_log2 * (int) x_adj_int;
+ long double ret = (__ieee754_powl (x_adj_mant, x_adj)
+ * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
+ * __ieee754_expl (-x_adj)
+ * __ieee754_sqrtl (2 * M_PIl / x_adj)
+ / prod);
+ exp_adj += x_eps * __ieee754_logl (x);
+ long double bsum = gamma_coeff[NCOEFF - 1];
+ long double x_adj2 = x_adj * x_adj;
+ for (size_t i = 1; i <= NCOEFF - 1; i++)
+ bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
+ exp_adj += bsum / x_adj;
+ return ret + ret * __expm1l (exp_adj);
+ }
+}
long double
__ieee754_gammal_r (long double x, int *signgamp)
{
- /* We don't have a real gamma implementation now. We'll use lgamma
- and the exp function. But due to the required boundary
- conditions we must check some values separately. */
int64_t hx;
u_int64_t lx;
if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
{
- /* Return value for x == 0 is NaN with invalid exception. */
+ /* Return value for x == 0 is Inf with divide by zero exception. */
*signgamp = 0;
- return x / x;
+ return 1.0 / x;
}
if (hx < 0 && (u_int64_t) hx < 0xffff000000000000ULL && __rintl (x) == x)
{
*signgamp = 0;
return (x - x) / (x - x);
}
+ if (hx == 0xffff000000000000ULL && lx == 0)
+ {
+ /* x == -Inf. According to ISO this is NaN. */
+ *signgamp = 0;
+ return x - x;
+ }
+ if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
+ {
+ /* Positive infinity (return positive infinity) or NaN (return
+ NaN). */
+ *signgamp = 0;
+ return x + x;
+ }
- /* XXX FIXME. */
- return __ieee754_expl (__ieee754_lgammal_r (x, signgamp));
+ if (x >= 1756.0L)
+ {
+ /* Overflow. */
+ *signgamp = 0;
+ return LDBL_MAX * LDBL_MAX;
+ }
+ else if (x > 0.0L)
+ {
+ *signgamp = 0;
+ int exp2_adj;
+ long double ret = gammal_positive (x, &exp2_adj);
+ return __scalbnl (ret, exp2_adj);
+ }
+ else if (x >= -LDBL_EPSILON / 4.0L)
+ {
+ *signgamp = 0;
+ return 1.0f / x;
+ }
+ else
+ {
+ long double tx = __truncl (x);
+ *signgamp = (tx == 2.0L * __truncl (tx / 2.0L)) ? -1 : 1;
+ if (x <= -1775.0L)
+ /* Underflow. */
+ return LDBL_MIN * LDBL_MIN;
+ long double frac = tx - x;
+ if (frac > 0.5L)
+ frac = 1.0L - frac;
+ long double sinpix = (frac <= 0.25L
+ ? __sinl (M_PIl * frac)
+ : __cosl (M_PIl * (0.5L - frac)));
+ int exp2_adj;
+ long double ret = M_PIl / (-x * sinpix
+ * gammal_positive (-x, &exp2_adj));
+ return __scalbnl (ret, -exp2_adj);
+ }
}
+strong_alias (__ieee754_gammal_r, __gammal_r_finite)
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