/* Implementation of gamma function according to ISO C.
- Copyright (C) 1997, 1999, 2001, 2004, 2011 Free Software Foundation, Inc.
+ Copyright (C) 1997-2019 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997.
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
+ <https://www.gnu.org/licenses/>. */
#include <math.h>
+#include <math-narrow-eval.h>
#include <math_private.h>
+#include <fenv_private.h>
+#include <math-underflow.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 double gamma_coeff[] =
+ {
+ 0x1.5555555555555p-4,
+ -0xb.60b60b60b60b8p-12,
+ 0x3.4034034034034p-12,
+ -0x2.7027027027028p-12,
+ 0x3.72a3c5631fe46p-12,
+ -0x7.daac36664f1f4p-12,
+ };
+
+#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
+
+/* Return gamma (X), for positive X less than 184, 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 double
+gamma_positive (double x, int *exp2_adj)
+{
+ int local_signgam;
+ if (x < 0.5)
+ {
+ *exp2_adj = 0;
+ return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x;
+ }
+ else if (x <= 1.5)
+ {
+ *exp2_adj = 0;
+ return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam));
+ }
+ else if (x < 6.5)
+ {
+ /* Adjust into the range for using exp (lgamma). */
+ *exp2_adj = 0;
+ double n = ceil (x - 1.5);
+ double x_adj = x - n;
+ double eps;
+ double prod = __gamma_product (x_adj, 0, n, &eps);
+ return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam))
+ * prod * (1.0 + eps));
+ }
+ else
+ {
+ double eps = 0;
+ double x_eps = 0;
+ double x_adj = x;
+ double prod = 1;
+ if (x < 12.0)
+ {
+ /* Adjust into the range for applying Stirling's
+ approximation. */
+ double n = ceil (12.0 - x);
+ x_adj = math_narrow_eval (x + n);
+ x_eps = (x - (x_adj - n));
+ prod = __gamma_product (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. */
+ double exp_adj = -eps;
+ double x_adj_int = round (x_adj);
+ double x_adj_frac = x_adj - x_adj_int;
+ int x_adj_log2;
+ double x_adj_mant = __frexp (x_adj, &x_adj_log2);
+ if (x_adj_mant < M_SQRT1_2)
+ {
+ x_adj_log2--;
+ x_adj_mant *= 2.0;
+ }
+ *exp2_adj = x_adj_log2 * (int) x_adj_int;
+ double ret = (__ieee754_pow (x_adj_mant, x_adj)
+ * __ieee754_exp2 (x_adj_log2 * x_adj_frac)
+ * __ieee754_exp (-x_adj)
+ * sqrt (2 * M_PI / x_adj)
+ / prod);
+ exp_adj += x_eps * __ieee754_log (x_adj);
+ double bsum = gamma_coeff[NCOEFF - 1];
+ 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 * __expm1 (exp_adj);
+ }
+}
double
__ieee754_gamma_r (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. */
int32_t hx;
- u_int32_t lx;
+ uint32_t lx;
+ double ret;
EXTRACT_WORDS (hx, lx, x);
- if (__builtin_expect (((hx & 0x7fffffff) | lx) == 0, 0))
+ if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0))
{
/* Return value for x == 0 is Inf with divide by zero exception. */
*signgamp = 0;
return 1.0 / x;
}
if (__builtin_expect (hx < 0, 0)
- && (u_int32_t) hx < 0xfff00000 && __rint (x) == x)
+ && (uint32_t) hx < 0xfff00000 && rint (x) == x)
{
/* Return value for integer x < 0 is NaN with invalid exception. */
*signgamp = 0;
return (x - x) / (x - x);
}
- if (__builtin_expect ((unsigned int) hx == 0xfff00000 && lx==0, 0))
+ if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0))
{
/* x == -Inf. According to ISO this is NaN. */
*signgamp = 0;
return x - x;
}
+ if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000))
+ {
+ /* Positive infinity (return positive infinity) or NaN (return
+ NaN). */
+ *signgamp = 0;
+ return x + x;
+ }
- /* XXX FIXME. */
- return __ieee754_exp (__ieee754_lgamma_r (x, signgamp));
+ if (x >= 172.0)
+ {
+ /* Overflow. */
+ *signgamp = 0;
+ ret = math_narrow_eval (DBL_MAX * DBL_MAX);
+ return ret;
+ }
+ else
+ {
+ SET_RESTORE_ROUND (FE_TONEAREST);
+ if (x > 0.0)
+ {
+ *signgamp = 0;
+ int exp2_adj;
+ double tret = gamma_positive (x, &exp2_adj);
+ ret = __scalbn (tret, exp2_adj);
+ }
+ else if (x >= -DBL_EPSILON / 4.0)
+ {
+ *signgamp = 0;
+ ret = 1.0 / x;
+ }
+ else
+ {
+ double tx = trunc (x);
+ *signgamp = (tx == 2.0 * trunc (tx / 2.0)) ? -1 : 1;
+ if (x <= -184.0)
+ /* Underflow. */
+ ret = DBL_MIN * DBL_MIN;
+ else
+ {
+ double frac = tx - x;
+ if (frac > 0.5)
+ frac = 1.0 - frac;
+ double sinpix = (frac <= 0.25
+ ? __sin (M_PI * frac)
+ : __cos (M_PI * (0.5 - frac)));
+ int exp2_adj;
+ double tret = M_PI / (-x * sinpix
+ * gamma_positive (-x, &exp2_adj));
+ ret = __scalbn (tret, -exp2_adj);
+ math_check_force_underflow_nonneg (ret);
+ }
+ }
+ ret = math_narrow_eval (ret);
+ }
+ if (isinf (ret) && x != 0)
+ {
+ if (*signgamp < 0)
+ {
+ ret = math_narrow_eval (-copysign (DBL_MAX, ret) * DBL_MAX);
+ ret = -ret;
+ }
+ else
+ ret = math_narrow_eval (copysign (DBL_MAX, ret) * DBL_MAX);
+ return ret;
+ }
+ else if (ret == 0)
+ {
+ if (*signgamp < 0)
+ {
+ ret = math_narrow_eval (-copysign (DBL_MIN, ret) * DBL_MIN);
+ ret = -ret;
+ }
+ else
+ ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
+ return ret;
+ }
+ else
+ return ret;
}
strong_alias (__ieee754_gamma_r, __gamma_r_finite)
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