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/* Implementation of gamma function according to ISO C.
Copyright (C) 1997-2021 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997.
The GNU C 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 GNU C 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 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>
#include <libm-alias-finite.h>
#include <mul_split.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 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 h1, l1, h2, l2;
mul_split (&h1, &l1, __ieee754_pow (x_adj_mant, x_adj),
__ieee754_exp2 (x_adj_log2 * x_adj_frac));
mul_split (&h2, &l2, __ieee754_exp (-x_adj), sqrt (2 * M_PI / x_adj));
mul_expansion (&h1, &l1, h1, l1, h2, l2);
/* Divide by prod * (1 + eps). */
div_expansion (&h1, &l1, h1, l1, prod, prod * eps);
double 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;
/* Now return (h1+l1) * exp(exp_adj), where exp_adj is small. */
l1 += h1 * __expm1 (exp_adj);
return h1 + l1;
}
}
double
__ieee754_gamma_r (double x, int *signgamp)
{
int32_t hx;
uint32_t lx;
double ret;
EXTRACT_WORDS (hx, lx, x);
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)
&& (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 (__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;
}
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 h1, l1, h2, l2;
h2 = gamma_positive (-x, &exp2_adj);
mul_split (&h1, &l1, sinpix, h2);
/* sinpix*gamma_positive(.) = h1 + l1 */
mul_split (&h2, &l2, h1, x);
/* h1*x = h2 + l2 */
/* (h1 + l1) * x = h1*x + l1*x = h2 + l2 + l1*x */
l2 += l1 * x;
/* x*sinpix*gamma_positive(.) ~ h2 + l2 */
h1 = 0x3.243f6a8885a3p+0; /* binary64 approximation of Pi */
l1 = 0x8.d313198a2e038p-56; /* |h1+l1-Pi| < 3e-33 */
/* Now we divide h1 + l1 by h2 + l2. */
div_expansion (&h1, &l1, h1, l1, h2, l2);
ret = __scalbn (-h1, -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;
}
libm_alias_finite (__ieee754_gamma_r, __gamma_r)
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