glibc/sysdeps/ieee754/ldbl-128/e_gammal_r.c

/* 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.

   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_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <float.h>
#include <libm-alias-finite.h>

/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
   approximation to gamma function.  */

static const _Float128 gamma_coeff[] =
  {
    L(0x1.5555555555555555555555555555p-4),
    L(-0xb.60b60b60b60b60b60b60b60b60b8p-12),
    L(0x3.4034034034034034034034034034p-12),
    L(-0x2.7027027027027027027027027028p-12),
    L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12),
    L(-0x7.daac36664f1f207daac36664f1f4p-12),
    L(0x1.a41a41a41a41a41a41a41a41a41ap-8),
    L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8),
    L(0x2.dfd2c703c0cfff430edfd2c703cp-4),
    L(-0x1.6476701181f39edbdb9ce625987dp+0),
    L(0xd.672219167002d3a7a9c886459cp+0),
    L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4),
    L(0x8.911a740da740da740da740da741p+8),
    L(-0x8.d0cc570e255bf59ff6eec24b49p+12),
  };

#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 _Float128
gammal_positive (_Float128 x, int *exp2_adj)
{
  int local_signgam;
  if (x < L(0.5))
    {
      *exp2_adj = 0;
      return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
    }
  else if (x <= L(1.5))
    {
      *exp2_adj = 0;
      return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
    }
  else if (x < L(12.5))
    {
      /* Adjust into the range for using exp (lgamma).  */
      *exp2_adj = 0;
      _Float128 n = ceill (x - L(1.5));
      _Float128 x_adj = x - n;
      _Float128 eps;
      _Float128 prod = __gamma_productl (x_adj, 0, n, &eps);
      return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
	      * prod * (1 + eps));
    }
  else
    {
      _Float128 eps = 0;
      _Float128 x_eps = 0;
      _Float128 x_adj = x;
      _Float128 prod = 1;
      if (x < 24)
	{
	  /* Adjust into the range for applying Stirling's
	     approximation.  */
	  _Float128 n = ceill (24 - 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.  */
      _Float128 exp_adj = -eps;
      _Float128 x_adj_int = roundl (x_adj);
      _Float128 x_adj_frac = x_adj - x_adj_int;
      int x_adj_log2;
      _Float128 x_adj_mant = __frexpl (x_adj, &x_adj_log2);
      if (x_adj_mant < M_SQRT1_2l)
	{
	  x_adj_log2--;
	  x_adj_mant *= 2;
	}
      *exp2_adj = x_adj_log2 * (int) x_adj_int;
      _Float128 ret = (__ieee754_powl (x_adj_mant, x_adj)
		       * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
		       * __ieee754_expl (-x_adj)
		       * sqrtl (2 * M_PIl / x_adj)
		       / prod);
      exp_adj += x_eps * __ieee754_logl (x_adj);
      _Float128 bsum = gamma_coeff[NCOEFF - 1];
      _Float128 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);
    }
}

_Float128
__ieee754_gammal_r (_Float128 x, int *signgamp)
{
  int64_t hx;
  uint64_t lx;
  _Float128 ret;

  GET_LDOUBLE_WORDS64 (hx, lx, x);

  if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
    {
      /* Return value for x == 0 is Inf with divide by zero exception.  */
      *signgamp = 0;
      return 1.0 / x;
    }
  if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintl (x) == x)
    {
      /* Return value for integer x < 0 is NaN with invalid exception.  */
      *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;
    }

  if (x >= 1756)
    {
      /* Overflow.  */
      *signgamp = 0;
      return LDBL_MAX * LDBL_MAX;
    }
  else
    {
      SET_RESTORE_ROUNDL (FE_TONEAREST);
      if (x > 0)
	{
	  *signgamp = 0;
	  int exp2_adj;
	  ret = gammal_positive (x, &exp2_adj);
	  ret = __scalbnl (ret, exp2_adj);
	}
      else if (x >= -LDBL_EPSILON / 4)
	{
	  *signgamp = 0;
	  ret = 1 / x;
	}
      else
	{
	  _Float128 tx = truncl (x);
	  *signgamp = (tx == 2 * truncl (tx / 2)) ? -1 : 1;
	  if (x <= -1775)
	    /* Underflow.  */
	    ret = LDBL_MIN * LDBL_MIN;
	  else
	    {
	      _Float128 frac = tx - x;
	      if (frac > L(0.5))
		frac = 1 - frac;
	      _Float128 sinpix = (frac <= L(0.25)
				  ? __sinl (M_PIl * frac)
				  : __cosl (M_PIl * (L(0.5) - frac)));
	      int exp2_adj;
	      ret = M_PIl / (-x * sinpix
			     * gammal_positive (-x, &exp2_adj));
	      ret = __scalbnl (ret, -exp2_adj);
	      math_check_force_underflow_nonneg (ret);
	    }
	}
    }
  if (isinf (ret) && x != 0)
    {
      if (*signgamp < 0)
	return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX);
      else
	return copysignl (LDBL_MAX, ret) * LDBL_MAX;
    }
  else if (ret == 0)
    {
      if (*signgamp < 0)
	return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN);
      else
	return copysignl (LDBL_MIN, ret) * LDBL_MIN;
    }
  else
    return ret;
}
libm_alias_finite (__ieee754_gammal_r, __gammal_r)