Improve tgamma accuracy (bugs 2546, 2560, 5159, 15426).

2 files changed, 181 insertions, 5 deletions

diff --git a/sysdeps/ieee754/ldbl-128ibm/e_gammal_r.c b/sysdeps/ieee754/ldbl-128ibm/e_gammal_r.c
index 52ade9e4a1..90d8e3f0d2 100644
--- a/sysdeps/ieee754/ldbl-128ibm/e_gammal_r.c
+++ b/sysdeps/ieee754/ldbl-128ibm/e_gammal_r.c
@@ -20,14 +20,107 @@
 
 #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[] =
+  {
+    0x1.555555555555555555555555558p-4L,
+    -0xb.60b60b60b60b60b60b60b60b6p-12L,
+    0x3.4034034034034034034034034p-12L,
+    -0x2.7027027027027027027027027p-12L,
+    0x3.72a3c5631fe46ae1d4e700dca9p-12L,
+    -0x7.daac36664f1f207daac36664f2p-12L,
+    0x1.a41a41a41a41a41a41a41a41a4p-8L,
+    -0x7.90a1b2c3d4e5f708192a3b4c5ep-8L,
+    0x2.dfd2c703c0cfff430edfd2c704p-4L,
+    -0x1.6476701181f39edbdb9ce625988p+0L,
+    0xd.672219167002d3a7a9c886459cp+0L,
+    -0x9.cd9292e6660d55b3f712eb9e08p+4L,
+    0x8.911a740da740da740da740da74p+8L,
+  };
+
+#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
+
+/* Return gamma (X), for positive X less than 191, 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 < 11.5L)
+    {
+      /* 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 < 23.0L)
+	{
+	  /* Adjust into the range for applying Stirling's
+	     approximation.  */
+	  long double n = __ceill (23.0L - 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;
 
@@ -51,8 +144,49 @@ __ieee754_gammal_r (long double x, int *signgamp)
       *signgamp = 0;
       return x - x;
     }
+  if ((hx & 0x7ff0000000000000ULL) == 0x7ff0000000000000ULL)
+    {
+      /* 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 >= 172.0L)
+    {
+      /* 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 >= -0x1p-110L)
+    {
+      *signgamp = 0;
+      return 1.0f / x;
+    }
+  else
+    {
+      long double tx = __truncl (x);
+      *signgamp = (tx == 2.0L * __truncl (tx / 2.0L)) ? -1 : 1;
+      if (x <= -191.0L)
+	/* 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)
diff --git a/sysdeps/ieee754/ldbl-128ibm/gamma_productl.c b/sysdeps/ieee754/ldbl-128ibm/gamma_productl.c
new file mode 100644
index 0000000000..7c6186d230
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-128ibm/gamma_productl.c
@@ -0,0 +1,42 @@
+/* Compute a product of X, X+1, ..., with an error estimate.
+   Copyright (C) 2013 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
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+
+/* Compute the product of X + X_EPS, X + X_EPS + 1, ..., X + X_EPS + N
+   - 1, in the form R * (1 + *EPS) where the return value R is an
+   approximation to the product and *EPS is set to indicate the
+   approximate error in the return value.  X is such that all the
+   values X + 1, ..., X + N - 1 are exactly representable, and X_EPS /
+   X is small enough that factors quadratic in it can be
+   neglected.  */
+
+long double
+__gamma_productl (long double x, long double x_eps, int n, long double *eps)
+{
+  long double ret = x;
+  *eps = x_eps / x;
+  for (int i = 1; i < n; i++)
+    {
+      *eps += x_eps / (x + i);
+      ret *= x + i;
+      /* FIXME: no error estimates for the multiplication.  */
+    }
+  return ret;
+}