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

1 files changed, 140 insertions, 5 deletions

diff --git a/sysdeps/ieee754/ldbl-128/e_gammal_r.c b/sysdeps/ieee754/ldbl-128/e_gammal_r.c
index b6da31c13e..e8d49e9872 100644
--- a/sysdeps/ieee754/ldbl-128/e_gammal_r.c
+++ b/sysdeps/ieee754/ldbl-128/e_gammal_r.c
@@ -20,14 +20,108 @@
 
 #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;
 
@@ -51,8 +145,49 @@ __ieee754_gammal_r (long double x, int *signgamp)
       *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)