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

1 files changed, 135 insertions, 5 deletions

diff --git a/sysdeps/ieee754/dbl-64/e_gamma_r.c b/sysdeps/ieee754/dbl-64/e_gamma_r.c
index 9873551757..5b17f7b5ad 100644
--- a/sysdeps/ieee754/dbl-64/e_gamma_r.c
+++ b/sysdeps/ieee754/dbl-64/e_gamma_r.c
@@ -19,14 +19,104 @@
 
 #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 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);
+#if FLT_EVAL_METHOD != 0
+	  volatile
+#endif
+	  double x_tmp = x + n;
+	  x_adj = x_tmp;
+	  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)
+		    * __ieee754_sqrt (2 * M_PI / x_adj)
+		    / prod);
+      exp_adj += x_eps * __ieee754_log (x);
+      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;
 
@@ -51,8 +141,48 @@ __ieee754_gamma_r (double x, int *signgamp)
       *signgamp = 0;
       return x - x;
     }
+  if (__builtin_expect ((hx & 0x7ff00000) == 0x7ff00000, 0))
+    {
+      /* 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;
+      return DBL_MAX * DBL_MAX;
+    }
+  else if (x > 0.0)
+    {
+      *signgamp = 0;
+      int exp2_adj;
+      double ret = gamma_positive (x, &exp2_adj);
+      return __scalbn (ret, exp2_adj);
+    }
+  else if (x >= -DBL_EPSILON / 4.0)
+    {
+      *signgamp = 0;
+      return 1.0 / x;
+    }
+  else
+    {
+      double tx = __trunc (x);
+      *signgamp = (tx == 2.0 * __trunc (tx / 2.0)) ? -1 : 1;
+      if (x <= -184.0)
+	/* Underflow.  */
+	return DBL_MIN * DBL_MIN;
+      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 ret = M_PI / (-x * sinpix * gamma_positive (-x, &exp2_adj));
+      return __scalbn (ret, -exp2_adj);
+    }
 }
 strong_alias (__ieee754_gamma_r, __gamma_r_finite)