Update.

	* sysdeps/ieee754/ldbl-96/e_lgammal_r.c: New file.
	Contributed by Stephen L. Moshier <moshier@na-net.ornl.gov>.

	* sysdeps/ieee754/ldbl-96/e_gammal_r.c: Fix handling of boundary cases.

	* sysdeps/ieee754/dbl-64/e_gamma_r.c: Always initialize sign variable.
	* sysdeps/ieee754/flt-32/e_gammaf_r.c: Likewise.

	* sysdeps/i386/fpu/libm-test-ulps: Adjust after addition of lgammal.

1 files changed, 418 insertions, 0 deletions

diff --git a/sysdeps/ieee754/ldbl-96/e_lgammal_r.c b/sysdeps/ieee754/ldbl-96/e_lgammal_r.c
new file mode 100644
index 0000000000..f39ef355a1
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-96/e_lgammal_r.c
@@ -0,0 +1,418 @@
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+/* Long double expansions contributed by
+   Stephen L. Moshier <moshier@na-net.ornl.gov>  */
+
+/* __ieee754_lgammal_r(x, signgamp)
+ * Reentrant version of the logarithm of the Gamma function
+ * with user provide pointer for the sign of Gamma(x).
+ *
+ * Method:
+ *   1. Argument Reduction for 0 < x <= 8
+ * 	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
+ * 	reduce x to a number in [1.5,2.5] by
+ * 		lgamma(1+s) = log(s) + lgamma(s)
+ *	for example,
+ *		lgamma(7.3) = log(6.3) + lgamma(6.3)
+ *			    = log(6.3*5.3) + lgamma(5.3)
+ *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
+ *   2. Polynomial approximation of lgamma around its
+ *	minimun ymin=1.461632144968362245 to maintain monotonicity.
+ *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
+ *		Let z = x-ymin;
+ *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
+ *   2. Rational approximation in the primary interval [2,3]
+ *	We use the following approximation:
+ *		s = x-2.0;
+ *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
+ *	Our algorithms are based on the following observation
+ *
+ *                             zeta(2)-1    2    zeta(3)-1    3
+ * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
+ *                                 2                 3
+ *
+ *	where Euler = 0.5771... is the Euler constant, which is very
+ *	close to 0.5.
+ *
+ *   3. For x>=8, we have
+ *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
+ *	(better formula:
+ *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
+ *	Let z = 1/x, then we approximation
+ *		f(z) = lgamma(x) - (x-0.5)(log(x)-1)
+ *	by
+ *	  			    3       5             11
+ *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
+ *
+ *   4. For negative x, since (G is gamma function)
+ *		-x*G(-x)*G(x) = pi/sin(pi*x),
+ * 	we have
+ * 		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
+ *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
+ *	Hence, for x<0, signgam = sign(sin(pi*x)) and
+ *		lgamma(x) = log(|Gamma(x)|)
+ *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
+ *	Note: one should avoid compute pi*(-x) directly in the
+ *	      computation of sin(pi*(-x)).
+ *
+ *   5. Special Cases
+ *		lgamma(2+s) ~ s*(1-Euler) for tiny s
+ *		lgamma(1)=lgamma(2)=0
+ *		lgamma(x) ~ -log(x) for tiny x
+ *		lgamma(0) = lgamma(inf) = inf
+ *	 	lgamma(-integer) = +-inf
+ *
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+#ifdef __STDC__
+static const long double
+#else
+static long double
+#endif
+  half = 0.5L,
+  one = 1.0L,
+  pi = 3.14159265358979323846264L,
+
+  /* lgam(1+x) = 0.5 x + x a(x)/b(x)
+     -0.268402099609375 <= x <= 0
+     peak relative error 6.6e-22 */
+  a0 = -6.343246574721079391729402781192128239938E2L,
+  a1 =  1.856560238672465796768677717168371401378E3L,
+  a2 =  2.404733102163746263689288466865843408429E3L,
+  a3 =  8.804188795790383497379532868917517596322E2L,
+  a4 =  1.135361354097447729740103745999661157426E2L,
+  a5 =  3.766956539107615557608581581190400021285E0L,
+
+  b0 =  8.214973713960928795704317259806842490498E3L,
+  b1 =  1.026343508841367384879065363925870888012E4L,
+  b2 =  4.553337477045763320522762343132210919277E3L,
+  b3 =  8.506975785032585797446253359230031874803E2L,
+  b4 =  6.042447899703295436820744186992189445813E1L,
+  /* b5 =  1.000000000000000000000000000000000000000E0 */
+
+
+  tc =  1.4616321449683623412626595423257213284682E0L,
+  tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
+/* tt = (tail of tf), i.e. tf + tt has extended precision. */
+  tt = 3.3649914684731379602768989080467587736363E-18L,
+  /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
+-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
+
+  /* lgam (x + tc) = tf + tt + x g(x)/h(x)
+     - 0.230003726999612341262659542325721328468 <= x
+     <= 0.2699962730003876587373404576742786715318
+     peak relative error 2.1e-21 */
+  g0 = 3.645529916721223331888305293534095553827E-18L,
+  g1 = 5.126654642791082497002594216163574795690E3L,
+  g2 = 8.828603575854624811911631336122070070327E3L,
+  g3 = 5.464186426932117031234820886525701595203E3L,
+  g4 = 1.455427403530884193180776558102868592293E3L,
+  g5 = 1.541735456969245924860307497029155838446E2L,
+  g6 = 4.335498275274822298341872707453445815118E0L,
+
+  h0 = 1.059584930106085509696730443974495979641E4L,
+  h1 =  2.147921653490043010629481226937850618860E4L,
+  h2 = 1.643014770044524804175197151958100656728E4L,
+  h3 =  5.869021995186925517228323497501767586078E3L,
+  h4 =  9.764244777714344488787381271643502742293E2L,
+  h5 =  6.442485441570592541741092969581997002349E1L,
+  /* h6 = 1.000000000000000000000000000000000000000E0 */
+
+
+  /* lgam (x+1) = -0.5 x + x u(x)/v(x)
+     -0.100006103515625 <= x <= 0.231639862060546875
+     peak relative error 1.3e-21 */
+  u0 = -8.886217500092090678492242071879342025627E1L,
+  u1 =  6.840109978129177639438792958320783599310E2L,
+  u2 =  2.042626104514127267855588786511809932433E3L,
+  u3 =  1.911723903442667422201651063009856064275E3L,
+  u4 =  7.447065275665887457628865263491667767695E2L,
+  u5 =  1.132256494121790736268471016493103952637E2L,
+  u6 =  4.484398885516614191003094714505960972894E0L,
+
+  v0 =  1.150830924194461522996462401210374632929E3L,
+  v1 =  3.399692260848747447377972081399737098610E3L,
+  v2 =  3.786631705644460255229513563657226008015E3L,
+  v3 =  1.966450123004478374557778781564114347876E3L,
+  v4 =  4.741359068914069299837355438370682773122E2L,
+  v5 =  4.508989649747184050907206782117647852364E1L,
+  /* v6 =  1.000000000000000000000000000000000000000E0 */
+
+
+  /* lgam (x+2) = .5 x + x s(x)/r(x)
+     0 <= x <= 1
+     peak relative error 7.2e-22 */
+  s0 =  1.454726263410661942989109455292824853344E6L,
+  s1 = -3.901428390086348447890408306153378922752E6L,
+  s2 = -6.573568698209374121847873064292963089438E6L,
+  s3 = -3.319055881485044417245964508099095984643E6L,
+  s4 = -7.094891568758439227560184618114707107977E5L,
+  s5 = -6.263426646464505837422314539808112478303E4L,
+  s6 = -1.684926520999477529949915657519454051529E3L,
+
+  r0 = -1.883978160734303518163008696712983134698E7L,
+  r1 = -2.815206082812062064902202753264922306830E7L,
+  r2 = -1.600245495251915899081846093343626358398E7L,
+  r3 = -4.310526301881305003489257052083370058799E6L,
+  r4 = -5.563807682263923279438235987186184968542E5L,
+  r5 = -3.027734654434169996032905158145259713083E4L,
+  r6 = -4.501995652861105629217250715790764371267E2L,
+  /* r6 =  1.000000000000000000000000000000000000000E0 */
+
+
+/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
+   x >= 8
+   Peak relative error 1.51e-21
+   w0 = LS2PI - 0.5 */
+  w0 =  4.189385332046727417803e-1L,
+  w1 =  8.333333333333331447505E-2L,
+  w2 = -2.777777777750349603440E-3L,
+  w3 =  7.936507795855070755671E-4L,
+  w4 = -5.952345851765688514613E-4L,
+  w5 =  8.412723297322498080632E-4L,
+  w6 = -1.880801938119376907179E-3L,
+  w7 =  4.885026142432270781165E-3L;
+
+#ifdef __STDC__
+static const long double zero = 0.0L;
+#else
+static long double zero = 0.0L;
+#endif
+
+#ifdef __STDC__
+static long double
+sin_pi (long double x)
+#else
+static long double
+sin_pi (x)
+     long double x;
+#endif
+{
+  long double y, z;
+  int n, ix;
+  u_int32_t se, i0, i1;
+
+  GET_LDOUBLE_WORDS (se, i0, i1, x);
+  ix = se & 0x7fff;
+
+  i1 = (ix << 16) | (i0 >> 16);
+  if (ix < 0x3ffd8000) /* 0.25 */
+    return __sinl (pi * x);
+  y = -x;			/* x is assume negative */
+
+  /*
+   * argument reduction, make sure inexact flag not raised if input
+   * is an integer
+   */
+  z = __floorl (y);
+  if (z != y)
+    {				/* inexact anyway */
+      y *= half;
+      y = 2.0 * (y - __floorl (y));	/* y = |x| mod 2.0 */
+      n = (int) (y * 4.0);
+    }
+  else
+    {
+      return (zero + zero);
+    }
+
+  switch (n)
+    {
+    case 0:
+      y = __sinl (pi * y);
+      break;
+    case 1:
+    case 2:
+      y = __cosl (pi * (half - y));
+      break;
+    case 3:
+    case 4:
+      y = __sinl (pi * (one - y));
+      break;
+    case 5:
+    case 6:
+      y = -__cosl (pi * (y - 1.5));
+      break;
+    default:
+      y = __sinl (pi * (y - 2.0));
+      break;
+    }
+  return -y;
+}
+
+
+#ifdef __STDC__
+long double
+__ieee754_lgammal_r (long double x, int *signgamp)
+#else
+long double
+__ieee754_lgammal_r (x, signgamp)
+     long double x;
+     int *signgamp;
+#endif
+{
+  long double t, y, z, nadj, p, p1, p2, q, r, w;
+  int i, ix;
+  u_int32_t se, i0, i1;
+
+  GET_LDOUBLE_WORDS (se, i0, i1, x);
+  ix = se & 0x7fff;
+
+  if ((ix | i0 | i1) == 0)
+    return one / fabsl (x);
+
+  ix = (ix << 16) | (i0 >> 16);
+
+  /* purge off +-inf, NaN, +-0, and negative arguments */
+  *signgamp = 1;
+  if (ix >= 0x7fff0000)
+    return x * x;
+
+  if (ix < 0x3fc08000) /* 2^-63 */
+    {				/* |x|<2**-63, return -log(|x|) */
+      if (se & 0x8000)
+	{
+	  *signgamp = -1;
+	  return -__ieee754_logl (-x);
+	}
+      else
+	return -__ieee754_logl (x);
+    }
+  if (se & 0x8000)
+    {
+      if (x == __floorl(x))
+	return x / zero;
+      t = sin_pi (x);
+      if (t == zero)
+	return one / fabsl (t);	/* -integer */
+      nadj = __ieee754_logl (pi / fabsl (t * x));
+      if (t < zero)
+	*signgamp = -1;
+      x = -x;
+    }
+
+  /* purge off 1 and 2 */
+  if ((((ix - 0x3fff8000) | i0 | i1) == 0)
+      || (((ix - 0x40008000) | i0 | i1) == 0))
+    r = 0;
+  else if (ix < 0x40008000) /* 2.0 */
+    {
+      /* x < 2.0 */
+      if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
+	{
+	  /* lgamma(x) = lgamma(x+1) - log(x) */
+	  r = -__ieee754_logl (x);
+	  if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
+	    {
+	      y = x - one;
+	      i = 0;
+	    }
+	  else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
+	    {
+	      y = x - (tc - one);
+	      i = 1;
+	    }
+	  else
+	    {
+	      /* x < 0.23 */
+	      y = x;
+	      i = 2;
+	    }
+	}
+      else
+	{
+	  r = zero;
+	  if (ix >= 0x3fffdda6) /* 1.73162841796875 */
+	    {
+	      /* [1.7316,2] */
+	      y = x - 2.0;
+	      i = 0;
+	    }
+	  else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
+	    {
+	      /* [1.23,1.73] */
+	      y = x - tc;
+	      i = 1;
+	    }
+	  else
+	    {
+	      /* [0.9, 1.23] */
+	      y = x - one;
+	      i = 2;
+	    }
+	}
+      switch (i)
+	{
+	case 0:
+	  p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
+	  p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
+	  r += half * y + y * p1/p2;
+	  break;
+	case 1:
+    p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
+    p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
+    p = tt + y * p1/p2;
+	  r += (tf + p);
+	  break;
+	case 2:
+ p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
+      p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
+	  r += (-half * y + p1 / p2);
+	}
+    }
+  else if (ix < 0x40028000) /* 8.0 */
+    {
+      /* x < 8.0 */
+      i = (int) x;
+      t = zero;
+      y = x - (double) i;
+  p = y *
+     (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
+  q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
+      r = half * y + p / q;
+      z = one;			/* lgamma(1+s) = log(s) + lgamma(s) */
+      switch (i)
+	{
+	case 7:
+	  z *= (y + 6.0);	/* FALLTHRU */
+	case 6:
+	  z *= (y + 5.0);	/* FALLTHRU */
+	case 5:
+	  z *= (y + 4.0);	/* FALLTHRU */
+	case 4:
+	  z *= (y + 3.0);	/* FALLTHRU */
+	case 3:
+	  z *= (y + 2.0);	/* FALLTHRU */
+	  r += __ieee754_logl (z);
+	  break;
+	}
+    }
+  else if (ix < 0x40418000) /* 2^66 */
+    {
+      /* 8.0 <= x < 2**66 */
+      t = __ieee754_logl (x);
+      z = one / x;
+      y = z * z;
+      w = w0 + z * (w1
+          + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
+      r = (x - half) * (t - one) + w;
+    }
+  else
+    /* 2**66 <= x <= inf */
+    r = x * (__ieee754_logl (x) - one);
+  if (se & 0x8000)
+    r = nadj - r;
+  return r;
+}