Implement fmal, some fma bugfixes

9 files changed, 492 insertions, 97 deletions

diff --git a/math/libm-test.inc b/math/libm-test.inc
index 1bec476fbf..ceed18d7fd 100644
--- a/math/libm-test.inc
+++ b/math/libm-test.inc
@@ -2787,8 +2787,24 @@ fma_test (void)
   TEST_fff_f (fma, minus_infty, plus_infty, plus_infty, nan_value, INVALID_EXCEPTION);
   TEST_fff_f (fma, plus_infty, minus_infty, plus_infty, nan_value, INVALID_EXCEPTION);
   TEST_fff_f (fma, minus_infty, minus_infty, minus_infty, nan_value, INVALID_EXCEPTION);
+  TEST_fff_f (fma, plus_infty, 3.5L, minus_infty, nan_value, INVALID_EXCEPTION);
+  TEST_fff_f (fma, minus_infty, -7.5L, minus_infty, nan_value, INVALID_EXCEPTION);
+  TEST_fff_f (fma, -13.5L, plus_infty, plus_infty, nan_value, INVALID_EXCEPTION);
+  TEST_fff_f (fma, minus_infty, 7.5L, plus_infty, nan_value, INVALID_EXCEPTION);
 
   TEST_fff_f (fma, 1.25L, 0.75L, 0.0625L, 1.0L);
+
+  FLOAT fltmax = CHOOSE (LDBL_MAX, DBL_MAX, FLT_MAX,
+			 LDBL_MAX, DBL_MAX, FLT_MAX);
+  TEST_fff_f (fma, -fltmax, -fltmax, minus_infty, minus_infty);
+  TEST_fff_f (fma, fltmax / 2, fltmax / 2, minus_infty, minus_infty);
+  TEST_fff_f (fma, -fltmax, fltmax, plus_infty, plus_infty);
+  TEST_fff_f (fma, fltmax / 2, -fltmax / 4, plus_infty, plus_infty);
+  TEST_fff_f (fma, plus_infty, 4, plus_infty, plus_infty);
+  TEST_fff_f (fma, 2, minus_infty, minus_infty, minus_infty);
+  TEST_fff_f (fma, minus_infty, minus_infty, plus_infty, plus_infty);
+  TEST_fff_f (fma, plus_infty, minus_infty, minus_infty, minus_infty);
+
 #if defined (TEST_FLOAT) && FLT_MANT_DIG == 24
   TEST_fff_f (fma, 0x1.7ff8p+13, 0x1.000002p+0, 0x1.ffffp-24, 0x1.7ff802p+13);
   TEST_fff_f (fma, 0x1.fffp+0, 0x1.00001p+0, -0x1.fffp+0, 0x1.fffp-20);
@@ -2818,6 +2834,15 @@ fma_test (void)
   TEST_fff_f (fma, -0x1.19cab66d73e17p-959, 0x1.c7108a8c5ff51p-107, -0x0.80b0ad65d9b64p-1022, -0x0.80b0ad65d9d59p-1022);
   TEST_fff_f (fma, -0x1.d2eaed6e8e9d3p-979, -0x1.4e066c62ac9ddp-63, -0x0.9245e6b003454p-1022, -0x0.9245c09c5fb5dp-1022);
   TEST_fff_f (fma, 0x1.153d650bb9f06p-907, 0x1.2d01230d48407p-125, -0x0.b278d5acfc3cp-1022, -0x0.b22757123bbe9p-1022);
+  TEST_fff_f (fma, -0x1.fffffffffffffp-711, 0x1.fffffffffffffp-275, 0x1.fffffe00007ffp-983, 0x1.7ffffe00007ffp-983);
+#endif
+#if defined (TEST_LDOUBLE) && LDBL_MANT_DIG == 64
+  TEST_fff_f (fma, -0x8.03fcp+3696L, 0xf.fffffffffffffffp-6140L, 0x8.3ffffffffffffffp-2450L, -0x8.01ecp-2440L);
+  TEST_fff_f (fma, 0x9.fcp+2033L, -0x8.000e1f000ff800fp-3613L, -0xf.fffffffffffc0ffp-1579L, -0xd.fc119fb093ed092p-1577L);
+  TEST_fff_f (fma, 0xc.7fc000003ffffffp-1194L, 0x8.1e0003fffffffffp+15327L, -0x8.fffep+14072L, 0xc.ae9f164020effffp+14136L);
+  TEST_fff_f (fma, -0x8.0001fc000000003p+1798L, 0xcp-2230L, 0x8.f7e000000000007p-468L, -0xc.0002f9ffee10404p-429L);
+  TEST_fff_f (fma, 0xc.0000000000007ffp+10130L, -0x8.000000000000001p+4430L, 0xc.07000000001ffffp+14513L, -0xb.fffffffffffd7e4p+14563L);
+  TEST_fff_f (fma, 0xb.ffffp-4777L, 0x8.000000fffffffffp-11612L, -0x0.3800fff8p-16385L, 0x5.c7fe80c7ffeffffp-16385L);
 #endif
 
   END (fma);
diff --git a/sysdeps/i386/fpu/s_fma.S b/sysdeps/i386/fpu/s_fma.S
deleted file mode 100644
index db4959ccfc..0000000000
--- a/sysdeps/i386/fpu/s_fma.S
+++ /dev/null
@@ -1,31 +0,0 @@
-/* Compute (X * Y) + Z as ternary operation.
-   Copyright (C) 1997, 1998 Free Software Foundation, Inc.
-   This file is part of the GNU C Library.
-   Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997.
-
-   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, write to the Free
-   Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
-   02111-1307 USA.  */
-
-#include <sysdep.h>
-
-	.text
-ENTRY(__fma)
-	fldl	4(%esp)		// x
-	fmull	12(%esp)	// x * y
-	fldl	20(%esp)	// z : x * y
-	faddp			// (x * y) + z
-	ret
-END(__fma)
-weak_alias (__fma, fma)
diff --git a/sysdeps/i386/fpu/s_fmaf.S b/sysdeps/i386/fpu/s_fmaf.S
deleted file mode 100644
index 5f875532a8..0000000000
--- a/sysdeps/i386/fpu/s_fmaf.S
+++ /dev/null
@@ -1,31 +0,0 @@
-/* Compute (X * Y) + Z as ternary operation.
-   Copyright (C) 1997 Free Software Foundation, Inc.
-   This file is part of the GNU C Library.
-   Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997.
-
-   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, write to the Free
-   Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
-   02111-1307 USA.  */
-
-#include <sysdep.h>
-
-	.text
-ENTRY(__fmaf)
-	flds	4(%esp)		// x
-	fmuls	8(%esp)		// x * y
-	flds	12(%esp)	// z : x * y
-	faddp			// (x * y) + z
-	ret
-END(__fmaf)
-weak_alias (__fmaf, fmaf)
diff --git a/sysdeps/i386/fpu/s_fmal.S b/sysdeps/i386/fpu/s_fmal.S
deleted file mode 100644
index 1837f848c0..0000000000
--- a/sysdeps/i386/fpu/s_fmal.S
+++ /dev/null
@@ -1,32 +0,0 @@
-/* Compute (X * Y) + Z as ternary operation.
-   Copyright (C) 1997 Free Software Foundation, Inc.
-   This file is part of the GNU C Library.
-   Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997.
-
-   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, write to the Free
-   Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
-   02111-1307 USA.  */
-
-#include <sysdep.h>
-
-	.text
-ENTRY(__fmal)
-	fldt	4(%esp)		// x
-	fldt	16(%esp)	// x : y
-	fmulp			// x * y
-	fldt	28(%esp)	// z : x * y
-	faddp			// (x * y) + z
-	ret
-END(__fmal)
-weak_alias (__fmal, fmal)
diff --git a/sysdeps/ieee754/dbl-64/s_fma.c b/sysdeps/ieee754/dbl-64/s_fma.c
index 911682eceb..3b0bfd5ce6 100644
--- a/sysdeps/ieee754/dbl-64/s_fma.c
+++ b/sysdeps/ieee754/dbl-64/s_fma.c
@@ -43,6 +43,12 @@ __fma (double x, double y, double z)
       || __builtin_expect (u.ieee.exponent + v.ieee.exponent
 			   <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
     {
+      /* If z is Inf, but x and y are finite, the result should be
+	 z rather than NaN.  */
+      if (w.ieee.exponent == 0x7ff
+	  && u.ieee.exponent != 0x7ff
+          && v.ieee.exponent != 0x7ff)
+	return (z + x) + y;
       /* If x or y or z is Inf/NaN, or if fma will certainly overflow,
 	 or if x * y is less than half of DBL_DENORM_MIN,
 	 compute as x * y + z.  */
@@ -165,6 +171,8 @@ __fma (double x, double y, double z)
     }
   else
     {
+      if ((u.ieee.mantissa1 & 1) == 0)
+	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
       v.d = a1 + u.d;
       int j = fetestexcept (FE_INEXACT) != 0;
       feupdateenv (&env);
diff --git a/sysdeps/ieee754/ldbl-128/s_fmal.c b/sysdeps/ieee754/ldbl-128/s_fmal.c
new file mode 100644
index 0000000000..9ec5ba9ee9
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-128/s_fmal.c
@@ -0,0 +1,221 @@
+/* Compute x * y + z as ternary operation.
+   Copyright (C) 2010 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+   Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.
+
+   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, write to the Free
+   Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
+   02111-1307 USA.  */
+
+#include <float.h>
+#include <math.h>
+#include <fenv.h>
+#include <ieee754.h>
+
+/* This implementation uses rounding to odd to avoid problems with
+   double rounding.  See a paper by Boldo and Melquiond:
+   http://www.lri.fr/~melquion/doc/08-tc.pdf  */
+
+long double
+__fmal (long double x, long double y, long double z)
+{
+  union ieee854_long_double u, v, w;
+  int adjust = 0;
+  u.d = x;
+  v.d = y;
+  w.d = z;
+  if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
+			>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
+			   - LDBL_MANT_DIG, 0)
+      || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
+      || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
+      || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
+      || __builtin_expect (u.ieee.exponent + v.ieee.exponent
+			   <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))
+    {
+      /* If z is Inf, but x and y are finite, the result should be
+	 z rather than NaN.  */
+      if (w.ieee.exponent == 0x7fff
+	  && u.ieee.exponent != 0x7fff
+          && v.ieee.exponent != 0x7fff)
+	return (z + x) + y;
+      /* If x or y or z is Inf/NaN, or if fma will certainly overflow,
+	 or if x * y is less than half of LDBL_DENORM_MIN,
+	 compute as x * y + z.  */
+      if (u.ieee.exponent == 0x7fff
+	  || v.ieee.exponent == 0x7fff
+	  || w.ieee.exponent == 0x7fff
+	  || u.ieee.exponent + v.ieee.exponent
+	     > 0x7fff + IEEE854_LONG_DOUBLE_BIAS
+	  || u.ieee.exponent + v.ieee.exponent
+	     < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
+	return x * y + z;
+      if (u.ieee.exponent + v.ieee.exponent
+	  >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
+	{
+	  /* Compute 1p-113 times smaller result and multiply
+	     at the end.  */
+	  if (u.ieee.exponent > v.ieee.exponent)
+	    u.ieee.exponent -= LDBL_MANT_DIG;
+	  else
+	    v.ieee.exponent -= LDBL_MANT_DIG;
+	  /* If x + y exponent is very large and z exponent is very small,
+	     it doesn't matter if we don't adjust it.  */
+	  if (w.ieee.exponent > LDBL_MANT_DIG)
+	    w.ieee.exponent -= LDBL_MANT_DIG;
+	  adjust = 1;
+	}
+      else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
+	{
+	  /* Similarly.
+	     If z exponent is very large and x and y exponents are
+	     very small, it doesn't matter if we don't adjust it.  */
+	  if (u.ieee.exponent > v.ieee.exponent)
+	    {
+	      if (u.ieee.exponent > LDBL_MANT_DIG)
+		u.ieee.exponent -= LDBL_MANT_DIG;
+	    }
+	  else if (v.ieee.exponent > LDBL_MANT_DIG)
+	    v.ieee.exponent -= LDBL_MANT_DIG;
+	  w.ieee.exponent -= LDBL_MANT_DIG;
+	  adjust = 1;
+	}
+      else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
+	{
+	  u.ieee.exponent -= LDBL_MANT_DIG;
+	  if (v.ieee.exponent)
+	    v.ieee.exponent += LDBL_MANT_DIG;
+	  else
+	    v.d *= 0x1p113L;
+	}
+      else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
+	{
+	  v.ieee.exponent -= LDBL_MANT_DIG;
+	  if (u.ieee.exponent)
+	    u.ieee.exponent += LDBL_MANT_DIG;
+	  else
+	    u.d *= 0x1p113L;
+	}
+      else /* if (u.ieee.exponent + v.ieee.exponent
+		  <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
+	{
+	  if (u.ieee.exponent > v.ieee.exponent)
+	    u.ieee.exponent += 2 * LDBL_MANT_DIG;
+	  else
+	    v.ieee.exponent += 2 * LDBL_MANT_DIG;
+	  if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 4)
+	    {
+	      if (w.ieee.exponent)
+		w.ieee.exponent += 2 * LDBL_MANT_DIG;
+	      else
+		w.d *= 0x1p226L;
+	      adjust = -1;
+	    }
+	  /* Otherwise x * y should just affect inexact
+	     and nothing else.  */
+	}
+      x = u.d;
+      y = v.d;
+      z = w.d;
+    }
+  /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
+#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
+  long double x1 = x * C;
+  long double y1 = y * C;
+  long double m1 = x * y;
+  x1 = (x - x1) + x1;
+  y1 = (y - y1) + y1;
+  long double x2 = x - x1;
+  long double y2 = y - y1;
+  long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
+
+  /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
+  long double a1 = z + m1;
+  long double t1 = a1 - z;
+  long double t2 = a1 - t1;
+  t1 = m1 - t1;
+  t2 = z - t2;
+  long double a2 = t1 + t2;
+
+  fenv_t env;
+  feholdexcept (&env);
+  fesetround (FE_TOWARDZERO);
+  /* Perform m2 + a2 addition with round to odd.  */
+  u.d = a2 + m2;
+
+  if (__builtin_expect (adjust == 0, 1))
+    {
+      if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
+	u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
+      feupdateenv (&env);
+      /* Result is a1 + u.d.  */
+      return a1 + u.d;
+    }
+  else if (__builtin_expect (adjust > 0, 1))
+    {
+      if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
+	u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
+      feupdateenv (&env);
+      /* Result is a1 + u.d, scaled up.  */
+      return (a1 + u.d) * 0x1p113L;
+    }
+  else
+    {
+      if ((u.ieee.mantissa3 & 1) == 0)
+	u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
+      v.d = a1 + u.d;
+      int j = fetestexcept (FE_INEXACT) != 0;
+      feupdateenv (&env);
+      /* Ensure the following computations are performed in default rounding
+	 mode instead of just reusing the round to zero computation.  */
+      asm volatile ("" : "=m" (u) : "m" (u));
+      /* If a1 + u.d is exact, the only rounding happens during
+	 scaling down.  */
+      if (j == 0)
+	return v.d * 0x1p-226L;
+      /* If result rounded to zero is not subnormal, no double
+	 rounding will occur.  */
+      if (v.ieee.exponent > 226)
+	return (a1 + u.d) * 0x1p-226L;
+      /* If v.d * 0x1p-226L with round to zero is a subnormal above
+	 or equal to LDBL_MIN / 2, then v.d * 0x1p-226L shifts mantissa
+	 down just by 1 bit, which means v.ieee.mantissa3 |= j would
+	 change the round bit, not sticky or guard bit.
+	 v.d * 0x1p-226L never normalizes by shifting up,
+	 so round bit plus sticky bit should be already enough
+	 for proper rounding.  */
+      if (v.ieee.exponent == 226)
+	{
+	  /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
+	     v.ieee.mantissa3 & 1 is the round bit and j is our sticky
+	     bit.  In round-to-nearest 001 rounds down like 00,
+	     011 rounds up, even though 01 rounds down (thus we need
+	     to adjust), 101 rounds down like 10 and 111 rounds up
+	     like 11.  */
+	  if ((v.ieee.mantissa3 & 3) == 1)
+	    {
+	      v.d *= 0x1p-226L;
+	      if (v.ieee.negative)
+		return v.d - 0x1p-16493L /* __LDBL_DENORM_MIN__ */;
+	      else
+		return v.d + 0x1p-16493L /* __LDBL_DENORM_MIN__ */;
+	    }
+	  else
+	    return v.d * 0x1p-226L;
+	}
+      v.ieee.mantissa3 |= j;
+      return v.d * 0x1p-226L;
+    }
+}
+weak_alias (__fmal, fmal)
diff --git a/sysdeps/ieee754/ldbl-64-128/s_fmal.c b/sysdeps/ieee754/ldbl-64-128/s_fmal.c
new file mode 100644
index 0000000000..218aa52b35
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-64-128/s_fmal.c
@@ -0,0 +1,5 @@
+#include <math_ldbl_opt.h>
+#undef weak_alias
+#define weak_alias(n,a)
+#include <sysdeps/ieee754/ldbl-128/s_fmal.c>
+long_double_symbol (libm, __fmal, fmal);
diff --git a/sysdeps/ieee754/ldbl-96/s_fma.c b/sysdeps/ieee754/ldbl-96/s_fma.c
index 90c6d1f29d..6c7e9d0d36 100644
--- a/sysdeps/ieee754/ldbl-96/s_fma.c
+++ b/sysdeps/ieee754/ldbl-96/s_fma.c
@@ -30,11 +30,20 @@
 double
 __fma (double x, double y, double z)
 {
+  if (__builtin_expect (isinf (z), 0))
+    {
+      /* If z is Inf, but x and y are finite, the result should be
+	 z rather than NaN.  */
+      if (finite (x) && finite (y))
+	return (z + x) + y;
+      return (x * y) + z;
+    }
+
   /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
 #define C ((1ULL << (LDBL_MANT_DIG + 1) / 2) + 1)
-  long double x1 = x * C;
-  long double y1 = y * C;
-  long double m1 = x * y;
+  long double x1 = (long double) x * C;
+  long double y1 = (long double) y * C;
+  long double m1 = (long double) x * y;
   x1 = (x - x1) + x1;
   y1 = (y - y1) + y1;
   long double x2 = x - x1;
diff --git a/sysdeps/ieee754/ldbl-96/s_fmal.c b/sysdeps/ieee754/ldbl-96/s_fmal.c
new file mode 100644
index 0000000000..340f02536c
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-96/s_fmal.c
@@ -0,0 +1,221 @@
+/* Compute x * y + z as ternary operation.
+   Copyright (C) 2010 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+   Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.
+
+   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, write to the Free
+   Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
+   02111-1307 USA.  */
+
+#include <float.h>
+#include <math.h>
+#include <fenv.h>
+#include <ieee754.h>
+
+/* This implementation uses rounding to odd to avoid problems with
+   double rounding.  See a paper by Boldo and Melquiond:
+   http://www.lri.fr/~melquion/doc/08-tc.pdf  */
+
+long double
+__fmal (long double x, long double y, long double z)
+{
+  union ieee854_long_double u, v, w;
+  int adjust = 0;
+  u.d = x;
+  v.d = y;
+  w.d = z;
+  if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
+			>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
+			   - LDBL_MANT_DIG, 0)
+      || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
+      || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
+      || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
+      || __builtin_expect (u.ieee.exponent + v.ieee.exponent
+			   <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))
+    {
+      /* If z is Inf, but x and y are finite, the result should be
+	 z rather than NaN.  */
+      if (w.ieee.exponent == 0x7fff
+	  && u.ieee.exponent != 0x7fff
+          && v.ieee.exponent != 0x7fff)
+	return (z + x) + y;
+      /* If x or y or z is Inf/NaN, or if fma will certainly overflow,
+	 or if x * y is less than half of LDBL_DENORM_MIN,
+	 compute as x * y + z.  */
+      if (u.ieee.exponent == 0x7fff
+	  || v.ieee.exponent == 0x7fff
+	  || w.ieee.exponent == 0x7fff
+	  || u.ieee.exponent + v.ieee.exponent
+	     > 0x7fff + IEEE854_LONG_DOUBLE_BIAS
+	  || u.ieee.exponent + v.ieee.exponent
+	     < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
+	return x * y + z;
+      if (u.ieee.exponent + v.ieee.exponent
+	  >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
+	{
+	  /* Compute 1p-64 times smaller result and multiply
+	     at the end.  */
+	  if (u.ieee.exponent > v.ieee.exponent)
+	    u.ieee.exponent -= LDBL_MANT_DIG;
+	  else
+	    v.ieee.exponent -= LDBL_MANT_DIG;
+	  /* If x + y exponent is very large and z exponent is very small,
+	     it doesn't matter if we don't adjust it.  */
+	  if (w.ieee.exponent > LDBL_MANT_DIG)
+	    w.ieee.exponent -= LDBL_MANT_DIG;
+	  adjust = 1;
+	}
+      else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
+	{
+	  /* Similarly.
+	     If z exponent is very large and x and y exponents are
+	     very small, it doesn't matter if we don't adjust it.  */
+	  if (u.ieee.exponent > v.ieee.exponent)
+	    {
+	      if (u.ieee.exponent > LDBL_MANT_DIG)
+		u.ieee.exponent -= LDBL_MANT_DIG;
+	    }
+	  else if (v.ieee.exponent > LDBL_MANT_DIG)
+	    v.ieee.exponent -= LDBL_MANT_DIG;
+	  w.ieee.exponent -= LDBL_MANT_DIG;
+	  adjust = 1;
+	}
+      else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
+	{
+	  u.ieee.exponent -= LDBL_MANT_DIG;
+	  if (v.ieee.exponent)
+	    v.ieee.exponent += LDBL_MANT_DIG;
+	  else
+	    v.d *= 0x1p64L;
+	}
+      else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
+	{
+	  v.ieee.exponent -= LDBL_MANT_DIG;
+	  if (u.ieee.exponent)
+	    u.ieee.exponent += LDBL_MANT_DIG;
+	  else
+	    u.d *= 0x1p64L;
+	}
+      else /* if (u.ieee.exponent + v.ieee.exponent
+		  <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
+	{
+	  if (u.ieee.exponent > v.ieee.exponent)
+	    u.ieee.exponent += 2 * LDBL_MANT_DIG;
+	  else
+	    v.ieee.exponent += 2 * LDBL_MANT_DIG;
+	  if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 4)
+	    {
+	      if (w.ieee.exponent)
+		w.ieee.exponent += 2 * LDBL_MANT_DIG;
+	      else
+		w.d *= 0x1p128L;
+	      adjust = -1;
+	    }
+	  /* Otherwise x * y should just affect inexact
+	     and nothing else.  */
+	}
+      x = u.d;
+      y = v.d;
+      z = w.d;
+    }
+  /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
+#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
+  long double x1 = x * C;
+  long double y1 = y * C;
+  long double m1 = x * y;
+  x1 = (x - x1) + x1;
+  y1 = (y - y1) + y1;
+  long double x2 = x - x1;
+  long double y2 = y - y1;
+  long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
+
+  /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
+  long double a1 = z + m1;
+  long double t1 = a1 - z;
+  long double t2 = a1 - t1;
+  t1 = m1 - t1;
+  t2 = z - t2;
+  long double a2 = t1 + t2;
+
+  fenv_t env;
+  feholdexcept (&env);
+  fesetround (FE_TOWARDZERO);
+  /* Perform m2 + a2 addition with round to odd.  */
+  u.d = a2 + m2;
+
+  if (__builtin_expect (adjust == 0, 1))
+    {
+      if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
+	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
+      feupdateenv (&env);
+      /* Result is a1 + u.d.  */
+      return a1 + u.d;
+    }
+  else if (__builtin_expect (adjust > 0, 1))
+    {
+      if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
+	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
+      feupdateenv (&env);
+      /* Result is a1 + u.d, scaled up.  */
+      return (a1 + u.d) * 0x1p64L;
+    }
+  else
+    {
+      if ((u.ieee.mantissa1 & 1) == 0)
+	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
+      v.d = a1 + u.d;
+      int j = fetestexcept (FE_INEXACT) != 0;
+      feupdateenv (&env);
+      /* Ensure the following computations are performed in default rounding
+	 mode instead of just reusing the round to zero computation.  */
+      asm volatile ("" : "=m" (u) : "m" (u));
+      /* If a1 + u.d is exact, the only rounding happens during
+	 scaling down.  */
+      if (j == 0)
+	return v.d * 0x1p-128L;
+      /* If result rounded to zero is not subnormal, no double
+	 rounding will occur.  */
+      if (v.ieee.exponent > 128)
+	return (a1 + u.d) * 0x1p-128L;
+      /* If v.d * 0x1p-128L with round to zero is a subnormal above
+	 or equal to LDBL_MIN / 2, then v.d * 0x1p-128L shifts mantissa
+	 down just by 1 bit, which means v.ieee.mantissa1 |= j would
+	 change the round bit, not sticky or guard bit.
+	 v.d * 0x1p-128L never normalizes by shifting up,
+	 so round bit plus sticky bit should be already enough
+	 for proper rounding.  */
+      if (v.ieee.exponent == 128)
+	{
+	  /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
+	     v.ieee.mantissa1 & 1 is the round bit and j is our sticky
+	     bit.  In round-to-nearest 001 rounds down like 00,
+	     011 rounds up, even though 01 rounds down (thus we need
+	     to adjust), 101 rounds down like 10 and 111 rounds up
+	     like 11.  */
+	  if ((v.ieee.mantissa1 & 3) == 1)
+	    {
+	      v.d *= 0x1p-128L;
+	      if (v.ieee.negative)
+		return v.d - 0x1p-16445L /* __LDBL_DENORM_MIN__ */;
+	      else
+		return v.d + 0x1p-16445L /* __LDBL_DENORM_MIN__ */;
+	    }
+	  else
+	    return v.d * 0x1p-128L;
+	}
+      v.ieee.mantissa1 |= j;
+      return v.d * 0x1p-128L;
+    }
+}
+weak_alias (__fmal, fmal)