[thirdparty/glibc.git] / sysdeps / ieee754 / dbl-64 / s_fma.c

/* Compute x * y + z as ternary operation.
   Copyright (C) 2010-2019 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, see
   <https://www.gnu.org/licenses/>.  */

#include <float.h>
#include <math.h>
#include <fenv.h>
#include <ieee754.h>
#include <math-barriers.h>
#include <fenv_private.h>
#include <libm-alias-double.h>
#include <tininess.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  */

double
__fma (double x, double y, double z)
{
  union ieee754_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
			>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
      || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
      || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
      || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
      || __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 z is zero and x are y are nonzero, compute the result
	 as x * y to avoid the wrong sign of a zero result if x * y
	 underflows to 0.  */
      if (z == 0 && x != 0 && y != 0)
	return x * y;
      /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
	 x * y + z.  */
      if (u.ieee.exponent == 0x7ff
	  || v.ieee.exponent == 0x7ff
	  || w.ieee.exponent == 0x7ff
	  || x == 0
	  || y == 0)
	return x * y + z;
      /* If fma will certainly overflow, compute as x * y.  */
      if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
	return x * y;
      /* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the
	 result nor whether there is underflow depends on its exact
	 value, only on its sign.  */
      if (u.ieee.exponent + v.ieee.exponent
	  < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
	{
	  int neg = u.ieee.negative ^ v.ieee.negative;
	  double tiny = neg ? -0x1p-1074 : 0x1p-1074;
	  if (w.ieee.exponent >= 3)
	    return tiny + z;
	  /* Scaling up, adding TINY and scaling down produces the
	     correct result, because in round-to-nearest mode adding
	     TINY has no effect and in other modes double rounding is
	     harmless.  But it may not produce required underflow
	     exceptions.  */
	  v.d = z * 0x1p54 + tiny;
	  if (TININESS_AFTER_ROUNDING
	      ? v.ieee.exponent < 55
	      : (w.ieee.exponent == 0
		 || (w.ieee.exponent == 1
		     && w.ieee.negative != neg
		     && w.ieee.mantissa1 == 0
		     && w.ieee.mantissa0 == 0)))
	    {
	      double force_underflow = x * y;
	      math_force_eval (force_underflow);
	    }
	  return v.d * 0x1p-54;
	}
      if (u.ieee.exponent + v.ieee.exponent
	  >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
	{
	  /* Compute 1p-53 times smaller result and multiply
	     at the end.  */
	  if (u.ieee.exponent > v.ieee.exponent)
	    u.ieee.exponent -= DBL_MANT_DIG;
	  else
	    v.ieee.exponent -= DBL_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 > DBL_MANT_DIG)
	    w.ieee.exponent -= DBL_MANT_DIG;
	  adjust = 1;
	}
      else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
	{
	  /* Similarly.
	     If z exponent is very large and x and y exponents are
	     very small, adjust them up to avoid spurious underflows,
	     rather than down.  */
	  if (u.ieee.exponent + v.ieee.exponent
	      <= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG)
	    {
	      if (u.ieee.exponent > v.ieee.exponent)
		u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	      else
		v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	    }
	  else if (u.ieee.exponent > v.ieee.exponent)
	    {
	      if (u.ieee.exponent > DBL_MANT_DIG)
		u.ieee.exponent -= DBL_MANT_DIG;
	    }
	  else if (v.ieee.exponent > DBL_MANT_DIG)
	    v.ieee.exponent -= DBL_MANT_DIG;
	  w.ieee.exponent -= DBL_MANT_DIG;
	  adjust = 1;
	}
      else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
	{
	  u.ieee.exponent -= DBL_MANT_DIG;
	  if (v.ieee.exponent)
	    v.ieee.exponent += DBL_MANT_DIG;
	  else
	    v.d *= 0x1p53;
	}
      else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
	{
	  v.ieee.exponent -= DBL_MANT_DIG;
	  if (u.ieee.exponent)
	    u.ieee.exponent += DBL_MANT_DIG;
	  else
	    u.d *= 0x1p53;
	}
      else /* if (u.ieee.exponent + v.ieee.exponent
		  <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
	{
	  if (u.ieee.exponent > v.ieee.exponent)
	    u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	  else
	    v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	  if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
	    {
	      if (w.ieee.exponent)
		w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	      else
		w.d *= 0x1p108;
	      adjust = -1;
	    }
	  /* Otherwise x * y should just affect inexact
	     and nothing else.  */
	}
      x = u.d;
      y = v.d;
      z = w.d;
    }

  /* Ensure correct sign of exact 0 + 0.  */
  if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
    {
      x = math_opt_barrier (x);
      return x * y + z;
    }

  fenv_t env;
  libc_feholdexcept_setround (&env, FE_TONEAREST);

  /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
  double x1 = x * C;
  double y1 = y * C;
  double m1 = x * y;
  x1 = (x - x1) + x1;
  y1 = (y - y1) + y1;
  double x2 = x - x1;
  double y2 = y - y1;
  double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;

  /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
  double a1 = z + m1;
  double t1 = a1 - z;
  double t2 = a1 - t1;
  t1 = m1 - t1;
  t2 = z - t2;
  double a2 = t1 + t2;
  /* Ensure the arithmetic is not scheduled after feclearexcept call.  */
  math_force_eval (m2);
  math_force_eval (a2);
  feclearexcept (FE_INEXACT);

  /* If the result is an exact zero, ensure it has the correct sign.  */
  if (a1 == 0 && m2 == 0)
    {
      libc_feupdateenv (&env);
      /* Ensure that round-to-nearest value of z + m1 is not reused.  */
      z = math_opt_barrier (z);
      return z + m1;
    }

  libc_fesetround (FE_TOWARDZERO);

  /* Perform m2 + a2 addition with round to odd.  */
  u.d = a2 + m2;

  if (__glibc_unlikely (adjust < 0))
    {
      if ((u.ieee.mantissa1 & 1) == 0)
	u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
      v.d = a1 + u.d;
      /* Ensure the addition is not scheduled after fetestexcept call.  */
      math_force_eval (v.d);
    }

  /* Reset rounding mode and test for inexact simultaneously.  */
  int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;

  if (__glibc_likely (adjust == 0))
    {
      if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
	u.ieee.mantissa1 |= j;
      /* Result is a1 + u.d.  */
      return a1 + u.d;
    }
  else if (__glibc_likely (adjust > 0))
    {
      if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
	u.ieee.mantissa1 |= j;
      /* Result is a1 + u.d, scaled up.  */
      return (a1 + u.d) * 0x1p53;
    }
  else
    {
      /* If a1 + u.d is exact, the only rounding happens during
	 scaling down.  */
      if (j == 0)
	return v.d * 0x1p-108;
      /* If result rounded to zero is not subnormal, no double
	 rounding will occur.  */
      if (v.ieee.exponent > 108)
	return (a1 + u.d) * 0x1p-108;
      /* If v.d * 0x1p-108 with round to zero is a subnormal above
	 or equal to DBL_MIN / 2, then v.d * 0x1p-108 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-108 never normalizes by shifting up,
	 so round bit plus sticky bit should be already enough
	 for proper rounding.  */
      if (v.ieee.exponent == 108)
	{
	  /* If the exponent would be in the normal range when
	     rounding to normal precision with unbounded exponent
	     range, the exact result is known and spurious underflows
	     must be avoided on systems detecting tininess after
	     rounding.  */
	  if (TININESS_AFTER_ROUNDING)
	    {
	      w.d = a1 + u.d;
	      if (w.ieee.exponent == 109)
		return w.d * 0x1p-108;
	    }
	  /* 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.  */
	  w.d = 0.0;
	  w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
	  w.ieee.negative = v.ieee.negative;
	  v.ieee.mantissa1 &= ~3U;
	  v.d *= 0x1p-108;
	  w.d *= 0x1p-2;
	  return v.d + w.d;
	}
      v.ieee.mantissa1 |= j;
      return v.d * 0x1p-108;
    }
}
#ifndef __fma
libm_alias_double (__fma, fma)
#endif
Commit	Line	Data
5e908464	1	/* Compute x * y + z as ternary operation.
04277e02	2	Copyright (C) 2010-2019 Free Software Foundation, Inc.
5e908464 JJ	3	This file is part of the GNU C Library.
	4	Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.
	5
	6	The GNU C Library is free software; you can redistribute it and/or
	7	modify it under the terms of the GNU Lesser General Public
	8	License as published by the Free Software Foundation; either
	9	version 2.1 of the License, or (at your option) any later version.
	10
	11	The GNU C Library is distributed in the hope that it will be useful,
	12	but WITHOUT ANY WARRANTY; without even the implied warranty of
	13	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
	14	Lesser General Public License for more details.
	15
	16	You should have received a copy of the GNU Lesser General Public
59ba27a6	17	License along with the GNU C Library; if not, see
5a82c748	18	<https://www.gnu.org/licenses/>. */
5e908464 JJ	19
	20	#include <float.h>
	21	#include <math.h>
	22	#include <fenv.h>
	23	#include <ieee754.h>
b4d5b8b0	24	#include <math-barriers.h>
70e2ba33	25	#include <fenv_private.h>
7e16a5d1	26	#include <libm-alias-double.h>
ef82f4da	27	#include <tininess.h>
5e908464 JJ	28
	29	/* This implementation uses rounding to odd to avoid problems with
	30	double rounding. See a paper by Boldo and Melquiond:
	31	http://www.lri.fr/~melquion/doc/08-tc.pdf */
	32
	33	double
	34	__fma (double x, double y, double z)
	35	{
	36	union ieee754_double u, v, w;
	37	int adjust = 0;
	38	u.d = x;
	39	v.d = y;
	40	w.d = z;
	41	if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
	42	>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
	43	\|\| __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
	44	\|\| __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
f3f7372d JJ	45	\|\| __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
	46	\|\| __builtin_expect (u.ieee.exponent + v.ieee.exponent
	47	<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
5e908464	48	{
3e692e05 JJ	49	/* If z is Inf, but x and y are finite, the result should be
	50	z rather than NaN. */
	51	if (w.ieee.exponent == 0x7ff
	52	&& u.ieee.exponent != 0x7ff
d9a8d0ab	53	&& v.ieee.exponent != 0x7ff)
3e692e05	54	return (z + x) + y;
bec749fd JM	55	/* If z is zero and x are y are nonzero, compute the result
	56	as x * y to avoid the wrong sign of a zero result if x * y
	57	underflows to 0. */
	58	if (z == 0 && x != 0 && y != 0)
	59	return x * y;
a0c2940d JM	60	/* If x or y or z is Inf/NaN, or if x * y is zero, compute as
a0c2940d JM	61	x * y + z. */
5e908464 JJ	62	if (u.ieee.exponent == 0x7ff
	63	\|\| v.ieee.exponent == 0x7ff
	64	\|\| w.ieee.exponent == 0x7ff
473611b2 JM	65	\|\| x == 0
473611b2 JM	66	\|\| y == 0)
5e908464	67	return x * y + z;
a0c2940d JM	68	/* If fma will certainly overflow, compute as x * y. */
	69	if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
	70	return x * y;
1f4dafa3	71	/* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the
473611b2 JM	72	result nor whether there is underflow depends on its exact
	73	value, only on its sign. */
	74	if (u.ieee.exponent + v.ieee.exponent
	75	< IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
	76	{
	77	int neg = u.ieee.negative ^ v.ieee.negative;
	78	double tiny = neg ? -0x1p-1074 : 0x1p-1074;
	79	if (w.ieee.exponent >= 3)
	80	return tiny + z;
	81	/* Scaling up, adding TINY and scaling down produces the
	82	correct result, because in round-to-nearest mode adding
	83	TINY has no effect and in other modes double rounding is
	84	harmless. But it may not produce required underflow
	85	exceptions. */
	86	v.d = z * 0x1p54 + tiny;
	87	if (TININESS_AFTER_ROUNDING
	88	? v.ieee.exponent < 55
	89	: (w.ieee.exponent == 0
	90	\|\| (w.ieee.exponent == 1
	91	&& w.ieee.negative != neg
	92	&& w.ieee.mantissa1 == 0
	93	&& w.ieee.mantissa0 == 0)))
	94	{
d96164c3 JM	95	double force_underflow = x * y;
d96164c3 JM	96	math_force_eval (force_underflow);
473611b2 JM	97	}
	98	return v.d * 0x1p-54;
	99	}
5e908464 JJ	100	if (u.ieee.exponent + v.ieee.exponent
	101	>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
	102	{
	103	/* Compute 1p-53 times smaller result and multiply
	104	at the end. */
	105	if (u.ieee.exponent > v.ieee.exponent)
	106	u.ieee.exponent -= DBL_MANT_DIG;
	107	else
	108	v.ieee.exponent -= DBL_MANT_DIG;
	109	/* If x + y exponent is very large and z exponent is very small,
	110	it doesn't matter if we don't adjust it. */
	111	if (w.ieee.exponent > DBL_MANT_DIG)
	112	w.ieee.exponent -= DBL_MANT_DIG;
	113	adjust = 1;
	114	}
	115	else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
	116	{
	117	/* Similarly.
	118	If z exponent is very large and x and y exponents are
82477c28 JM	119	very small, adjust them up to avoid spurious underflows,
	120	rather than down. */
	121	if (u.ieee.exponent + v.ieee.exponent
739babd7	122	<= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG)
82477c28 JM	123	{
	124	if (u.ieee.exponent > v.ieee.exponent)
	125	u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	126	else
	127	v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
	128	}
	129	else if (u.ieee.exponent > v.ieee.exponent)
5e908464 JJ	130	{
	131	if (u.ieee.exponent > DBL_MANT_DIG)
	132	u.ieee.exponent -= DBL_MANT_DIG;
	133	}
	134	else if (v.ieee.exponent > DBL_MANT_DIG)
	135	v.ieee.exponent -= DBL_MANT_DIG;
	136	w.ieee.exponent -= DBL_MANT_DIG;
	137	adjust = 1;
	138	}
	139	else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
	140	{
	141	u.ieee.exponent -= DBL_MANT_DIG;
	142	if (v.ieee.exponent)
	143	v.ieee.exponent += DBL_MANT_DIG;
	144	else
	145	v.d *= 0x1p53;
	146	}
f3f7372d	147	else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
5e908464 JJ	148	{
	149	v.ieee.exponent -= DBL_MANT_DIG;
	150	if (u.ieee.exponent)
	151	u.ieee.exponent += DBL_MANT_DIG;
	152	else
	153	u.d *= 0x1p53;
	154	}
f3f7372d JJ	155	else /* if (u.ieee.exponent + v.ieee.exponent
	156	<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
	157	{
	158	if (u.ieee.exponent > v.ieee.exponent)
82477c28	159	u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
f3f7372d	160	else
82477c28 JM	161	v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
82477c28 JM	162	if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
f3f7372d JJ	163	{
f3f7372d JJ	164	if (w.ieee.exponent)
82477c28	165	w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
f3f7372d	166	else
82477c28	167	w.d *= 0x1p108;
f3f7372d JJ	168	adjust = -1;
	169	}
	170	/* Otherwise x * y should just affect inexact
	171	and nothing else. */
	172	}
5e908464 JJ	173	x = u.d;
	174	y = v.d;
	175	z = w.d;
	176	}
8ec5b013 JM	177
8ec5b013 JM	178	/* Ensure correct sign of exact 0 + 0. */
a1ffb40e	179	if (__glibc_unlikely ((x == 0 \|\| y == 0) && z == 0))
09245377 L	180	{
	181	x = math_opt_barrier (x);
	182	return x * y + z;
	183	}
8ec5b013	184
5b5b04d6 JM	185	fenv_t env;
	186	libc_feholdexcept_setround (&env, FE_TONEAREST);
	187
5e908464 JJ	188	/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
	189	#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
	190	double x1 = x * C;
	191	double y1 = y * C;
	192	double m1 = x * y;
	193	x1 = (x - x1) + x1;
	194	y1 = (y - y1) + y1;
	195	double x2 = x - x1;
	196	double y2 = y - y1;
	197	double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
	198
	199	/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
	200	double a1 = z + m1;
	201	double t1 = a1 - z;
	202	double t2 = a1 - t1;
	203	t1 = m1 - t1;
	204	t2 = z - t2;
	205	double a2 = t1 + t2;
4896f049 RH	206	/* Ensure the arithmetic is not scheduled after feclearexcept call. */
	207	math_force_eval (m2);
	208	math_force_eval (a2);
5b5b04d6	209	feclearexcept (FE_INEXACT);
5e908464	210
4896f049	211	/* If the result is an exact zero, ensure it has the correct sign. */
5b5b04d6 JM	212	if (a1 == 0 && m2 == 0)
	213	{
	214	libc_feupdateenv (&env);
4896f049 RH	215	/* Ensure that round-to-nearest value of z + m1 is not reused. */
4896f049 RH	216	z = math_opt_barrier (z);
5b5b04d6 JM	217	return z + m1;
	218	}
	219
	220	libc_fesetround (FE_TOWARDZERO);
0fe0f1f8	221
5e908464 JJ	222	/* Perform m2 + a2 addition with round to odd. */
5e908464 JJ	223	u.d = a2 + m2;
5e908464	224
a1ffb40e	225	if (__glibc_unlikely (adjust < 0))
0fe0f1f8 RH	226	{
	227	if ((u.ieee.mantissa1 & 1) == 0)
	228	u.ieee.mantissa1 \|= libc_fetestexcept (FE_INEXACT) != 0;
	229	v.d = a1 + u.d;
efb73488 AJ	230	/* Ensure the addition is not scheduled after fetestexcept call. */
efb73488 AJ	231	math_force_eval (v.d);
0fe0f1f8 RH	232	}
	233
	234	/* Reset rounding mode and test for inexact simultaneously. */
	235	int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
	236
a1ffb40e	237	if (__glibc_likely (adjust == 0))
f3f7372d JJ	238	{
f3f7372d JJ	239	if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
0fe0f1f8	240	u.ieee.mantissa1 \|= j;
f3f7372d JJ	241	/* Result is a1 + u.d. */
	242	return a1 + u.d;
	243	}
a1ffb40e	244	else if (__glibc_likely (adjust > 0))
f3f7372d JJ	245	{
f3f7372d JJ	246	if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
0fe0f1f8	247	u.ieee.mantissa1 \|= j;
f3f7372d JJ	248	/* Result is a1 + u.d, scaled up. */
	249	return (a1 + u.d) * 0x1p53;
	250	}
	251	else
	252	{
f3f7372d JJ	253	/* If a1 + u.d is exact, the only rounding happens during
	254	scaling down. */
	255	if (j == 0)
82477c28	256	return v.d * 0x1p-108;
f3f7372d JJ	257	/* If result rounded to zero is not subnormal, no double
f3f7372d JJ	258	rounding will occur. */
82477c28 JM	259	if (v.ieee.exponent > 108)
	260	return (a1 + u.d) * 0x1p-108;
	261	/* If v.d * 0x1p-108 with round to zero is a subnormal above
	262	or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
f3f7372d JJ	263	down just by 1 bit, which means v.ieee.mantissa1 \|= j would
f3f7372d JJ	264	change the round bit, not sticky or guard bit.
82477c28	265	v.d * 0x1p-108 never normalizes by shifting up,
f3f7372d JJ	266	so round bit plus sticky bit should be already enough
f3f7372d JJ	267	for proper rounding. */
82477c28	268	if (v.ieee.exponent == 108)
f3f7372d	269	{
ef82f4da JM	270	/* If the exponent would be in the normal range when
	271	rounding to normal precision with unbounded exponent
	272	range, the exact result is known and spurious underflows
	273	must be avoided on systems detecting tininess after
	274	rounding. */
	275	if (TININESS_AFTER_ROUNDING)
	276	{
	277	w.d = a1 + u.d;
82477c28 JM	278	if (w.ieee.exponent == 109)
82477c28 JM	279	return w.d * 0x1p-108;
ef82f4da	280	}
f3f7372d JJ	281	/* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
f3f7372d JJ	282	v.ieee.mantissa1 & 1 is the round bit and j is our sticky
8627a232 JM	283	bit. */
	284	w.d = 0.0;
	285	w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) \| j;
	286	w.ieee.negative = v.ieee.negative;
	287	v.ieee.mantissa1 &= ~3U;
82477c28	288	v.d *= 0x1p-108;
8627a232 JM	289	w.d *= 0x1p-2;
8627a232 JM	290	return v.d + w.d;
f3f7372d JJ	291	}
f3f7372d JJ	292	v.ieee.mantissa1 \|= j;
82477c28	293	return v.d * 0x1p-108;
f3f7372d	294	}
5e908464 JJ	295	}
5e908464 JJ	296	#ifndef __fma
7e16a5d1	297	libm_alias_double (__fma, fma)
5e908464	298	#endif