libquadmath/math/fmaq.c - rust-lang/gcc - Git at Google

 /* Compute x * y + z as ternary operation.
    Copyright (C) 2010-2018 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
    <http://www.gnu.org/licenses/>.  */

 #include "quadmath-imp.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  */

 __float128
 fmaq (__float128 x, __float128 y, __float128 z)
 {
   ieee854_float128 u, v, w;
   int adjust = 0;
   u.value = x;
   v.value = y;
   w.value = z;
   if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
 			>= 0x7fff + IEEE854_FLOAT128_BIAS
 			   - FLT128_MANT_DIG, 0)
       || __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
       || __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
       || __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
       || __builtin_expect (u.ieee.exponent + v.ieee.exponent
 			   <= IEEE854_FLOAT128_BIAS + FLT128_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 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 == 0x7fff
 	  || v.ieee.exponent == 0x7fff
 	  || w.ieee.exponent == 0x7fff
 	  || x == 0
 	  || y == 0)
 	return x * y + z;
       /* If fma will certainly overflow, compute as x * y.  */
       if (u.ieee.exponent + v.ieee.exponent
 	  > 0x7fff + IEEE854_FLOAT128_BIAS)
 	return x * y;
       /* If x * y is less than 1/4 of FLT128_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
 	  < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2)
 	{
 	  int neg = u.ieee.negative ^ v.ieee.negative;
 	  __float128 tiny = neg ? -0x1p-16494Q : 0x1p-16494Q;
 	  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.value = z * 0x1p114Q + tiny;
 	  if (TININESS_AFTER_ROUNDING
 	      ? v.ieee.exponent < 115
 	      : (w.ieee.exponent == 0
 		 || (w.ieee.exponent == 1
 		     && w.ieee.negative != neg
 		     && w.ieee.mantissa3 == 0
 		     && w.ieee.mantissa2 == 0
 		     && w.ieee.mantissa1 == 0
 		     && w.ieee.mantissa0 == 0)))
 	    {
 	      __float128 force_underflow = x * y;
 	      math_force_eval (force_underflow);
 	    }
 	  return v.value * 0x1p-114Q;
 	}
       if (u.ieee.exponent + v.ieee.exponent
 	  >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG)
 	{
 	  /* Compute 1p-113 times smaller result and multiply
 	     at the end.  */
 	  if (u.ieee.exponent > v.ieee.exponent)
 	    u.ieee.exponent -= FLT128_MANT_DIG;
 	  else
 	    v.ieee.exponent -= FLT128_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 > FLT128_MANT_DIG)
 	    w.ieee.exponent -= FLT128_MANT_DIG;
 	  adjust = 1;
 	}
       else if (w.ieee.exponent >= 0x7fff - FLT128_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
 	      <= IEEE854_FLOAT128_BIAS + 2 * FLT128_MANT_DIG)
 	    {
 	      if (u.ieee.exponent > v.ieee.exponent)
 		u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
 	      else
 		v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
 	    }
 	  else if (u.ieee.exponent > v.ieee.exponent)
 	    {
 	      if (u.ieee.exponent > FLT128_MANT_DIG)
 		u.ieee.exponent -= FLT128_MANT_DIG;
 	    }
 	  else if (v.ieee.exponent > FLT128_MANT_DIG)
 	    v.ieee.exponent -= FLT128_MANT_DIG;
 	  w.ieee.exponent -= FLT128_MANT_DIG;
 	  adjust = 1;
 	}
       else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
 	{
 	  u.ieee.exponent -= FLT128_MANT_DIG;
 	  if (v.ieee.exponent)
 	    v.ieee.exponent += FLT128_MANT_DIG;
 	  else
 	    v.value *= 0x1p113Q;
 	}
       else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
 	{
 	  v.ieee.exponent -= FLT128_MANT_DIG;
 	  if (u.ieee.exponent)
 	    u.ieee.exponent += FLT128_MANT_DIG;
 	  else
 	    u.value *= 0x1p113Q;
 	}
       else /* if (u.ieee.exponent + v.ieee.exponent
 		  <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */
 	{
 	  if (u.ieee.exponent > v.ieee.exponent)
 	    u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
 	  else
 	    v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
 	  if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 6)
 	    {
 	      if (w.ieee.exponent)
 		w.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
 	      else
 		w.value *= 0x1p228Q;
 	      adjust = -1;
 	    }
 	  /* Otherwise x * y should just affect inexact
 	     and nothing else.  */
 	}
       x = u.value;
       y = v.value;
       z = w.value;
     }

   /* 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;
   feholdexcept (&env);
   fesetround (FE_TONEAREST);

   /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
 #define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1)
   __float128 x1 = x * C;
   __float128 y1 = y * C;
   __float128 m1 = x * y;
   x1 = (x - x1) + x1;
   y1 = (y - y1) + y1;
   __float128 x2 = x - x1;
   __float128 y2 = y - y1;
   __float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;

   /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
   __float128 a1 = z + m1;
   __float128 t1 = a1 - z;
   __float128 t2 = a1 - t1;
   t1 = m1 - t1;
   t2 = z - t2;
   __float128 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)
     {
       feupdateenv (&env);
       /* Ensure that round-to-nearest value of z + m1 is not reused.  */
       z = math_opt_barrier (z);
       return z + m1;
     }

   fesetround (FE_TOWARDZERO);
   /* Perform m2 + a2 addition with round to odd.  */
   u.value = a2 + m2;

   if (__glibc_likely (adjust == 0))
     {
       if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
 	u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
       feupdateenv (&env);
       /* Result is a1 + u.value.  */
       return a1 + u.value;
     }
   else if (__glibc_likely (adjust > 0))
     {
       if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
 	u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
       feupdateenv (&env);
       /* Result is a1 + u.value, scaled up.  */
       return (a1 + u.value) * 0x1p113Q;
     }
   else
     {
       if ((u.ieee.mantissa3 & 1) == 0)
 	u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
       v.value = a1 + u.value;
       /* Ensure the addition is not scheduled after fetestexcept call.  */
       math_force_eval (v.value);
       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.value is exact, the only rounding happens during
 	 scaling down.  */
       if (j == 0)
 	return v.value * 0x1p-228Q;
       /* If result rounded to zero is not subnormal, no double
 	 rounding will occur.  */
       if (v.ieee.exponent > 228)
 	return (a1 + u.value) * 0x1p-228Q;
       /* If v.value * 0x1p-228L with round to zero is a subnormal above
 	 or equal to FLT128_MIN / 2, then v.value * 0x1p-228L shifts mantissa
 	 down just by 1 bit, which means v.ieee.mantissa3 |= j would
 	 change the round bit, not sticky or guard bit.
 	 v.value * 0x1p-228L never normalizes by shifting up,
 	 so round bit plus sticky bit should be already enough
 	 for proper rounding.  */
       if (v.ieee.exponent == 228)
 	{
 	  /* 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.value = a1 + u.value;
 	      if (w.ieee.exponent == 229)
 		return w.value * 0x1p-228Q;
 	    }
 	  /* 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.  */
 	  w.value = 0;
 	  w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
 	  w.ieee.negative = v.ieee.negative;
 	  v.ieee.mantissa3 &= ~3U;
 	  v.value *= 0x1p-228Q;
 	  w.value *= 0x1p-2Q;
 	  return v.value + w.value;
 	}
       v.ieee.mantissa3 |= j;
       return v.value * 0x1p-228Q;
     }
 }
	/* Compute x * y + z as ternary operation.
	Copyright (C) 2010-2018 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
	<http://www.gnu.org/licenses/>. */

	#include "quadmath-imp.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 */

	__float128
	fmaq (__float128 x, __float128 y, __float128 z)
	{
	ieee854_float128 u, v, w;
	int adjust = 0;
	u.value = x;
	v.value = y;
	w.value = z;
	if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
	>= 0x7fff + IEEE854_FLOAT128_BIAS
	- FLT128_MANT_DIG, 0)
	\|\| __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
	\|\| __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
	\|\| __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
	\|\| __builtin_expect (u.ieee.exponent + v.ieee.exponent
	<= IEEE854_FLOAT128_BIAS + FLT128_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 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 == 0x7fff
	\|\| v.ieee.exponent == 0x7fff
	\|\| w.ieee.exponent == 0x7fff
	\|\| x == 0
	\|\| y == 0)
	return x * y + z;
	/* If fma will certainly overflow, compute as x * y. */
	if (u.ieee.exponent + v.ieee.exponent
	> 0x7fff + IEEE854_FLOAT128_BIAS)
	return x * y;
	/* If x * y is less than 1/4 of FLT128_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
	< IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2)
	{
	int neg = u.ieee.negative ^ v.ieee.negative;
	__float128 tiny = neg ? -0x1p-16494Q : 0x1p-16494Q;
	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.value = z * 0x1p114Q + tiny;
	if (TININESS_AFTER_ROUNDING
	? v.ieee.exponent < 115
	: (w.ieee.exponent == 0
	\|\| (w.ieee.exponent == 1
	&& w.ieee.negative != neg
	&& w.ieee.mantissa3 == 0
	&& w.ieee.mantissa2 == 0
	&& w.ieee.mantissa1 == 0
	&& w.ieee.mantissa0 == 0)))
	{
	__float128 force_underflow = x * y;
	math_force_eval (force_underflow);
	}
	return v.value * 0x1p-114Q;
	}
	if (u.ieee.exponent + v.ieee.exponent
	>= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG)
	{
	/* Compute 1p-113 times smaller result and multiply
	at the end. */
	if (u.ieee.exponent > v.ieee.exponent)
	u.ieee.exponent -= FLT128_MANT_DIG;
	else
	v.ieee.exponent -= FLT128_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 > FLT128_MANT_DIG)
	w.ieee.exponent -= FLT128_MANT_DIG;
	adjust = 1;
	}
	else if (w.ieee.exponent >= 0x7fff - FLT128_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
	<= IEEE854_FLOAT128_BIAS + 2 * FLT128_MANT_DIG)
	{
	if (u.ieee.exponent > v.ieee.exponent)
	u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
	else
	v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
	}
	else if (u.ieee.exponent > v.ieee.exponent)
	{
	if (u.ieee.exponent > FLT128_MANT_DIG)
	u.ieee.exponent -= FLT128_MANT_DIG;
	}
	else if (v.ieee.exponent > FLT128_MANT_DIG)
	v.ieee.exponent -= FLT128_MANT_DIG;
	w.ieee.exponent -= FLT128_MANT_DIG;
	adjust = 1;
	}
	else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
	{
	u.ieee.exponent -= FLT128_MANT_DIG;
	if (v.ieee.exponent)
	v.ieee.exponent += FLT128_MANT_DIG;
	else
	v.value *= 0x1p113Q;
	}
	else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
	{
	v.ieee.exponent -= FLT128_MANT_DIG;
	if (u.ieee.exponent)
	u.ieee.exponent += FLT128_MANT_DIG;
	else
	u.value *= 0x1p113Q;
	}
	else /* if (u.ieee.exponent + v.ieee.exponent
	<= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */
	{
	if (u.ieee.exponent > v.ieee.exponent)
	u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
	else
	v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
	if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 6)
	{
	if (w.ieee.exponent)
	w.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
	else
	w.value *= 0x1p228Q;
	adjust = -1;
	}
	/* Otherwise x * y should just affect inexact
	and nothing else. */
	}
	x = u.value;
	y = v.value;
	z = w.value;
	}

	/* 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;
	feholdexcept (&env);
	fesetround (FE_TONEAREST);

	/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
	#define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1)
	__float128 x1 = x * C;
	__float128 y1 = y * C;
	__float128 m1 = x * y;
	x1 = (x - x1) + x1;
	y1 = (y - y1) + y1;
	__float128 x2 = x - x1;
	__float128 y2 = y - y1;
	__float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;

	/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
	__float128 a1 = z + m1;
	__float128 t1 = a1 - z;
	__float128 t2 = a1 - t1;
	t1 = m1 - t1;
	t2 = z - t2;
	__float128 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)
	{
	feupdateenv (&env);
	/* Ensure that round-to-nearest value of z + m1 is not reused. */
	z = math_opt_barrier (z);
	return z + m1;
	}

	fesetround (FE_TOWARDZERO);
	/* Perform m2 + a2 addition with round to odd. */
	u.value = a2 + m2;

	if (__glibc_likely (adjust == 0))
	{
	if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
	u.ieee.mantissa3 \|= fetestexcept (FE_INEXACT) != 0;
	feupdateenv (&env);
	/* Result is a1 + u.value. */
	return a1 + u.value;
	}
	else if (__glibc_likely (adjust > 0))
	{
	if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
	u.ieee.mantissa3 \|= fetestexcept (FE_INEXACT) != 0;
	feupdateenv (&env);
	/* Result is a1 + u.value, scaled up. */
	return (a1 + u.value) * 0x1p113Q;
	}
	else
	{
	if ((u.ieee.mantissa3 & 1) == 0)
	u.ieee.mantissa3 \|= fetestexcept (FE_INEXACT) != 0;
	v.value = a1 + u.value;
	/* Ensure the addition is not scheduled after fetestexcept call. */
	math_force_eval (v.value);
	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.value is exact, the only rounding happens during
	scaling down. */
	if (j == 0)
	return v.value * 0x1p-228Q;
	/* If result rounded to zero is not subnormal, no double
	rounding will occur. */
	if (v.ieee.exponent > 228)
	return (a1 + u.value) * 0x1p-228Q;
	/* If v.value * 0x1p-228L with round to zero is a subnormal above
	or equal to FLT128_MIN / 2, then v.value * 0x1p-228L shifts mantissa
	down just by 1 bit, which means v.ieee.mantissa3 \|= j would
	change the round bit, not sticky or guard bit.
	v.value * 0x1p-228L never normalizes by shifting up,
	so round bit plus sticky bit should be already enough
	for proper rounding. */
	if (v.ieee.exponent == 228)
	{
	/* 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.value = a1 + u.value;
	if (w.ieee.exponent == 229)
	return w.value * 0x1p-228Q;
	}
	/* 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. */
	w.value = 0;
	w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) \| j;
	w.ieee.negative = v.ieee.negative;
	v.ieee.mantissa3 &= ~3U;
	v.value *= 0x1p-228Q;
	w.value *= 0x1p-2Q;
	return v.value + w.value;
	}
	v.ieee.mantissa3 \|= j;
	return v.value * 0x1p-228Q;
	}
	}