gcc/libquadmath/math/fmaq.c

/* Compute x * y + z as ternary operation.
   Copyright (C) 2010-2012 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 "quadmath-imp.h"
#include <math.h>
#include <float.h>
#ifdef HAVE_FENV_H
# include <fenv.h>
# if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETROUND \
     && defined HAVE_FEUPDATEENV && defined HAVE_FETESTEXCEPT \
     && defined FE_TOWARDZERO && defined FE_INEXACT
#  define USE_FENV_H
# endif
#endif

/* 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 fma will certainly overflow,
	 or if x * y is less than half of FLT128_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_FLOAT128_BIAS
	  || u.ieee.exponent + v.ieee.exponent
	     < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2)
	return x * y + z;
      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, it doesn't matter if we don't adjust it.  */
	  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;
	  else
	    v.ieee.exponent += 2 * FLT128_MANT_DIG;
	  if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 4)
	    {
	      if (w.ieee.exponent)
		w.ieee.exponent += 2 * FLT128_MANT_DIG;
	      else
		w.value *= 0x1p226Q;
	      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 (__builtin_expect ((x == 0 || y == 0) && z == 0, 0))
    return x * y + z;

  /* 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;

#ifdef USE_FENV_H
  fenv_t env;
  feholdexcept (&env);
  fesetround (FE_TOWARDZERO);
#endif
  /* Perform m2 + a2 addition with round to odd.  */
  u.value = a2 + m2;

  if (__builtin_expect (adjust == 0, 1))
    {
#ifdef USE_FENV_H
      if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff)
	u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
      feupdateenv (&env);
#endif
      /* Result is a1 + u.value.  */
      return a1 + u.value;
    }
  else if (__builtin_expect (adjust > 0, 1))
    {
#ifdef USE_FENV_H
      if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff)
	u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
      feupdateenv (&env);
#endif
      /* Result is a1 + u.value, scaled up.  */
      return (a1 + u.value) * 0x1p113Q;
    }
  else
    {
#ifdef USE_FENV_H
      if ((u.ieee.mant_low & 1) == 0)
	u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
#endif
      v.value = a1 + u.value;
      /* Ensure the addition is not scheduled after fetestexcept call.  */
      asm volatile ("" : : "m" (v.value));
#ifdef USE_FENV_H
      int j = fetestexcept (FE_INEXACT) != 0;
      feupdateenv (&env);
#else
      int j = 0;
#endif
      /* 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-226Q;
      /* If result rounded to zero is not subnormal, no double
	 rounding will occur.  */
      if (v.ieee.exponent > 226)
	return (a1 + u.value) * 0x1p-226Q;
      /* If v.value * 0x1p-226Q with round to zero is a subnormal above
	 or equal to FLT128_MIN / 2, then v.value * 0x1p-226Q shifts mantissa
	 down just by 1 bit, which means v.ieee.mant_low |= j would
	 change the round bit, not sticky or guard bit.
	 v.value * 0x1p-226Q never normalizes by shifting up,
	 so round bit plus sticky bit should be already enough
	 for proper rounding.  */
      if (v.ieee.exponent == 226)
	{
         /* 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 == 227)
               return w.value * 0x1p-226L;
           }
	  /* v.ieee.mant_low & 2 is LSB bit of the result before rounding,
	     v.ieee.mant_low & 1 is the round bit and j is our sticky
	     bit. */
	  w.value = 0.0Q;
	  w.ieee.mant_low = ((v.ieee.mant_low & 3) << 1) | j;
	  w.ieee.negative = v.ieee.negative;
	  v.ieee.mant_low &= ~3U;
	  v.value *= 0x1p-226L;
	  w.value *= 0x1p-2L;
	  return v.value + w.value;
	}
      v.ieee.mant_low |= j;
      return v.value * 0x1p-226Q;
    }
}