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/* Compute x * y + z as ternary operation.
Copyright (C) 2010, 2011 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 <float.h>
#include <math.h>
#include <fenv.h>
#include <ieee754.h>
#include <math_private.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 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. */
if (u.ieee.exponent == 0x7ff
|| v.ieee.exponent == 0x7ff
|| w.ieee.exponent == 0x7ff
|| u.ieee.exponent + v.ieee.exponent
> 0x7ff + IEEE754_DOUBLE_BIAS
|| u.ieee.exponent + v.ieee.exponent
< IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
return x * y + z;
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, it doesn't matter if we don't adjust it. */
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;
else
v.ieee.exponent += 2 * DBL_MANT_DIG;
if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 4)
{
if (w.ieee.exponent)
w.ieee.exponent += 2 * DBL_MANT_DIG;
else
w.d *= 0x1p106;
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 ((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;
fenv_t env;
libc_feholdexcept_setround (&env, 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 != 0x7ff)
u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
libc_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 != 0x7ff)
u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
libc_feupdateenv (&env);
/* Result is a1 + u.d, scaled up. */
return (a1 + u.d) * 0x1p53;
}
else
{
if ((u.ieee.mantissa1 & 1) == 0)
u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
v.d = a1 + u.d;
int j = libc_fetestexcept (FE_INEXACT) != 0;
libc_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-106;
/* If result rounded to zero is not subnormal, no double
rounding will occur. */
if (v.ieee.exponent > 106)
return (a1 + u.d) * 0x1p-106;
/* If v.d * 0x1p-106 with round to zero is a subnormal above
or equal to DBL_MIN / 2, then v.d * 0x1p-106 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-106 never normalizes by shifting up,
so round bit plus sticky bit should be already enough
for proper rounding. */
if (v.ieee.exponent == 106)
{
/* 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-106;
if (v.ieee.negative)
return v.d - 0x1p-1074 /* __DBL_DENORM_MIN__ */;
else
return v.d + 0x1p-1074 /* __DBL_DENORM_MIN__ */;
}
else
return v.d * 0x1p-106;
}
v.ieee.mantissa1 |= j;
return v.d * 0x1p-106;
}
}
#ifndef __fma
weak_alias (__fma, fma)
#endif
#ifdef NO_LONG_DOUBLE
strong_alias (__fma, __fmal)
weak_alias (__fmal, fmal)
#endif
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