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/* 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, see
<http://www.gnu.org/licenses/>. */
#include <float.h>
#include <math.h>
#include <fenv.h>
#include <ieee754.h>
#include <math_private.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 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;
}
/* 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 ((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, 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 (__builtin_expect (adjust == 0, 1))
{
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 (__builtin_expect (adjust > 0, 1))
{
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-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)
{
/* 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 == 107)
return w.d * 0x1p-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. */
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-106;
w.d *= 0x1p-2;
return v.d + w.d;
}
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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