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/* Compute x * y + z as ternary operation.
Copyright (C) 2010-2020 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 <libm-alias-ldouble.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 */
long double
__fmal (long double x, long double y, long double z)
{
union ieee854_long_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
>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
- LDBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
|| __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
|| __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent + v.ieee.exponent
<= IEEE854_LONG_DOUBLE_BIAS + LDBL_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_LONG_DOUBLE_BIAS)
return x * y;
/* If x * y is less than 1/4 of LDBL_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_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
{
int neg = u.ieee.negative ^ v.ieee.negative;
long double tiny = neg ? -0x1p-16445L : 0x1p-16445L;
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 * 0x1p65L + tiny;
if (TININESS_AFTER_ROUNDING
? v.ieee.exponent < 66
: (w.ieee.exponent == 0
|| (w.ieee.exponent == 1
&& w.ieee.negative != neg
&& w.ieee.mantissa1 == 0
&& w.ieee.mantissa0 == 0x80000000)))
{
long double force_underflow = x * y;
math_force_eval (force_underflow);
}
return v.d * 0x1p-65L;
}
if (u.ieee.exponent + v.ieee.exponent
>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
{
/* Compute 1p-64 times smaller result and multiply
at the end. */
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent -= LDBL_MANT_DIG;
else
v.ieee.exponent -= LDBL_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 > LDBL_MANT_DIG)
w.ieee.exponent -= LDBL_MANT_DIG;
adjust = 1;
}
else if (w.ieee.exponent >= 0x7fff - LDBL_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_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
else
v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
}
else if (u.ieee.exponent > v.ieee.exponent)
{
if (u.ieee.exponent > LDBL_MANT_DIG)
u.ieee.exponent -= LDBL_MANT_DIG;
}
else if (v.ieee.exponent > LDBL_MANT_DIG)
v.ieee.exponent -= LDBL_MANT_DIG;
w.ieee.exponent -= LDBL_MANT_DIG;
adjust = 1;
}
else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
{
u.ieee.exponent -= LDBL_MANT_DIG;
if (v.ieee.exponent)
v.ieee.exponent += LDBL_MANT_DIG;
else
v.d *= 0x1p64L;
}
else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
{
v.ieee.exponent -= LDBL_MANT_DIG;
if (u.ieee.exponent)
u.ieee.exponent += LDBL_MANT_DIG;
else
u.d *= 0x1p64L;
}
else /* if (u.ieee.exponent + v.ieee.exponent
<= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
else
v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
{
if (w.ieee.exponent)
w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
else
w.d *= 0x1p130L;
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;
feholdexcept (&env);
fesetround (FE_TONEAREST);
/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
long double x1 = x * C;
long double y1 = y * C;
long double m1 = x * y;
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
long double x2 = x - x1;
long double y2 = y - y1;
long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
long double a1 = z + m1;
long double t1 = a1 - z;
long double t2 = a1 - t1;
t1 = m1 - t1;
t2 = z - t2;
long 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)
{
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.d = a2 + m2;
if (__glibc_likely (adjust == 0))
{
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
feupdateenv (&env);
/* Result is a1 + u.d. */
return a1 + u.d;
}
else if (__glibc_likely (adjust > 0))
{
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
feupdateenv (&env);
/* Result is a1 + u.d, scaled up. */
return (a1 + u.d) * 0x1p64L;
}
else
{
if ((u.ieee.mantissa1 & 1) == 0)
u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
v.d = a1 + u.d;
/* Ensure the addition is not scheduled after fetestexcept call. */
math_force_eval (v.d);
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.d is exact, the only rounding happens during
scaling down. */
if (j == 0)
return v.d * 0x1p-130L;
/* If result rounded to zero is not subnormal, no double
rounding will occur. */
if (v.ieee.exponent > 130)
return (a1 + u.d) * 0x1p-130L;
/* If v.d * 0x1p-130L with round to zero is a subnormal above
or equal to LDBL_MIN / 2, then v.d * 0x1p-130L 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-130L never normalizes by shifting up,
so round bit plus sticky bit should be already enough
for proper rounding. */
if (v.ieee.exponent == 130)
{
/* 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 == 131)
return w.d * 0x1p-130L;
}
/* 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 = 0.0L;
w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
w.ieee.negative = v.ieee.negative;
v.ieee.mantissa1 &= ~3U;
v.d *= 0x1p-130L;
w.d *= 0x1p-2L;
return v.d + w.d;
}
v.ieee.mantissa1 |= j;
return v.d * 0x1p-130L;
}
}
libm_alias_ldouble (__fma, fma)
|
