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/* Compute x * y + z as ternary operation.
Copyright (C) 2011-2020 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by David Flaherty <flaherty@linux.vnet.ibm.com>.
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 <fenv.h>
#include <float.h>
#include <math.h>
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <math_ldbl_opt.h>
#include <mul_split.h>
#include <stdlib.h>
/* Calculate X + Y exactly and store the result in *HI + *LO. It is
given that |X| >= |Y| and the values are small enough that no
overflow occurs. */
static void
add_split (double *hi, double *lo, double x, double y)
{
/* Apply Dekker's algorithm. */
*hi = x + y;
*lo = (x - *hi) + y;
}
/* Value with extended range, used in intermediate computations. */
typedef struct
{
/* Value in [0.5, 1), as from frexp, or 0. */
double val;
/* Exponent of power of 2 it is multiplied by, or 0 for zero. */
int exp;
} ext_val;
/* Store D as an ext_val value. */
static void
store_ext_val (ext_val *v, double d)
{
v->val = __frexp (d, &v->exp);
}
/* Store X * Y as ext_val values *V0 and *V1. */
static void
mul_ext_val (ext_val *v0, ext_val *v1, double x, double y)
{
int xexp, yexp;
x = __frexp (x, &xexp);
y = __frexp (y, &yexp);
double hi, lo;
mul_split (&hi, &lo, x, y);
store_ext_val (v0, hi);
if (hi != 0)
v0->exp += xexp + yexp;
store_ext_val (v1, lo);
if (lo != 0)
v1->exp += xexp + yexp;
}
/* Compare absolute values of ext_val values pointed to by P and Q for
qsort. */
static int
compare (const void *p, const void *q)
{
const ext_val *pe = p;
const ext_val *qe = q;
if (pe->val == 0)
return qe->val == 0 ? 0 : -1;
else if (qe->val == 0)
return 1;
else if (pe->exp < qe->exp)
return -1;
else if (pe->exp > qe->exp)
return 1;
else
{
double pd = fabs (pe->val);
double qd = fabs (qe->val);
if (pd < qd)
return -1;
else if (pd == qd)
return 0;
else
return 1;
}
}
/* Calculate *X + *Y exactly, storing the high part in *X (rounded to
nearest) and the low part in *Y. It is given that |X| >= |Y|. */
static void
add_split_ext (ext_val *x, ext_val *y)
{
int xexp = x->exp, yexp = y->exp;
if (y->val == 0 || xexp - yexp > 53)
return;
double hi = x->val;
double lo = __scalbn (y->val, yexp - xexp);
add_split (&hi, &lo, hi, lo);
store_ext_val (x, hi);
if (hi != 0)
x->exp += xexp;
store_ext_val (y, lo);
if (lo != 0)
y->exp += xexp;
}
long double
__fmal (long double x, long double y, long double z)
{
double xhi, xlo, yhi, ylo, zhi, zlo;
int64_t hx, hy, hz;
int xexp, yexp, zexp;
double scale_val;
int scale_exp;
ldbl_unpack (x, &xhi, &xlo);
EXTRACT_WORDS64 (hx, xhi);
xexp = (hx & 0x7ff0000000000000LL) >> 52;
ldbl_unpack (y, &yhi, &ylo);
EXTRACT_WORDS64 (hy, yhi);
yexp = (hy & 0x7ff0000000000000LL) >> 52;
ldbl_unpack (z, &zhi, &zlo);
EXTRACT_WORDS64 (hz, zhi);
zexp = (hz & 0x7ff0000000000000LL) >> 52;
/* If z is Inf or NaN, but x and y are finite, avoid any exceptions
from computing x * y. */
if (zexp == 0x7ff && xexp != 0x7ff && yexp != 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 x * y is zero, compute as x * y
+ z. */
if (xexp == 0x7ff || yexp == 0x7ff || zexp == 0x7ff
|| x == 0 || y == 0)
return (x * y) + z;
{
SET_RESTORE_ROUND (FE_TONEAREST);
ext_val vals[10];
store_ext_val (&vals[0], zhi);
store_ext_val (&vals[1], zlo);
mul_ext_val (&vals[2], &vals[3], xhi, yhi);
mul_ext_val (&vals[4], &vals[5], xhi, ylo);
mul_ext_val (&vals[6], &vals[7], xlo, yhi);
mul_ext_val (&vals[8], &vals[9], xlo, ylo);
qsort (vals, 10, sizeof (ext_val), compare);
/* Add up the values so that each element of VALS has absolute
value at most equal to the last set bit of the next nonzero
element. */
for (size_t i = 0; i <= 8; i++)
{
add_split_ext (&vals[i + 1], &vals[i]);
qsort (vals + i + 1, 9 - i, sizeof (ext_val), compare);
}
/* Add up the values in the other direction, so that each element
of VALS has absolute value less than 5ulp of the next
value. */
size_t dstpos = 9;
for (size_t i = 1; i <= 9; i++)
{
if (vals[dstpos].val == 0)
{
vals[dstpos] = vals[9 - i];
vals[9 - i].val = 0;
vals[9 - i].exp = 0;
}
else
{
add_split_ext (&vals[dstpos], &vals[9 - i]);
if (vals[9 - i].val != 0)
{
if (9 - i < dstpos - 1)
{
vals[dstpos - 1] = vals[9 - i];
vals[9 - i].val = 0;
vals[9 - i].exp = 0;
}
dstpos--;
}
}
}
/* If the result is an exact zero, it results from adding two
values with opposite signs; recompute in the original rounding
mode. */
if (vals[9].val == 0)
goto zero_out;
/* Adding the top three values will now give a result as accurate
as the underlying long double arithmetic. */
add_split_ext (&vals[9], &vals[8]);
if (compare (&vals[8], &vals[7]) < 0)
{
ext_val tmp = vals[7];
vals[7] = vals[8];
vals[8] = tmp;
}
add_split_ext (&vals[8], &vals[7]);
add_split_ext (&vals[9], &vals[8]);
if (vals[9].exp > DBL_MAX_EXP || vals[9].exp < DBL_MIN_EXP)
{
/* Overflow or underflow, with the result depending on the
original rounding mode, but not on the low part computed
here. */
scale_val = vals[9].val;
scale_exp = vals[9].exp;
goto scale_out;
}
double hi = __scalbn (vals[9].val, vals[9].exp);
double lo = __scalbn (vals[8].val, vals[8].exp);
/* It is possible that the low part became subnormal and was
rounded so that the result is no longer canonical. */
ldbl_canonicalize (&hi, &lo);
long double ret = ldbl_pack (hi, lo);
math_check_force_underflow (ret);
return ret;
}
scale_out:
scale_val = math_opt_barrier (scale_val);
scale_val = __scalbn (scale_val, scale_exp);
if (fabs (scale_val) == DBL_MAX)
return copysignl (LDBL_MAX, scale_val);
math_check_force_underflow (scale_val);
return scale_val;
zero_out:;
double zero = 0.0;
zero = math_opt_barrier (zero);
return zero - zero;
}
#if IS_IN (libm)
long_double_symbol (libm, __fmal, fmal);
#else
long_double_symbol (libc, __fmal, fmal);
#endif
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