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/* Compute x^2 + y^2 - 1, without large cancellation error.
Copyright (C) 2012-2013 Free Software Foundation, Inc.
This file is part of the GNU C Library.
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 "quadmath-imp.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 inline void
add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
{
/* Apply Dekker's algorithm. */
*hi = x + y;
*lo = (x - *hi) + y;
}
/* Calculate X * Y exactly and store the result in *HI + *LO. It is
given that the values are small enough that no overflow occurs and
large enough (or zero) that no underflow occurs. */
static inline void
mul_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
{
/* Fast built-in fused multiply-add. */
*hi = x * y;
*lo = fmaq (x, y, -*hi);
}
/* Compare absolute values of floating-point values pointed to by P
and Q for qsort. */
static int
compare (const void *p, const void *q)
{
__float128 pld = fabsq (*(const __float128 *) p);
__float128 qld = fabsq (*(const __float128 *) q);
if (pld < qld)
return -1;
else if (pld == qld)
return 0;
else
return 1;
}
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
0.75 or Y >= 0.5. */
__float128
__quadmath_x2y2m1q (__float128 x, __float128 y)
{
__float128 vals[4];
size_t i;
/* FIXME: SET_RESTORE_ROUNDL (FE_TONEAREST); */
mul_split (&vals[1], &vals[0], x, x);
mul_split (&vals[3], &vals[2], y, y);
if (x >= 0.75Q)
vals[1] -= 1.0Q;
else
{
vals[1] -= 0.5Q;
vals[3] -= 0.5Q;
}
qsort (vals, 4, sizeof (__float128), 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 (i = 0; i <= 2; i++)
{
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
qsort (vals + i + 1, 3 - i, sizeof (__float128), compare);
}
/* Now any error from this addition will be small. */
return vals[3] + vals[2] + vals[1] + vals[0];
}
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