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/* Compute x^2 + y^2 - 1, without large cancellation error.
Copyright (C) 2012-2024 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
<https://www.gnu.org/licenses/>. */
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
#include <mul_splitl.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;
}
/* 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 = fabsl (*(const _Float128 *) p);
_Float128 qld = fabsl (*(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 X^2 + Y^2 >=
0.5. */
_Float128
__x2y2m1l (_Float128 x, _Float128 y)
{
_Float128 vals[5];
SET_RESTORE_ROUNDL (FE_TONEAREST);
mul_splitl (&vals[1], &vals[0], x, x);
mul_splitl (&vals[3], &vals[2], y, y);
vals[4] = -1;
qsort (vals, 5, 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 (size_t i = 0; i <= 3; i++)
{
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
qsort (vals + i + 1, 4 - i, sizeof (_Float128), compare);
}
/* Now any error from this addition will be small. */
return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
}
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