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/* Compute x^2 + y^2 - 1, without large cancellation error.
Copyright (C) 2012 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 <math.h>
#include <math_private.h>
#include <float.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 (double *hi, double *lo, double x, double 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 (double *hi, double *lo, double x, double y)
{
#ifdef __FP_FAST_FMA
/* Fast built-in fused multiply-add. */
*hi = x * y;
*lo = __builtin_fma (x, y, -*hi);
#elif defined FP_FAST_FMA
/* Fast library fused multiply-add, compiler before GCC 4.6. */
*hi = x * y;
*lo = __fma (x, y, -*hi);
#else
/* Apply Dekker's algorithm. */
*hi = x * y;
# define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
double x1 = x * C;
double y1 = y * C;
# undef C
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
double x2 = x - x1;
double y2 = y - y1;
*lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
#endif
}
/* Compare absolute values of floating-point values pointed to by P
and Q for qsort. */
static int
compare (const void *p, const void *q)
{
double pd = fabs (*(const double *) p);
double qd = fabs (*(const double *) q);
if (pd < qd)
return -1;
else if (pd == qd)
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. */
long double
__x2y2m1l (long double x, long double y)
{
double vals[12];
SET_RESTORE_ROUND (FE_TONEAREST);
union ibm_extended_long_double xu, yu;
xu.d = x;
yu.d = y;
if (fabs (xu.dd[1]) < 0x1p-500)
xu.dd[1] = 0.0;
if (fabs (yu.dd[1]) < 0x1p-500)
yu.dd[1] = 0.0;
mul_split (&vals[1], &vals[0], xu.dd[0], xu.dd[0]);
mul_split (&vals[3], &vals[2], xu.dd[0], xu.dd[1]);
vals[2] *= 2.0;
vals[3] *= 2.0;
mul_split (&vals[5], &vals[4], xu.dd[1], xu.dd[1]);
mul_split (&vals[7], &vals[6], yu.dd[0], yu.dd[0]);
mul_split (&vals[9], &vals[8], yu.dd[0], yu.dd[1]);
vals[8] *= 2.0;
vals[9] *= 2.0;
mul_split (&vals[11], &vals[10], yu.dd[1], yu.dd[1]);
if (xu.dd[0] >= 0.75)
vals[1] -= 1.0;
else
{
vals[1] -= 0.5;
vals[7] -= 0.5;
}
qsort (vals, 12, sizeof (double), 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 <= 10; i++)
{
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
qsort (vals + i + 1, 11 - i, sizeof (double), compare);
}
/* Now any error from this addition will be small. */
long double retval = (long double) vals[11];
for (size_t i = 10; i != (size_t) -1; i--)
retval += (long double) vals[i];
return retval;
}
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