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106 lines
3.1 KiB
C
106 lines
3.1 KiB
C
/* Compute x^2 + y^2 - 1, without large cancellation error.
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Copyright (C) 2012-2016 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<http://www.gnu.org/licenses/>. */
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#include <math.h>
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#include <math_private.h>
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#include <float.h>
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#include <stdlib.h>
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/* Calculate X + Y exactly and store the result in *HI + *LO. It is
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given that |X| >= |Y| and the values are small enough that no
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overflow occurs. */
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static inline void
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add_split (double *hi, double *lo, double x, double y)
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{
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/* Apply Dekker's algorithm. */
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*hi = x + y;
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*lo = (x - *hi) + y;
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}
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/* Calculate X * Y exactly and store the result in *HI + *LO. It is
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given that the values are small enough that no overflow occurs and
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large enough (or zero) that no underflow occurs. */
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static void
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mul_split (double *hi, double *lo, double x, double y)
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{
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#ifdef __FP_FAST_FMA
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/* Fast built-in fused multiply-add. */
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*hi = x * y;
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*lo = __builtin_fma (x, y, -*hi);
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#elif defined FP_FAST_FMA
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/* Fast library fused multiply-add, compiler before GCC 4.6. */
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*hi = x * y;
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*lo = __fma (x, y, -*hi);
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#else
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/* Apply Dekker's algorithm. */
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*hi = x * y;
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# define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
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double x1 = x * C;
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double y1 = y * C;
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# undef C
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x1 = (x - x1) + x1;
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y1 = (y - y1) + y1;
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double x2 = x - x1;
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double y2 = y - y1;
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*lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
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#endif
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}
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/* Compare absolute values of floating-point values pointed to by P
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and Q for qsort. */
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static int
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compare (const void *p, const void *q)
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{
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double pd = fabs (*(const double *) p);
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double qd = fabs (*(const double *) q);
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if (pd < qd)
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return -1;
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else if (pd == qd)
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return 0;
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else
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return 1;
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}
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/* Return X^2 + Y^2 - 1, computed without large cancellation error.
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It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
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0.5. */
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double
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__x2y2m1 (double x, double y)
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{
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double vals[5];
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SET_RESTORE_ROUND (FE_TONEAREST);
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mul_split (&vals[1], &vals[0], x, x);
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mul_split (&vals[3], &vals[2], y, y);
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vals[4] = -1.0;
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qsort (vals, 5, sizeof (double), compare);
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/* Add up the values so that each element of VALS has absolute value
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at most equal to the last set bit of the next nonzero
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element. */
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for (size_t i = 0; i <= 3; i++)
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{
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add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
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qsort (vals + i + 1, 4 - i, sizeof (double), compare);
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}
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/* Now any error from this addition will be small. */
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return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
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}
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