1 /* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012-2014 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <http://www.gnu.org/licenses/>. */
20 #include <math_private.h>
24 /* Calculate X + Y exactly and store the result in *HI + *LO. It is
25 given that |X| >= |Y| and the values are small enough that no
29 add_split (double *hi, double *lo, double x, double y)
31 /* Apply Dekker's algorithm. */
36 /* Calculate X * Y exactly and store the result in *HI + *LO. It is
37 given that the values are small enough that no overflow occurs and
38 large enough (or zero) that no underflow occurs. */
41 mul_split (double *hi, double *lo, double x, double y)
44 /* Fast built-in fused multiply-add. */
46 *lo = __builtin_fma (x, y, -*hi);
47 #elif defined FP_FAST_FMA
48 /* Fast library fused multiply-add, compiler before GCC 4.6. */
50 *lo = __fma (x, y, -*hi);
52 /* Apply Dekker's algorithm. */
54 # define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
62 *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
66 /* Compare absolute values of floating-point values pointed to by P
70 compare (const void *p, const void *q)
72 double pd = fabs (*(const double *) p);
73 double qd = fabs (*(const double *) q);
82 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
83 It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
87 __x2y2m1l (long double x, long double y)
90 SET_RESTORE_ROUND (FE_TONEAREST);
91 union ibm_extended_long_double xu, yu;
94 if (fabs (xu.d[1].d) < 0x1p-500)
96 if (fabs (yu.d[1].d) < 0x1p-500)
98 mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d);
99 mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d);
102 mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d);
103 mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d);
104 mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d);
107 mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
108 if (xu.d[0].d >= 0.75)
115 qsort (vals, 12, sizeof (double), compare);
116 /* Add up the values so that each element of VALS has absolute value
117 at most equal to the last set bit of the next nonzero
119 for (size_t i = 0; i <= 10; i++)
121 add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
122 qsort (vals + i + 1, 11 - i, sizeof (double), compare);
124 /* Now any error from this addition will be small. */
125 long double retval = (long double) vals[11];
126 for (size_t i = 10; i != (size_t) -1; i--)
127 retval += (long double) vals[i];