1/* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012-2022 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
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.
9
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.
14
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 <https://www.gnu.org/licenses/>. */
18
19#include <math.h>
20#include <math_private.h>
21#include <fenv_private.h>
22#include <mul_split.h>
23#include <stdlib.h>
24
25/* Calculate X + Y exactly and store the result in *HI + *LO. It is
26 given that |X| >= |Y| and the values are small enough that no
27 overflow occurs. */
28
29static inline void
30add_split (double *hi, double *lo, double x, double y)
31{
32 /* Apply Dekker's algorithm. */
33 *hi = x + y;
34 *lo = (x - *hi) + y;
35}
36
37/* Compare absolute values of floating-point values pointed to by P
38 and Q for qsort. */
39
40static int
41compare (const void *p, const void *q)
42{
43 double pd = fabs (x: *(const double *) p);
44 double qd = fabs (x: *(const double *) q);
45 if (pd < qd)
46 return -1;
47 else if (pd == qd)
48 return 0;
49 else
50 return 1;
51}
52
53/* Return X^2 + Y^2 - 1, computed without large cancellation error.
54 It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
55 0.5. */
56
57double
58__x2y2m1 (double x, double y)
59{
60 double vals[5];
61 SET_RESTORE_ROUND (FE_TONEAREST);
62 mul_split (hi: &vals[1], lo: &vals[0], x, y: x);
63 mul_split (hi: &vals[3], lo: &vals[2], x: y, y);
64 vals[4] = -1.0;
65 qsort (base: vals, nmemb: 5, size: sizeof (double), compar: compare);
66 /* Add up the values so that each element of VALS has absolute value
67 at most equal to the last set bit of the next nonzero
68 element. */
69 for (size_t i = 0; i <= 3; i++)
70 {
71 add_split (hi: &vals[i + 1], lo: &vals[i], x: vals[i + 1], y: vals[i]);
72 qsort (base: vals + i + 1, nmemb: 4 - i, size: sizeof (double), compar: compare);
73 }
74 /* Now any error from this addition will be small. */
75 return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
76}
77

source code of glibc/sysdeps/ieee754/dbl-64/x2y2m1.c