1/* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012-2018 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 <http://www.gnu.org/licenses/>. */
18
19#include <math.h>
20#include <math_private.h>
21#include <mul_split.h>
22#include <stdlib.h>
23
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
26 overflow occurs. */
27
28static inline void
29add_split (double *hi, double *lo, double x, double y)
30{
31 /* Apply Dekker's algorithm. */
32 *hi = x + y;
33 *lo = (x - *hi) + y;
34}
35
36/* Compare absolute values of floating-point values pointed to by P
37 and Q for qsort. */
38
39static int
40compare (const void *p, const void *q)
41{
42 double pd = fabs (*(const double *) p);
43 double qd = fabs (*(const double *) q);
44 if (pd < qd)
45 return -1;
46 else if (pd == qd)
47 return 0;
48 else
49 return 1;
50}
51
52/* Return X^2 + Y^2 - 1, computed without large cancellation error.
53 It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
54 0.5. */
55
56double
57__x2y2m1 (double x, double y)
58{
59 double vals[5];
60 SET_RESTORE_ROUND (FE_TONEAREST);
61 mul_split (&vals[1], &vals[0], x, x);
62 mul_split (&vals[3], &vals[2], y, y);
63 vals[4] = -1.0;
64 qsort (vals, 5, sizeof (double), compare);
65 /* Add up the values so that each element of VALS has absolute value
66 at most equal to the last set bit of the next nonzero
67 element. */
68 for (size_t i = 0; i <= 3; i++)
69 {
70 add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
71 qsort (vals + i + 1, 4 - i, sizeof (double), compare);
72 }
73 /* Now any error from this addition will be small. */
74 return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
75}
76