1/* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012-2021 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_splitl.h>
23#include <stdlib.h>
24
25
26/* Calculate X + Y exactly and store the result in *HI + *LO. It is
27 given that |X| >= |Y| and the values are small enough that no
28 overflow occurs. */
29
30static inline void
31add_split (_Float128 *hi, _Float128 *lo, _Float128 x, _Float128 y)
32{
33 /* Apply Dekker's algorithm. */
34 *hi = x + y;
35 *lo = (x - *hi) + y;
36}
37
38/* Compare absolute values of floating-point values pointed to by P
39 and Q for qsort. */
40
41static int
42compare (const void *p, const void *q)
43{
44 _Float128 pld = fabsl (*(const _Float128 *) p);
45 _Float128 qld = fabsl (*(const _Float128 *) q);
46 if (pld < qld)
47 return -1;
48 else if (pld == qld)
49 return 0;
50 else
51 return 1;
52}
53
54/* Return X^2 + Y^2 - 1, computed without large cancellation error.
55 It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
56 0.5. */
57
58_Float128
59__x2y2m1l (_Float128 x, _Float128 y)
60{
61 _Float128 vals[5];
62 SET_RESTORE_ROUNDL (FE_TONEAREST);
63 mul_splitl (&vals[1], &vals[0], x, x);
64 mul_splitl (&vals[3], &vals[2], y, y);
65 vals[4] = -1;
66 qsort (vals, 5, sizeof (_Float128), compare);
67 /* Add up the values so that each element of VALS has absolute value
68 at most equal to the last set bit of the next nonzero
69 element. */
70 for (size_t i = 0; i <= 3; i++)
71 {
72 add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
73 qsort (vals + i + 1, 4 - i, sizeof (_Float128), compare);
74 }
75 /* Now any error from this addition will be small. */
76 return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
77}
78