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
57long double
58__x2y2m1l (long double x, long double y)
59{
60 double vals[13];
61 SET_RESTORE_ROUND (FE_TONEAREST);
62 union ibm_extended_long_double xu, yu;
63 xu.ld = x;
64 yu.ld = y;
65 if (fabs (x: xu.d[1].d) < 0x1p-500)
66 xu.d[1].d = 0.0;
67 if (fabs (x: yu.d[1].d) < 0x1p-500)
68 yu.d[1].d = 0.0;
69 mul_split (hi: &vals[1], lo: &vals[0], x: xu.d[0].d, y: xu.d[0].d);
70 mul_split (hi: &vals[3], lo: &vals[2], x: xu.d[0].d, y: xu.d[1].d);
71 vals[2] *= 2.0;
72 vals[3] *= 2.0;
73 mul_split (hi: &vals[5], lo: &vals[4], x: xu.d[1].d, y: xu.d[1].d);
74 mul_split (hi: &vals[7], lo: &vals[6], x: yu.d[0].d, y: yu.d[0].d);
75 mul_split (hi: &vals[9], lo: &vals[8], x: yu.d[0].d, y: yu.d[1].d);
76 vals[8] *= 2.0;
77 vals[9] *= 2.0;
78 mul_split (hi: &vals[11], lo: &vals[10], x: yu.d[1].d, y: yu.d[1].d);
79 vals[12] = -1.0;
80 qsort (base: vals, nmemb: 13, size: sizeof (double), compar: compare);
81 /* Add up the values so that each element of VALS has absolute value
82 at most equal to the last set bit of the next nonzero
83 element. */
84 for (size_t i = 0; i <= 11; i++)
85 {
86 add_split (hi: &vals[i + 1], lo: &vals[i], x: vals[i + 1], y: vals[i]);
87 qsort (base: vals + i + 1, nmemb: 12 - i, size: sizeof (double), compar: compare);
88 }
89 /* Now any error from this addition will be small. */
90 long double retval = (long double) vals[12];
91 for (size_t i = 11; i != (size_t) -1; i--)
92 retval += (long double) vals[i];
93 return retval;
94}
95

source code of glibc/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c