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 /* 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
29 static inline void
add_split(long double * hi,long double * lo,long double x,long double y)30 add_split (long double *hi, long double *lo, long double x, long 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
40 static int
compare(const void * p,const void * q)41 compare (const void *p, const void *q)
42 {
43 long double pld = fabsl (*(const long double *) p);
44 long double qld = fabsl (*(const long double *) q);
45 if (pld < qld)
46 return -1;
47 else if (pld == qld)
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
57 long double
__x2y2m1l(long double x,long double y)58 __x2y2m1l (long double x, long double y)
59 {
60 long double vals[5];
61 SET_RESTORE_ROUNDL (FE_TONEAREST);
62 mul_splitl (&vals[1], &vals[0], x, x);
63 mul_splitl (&vals[3], &vals[2], y, y);
64 vals[4] = -1.0L;
65 qsort (vals, 5, sizeof (long double), 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 (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
72 qsort (vals + i + 1, 4 - i, sizeof (long double), 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