1/* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012 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 "quadmath-imp.h"
20#include <stdlib.h>
21
22/* Calculate X + Y exactly and store the result in *HI + *LO. It is
23 given that |X| >= |Y| and the values are small enough that no
24 overflow occurs. */
25
26static inline void
27add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
28{
29 /* Apply Dekker's algorithm. */
30 *hi = x + y;
31 *lo = (x - *hi) + y;
32}
33
34/* Calculate X * Y exactly and store the result in *HI + *LO. It is
35 given that the values are small enough that no overflow occurs and
36 large enough (or zero) that no underflow occurs. */
37
38static inline void
39mul_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
40{
41 /* Fast built-in fused multiply-add. */
42 *hi = x * y;
43 *lo = fmaq (x, y, -*hi);
44}
45
46/* Compare absolute values of floating-point values pointed to by P
47 and Q for qsort. */
48
49static int
50compare (const void *p, const void *q)
51{
52 __float128 pld = fabsq (*(const __float128 *) p);
53 __float128 qld = fabsq (*(const __float128 *) q);
54 if (pld < qld)
55 return -1;
56 else if (pld == qld)
57 return 0;
58 else
59 return 1;
60}
61
62/* Return X^2 + Y^2 - 1, computed without large cancellation error.
63 It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
64 0.75 or Y >= 0.5. */
65
66__float128
67__quadmath_x2y2m1q (__float128 x, __float128 y)
68{
69 __float128 vals[4];
70 size_t i;
71
72 /* FIXME: SET_RESTORE_ROUNDL (FE_TONEAREST); */
73 mul_split (&vals[1], &vals[0], x, x);
74 mul_split (&vals[3], &vals[2], y, y);
75 if (x >= 0.75Q)
76 vals[1] -= 1.0Q;
77 else
78 {
79 vals[1] -= 0.5Q;
80 vals[3] -= 0.5Q;
81 }
82 qsort (vals, 4, sizeof (__float128), compare);
83 /* Add up the values so that each element of VALS has absolute value
84 at most equal to the last set bit of the next nonzero
85 element. */
86 for (i = 0; i <= 2; i++)
87 {
88 add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
89 qsort (vals + i + 1, 3 - i, sizeof (__float128), compare);
90 }
91 /* Now any error from this addition will be small. */
92 return vals[3] + vals[2] + vals[1] + vals[0];
93}
94