1 | /* Compute x^2 + y^2 - 1, without large cancellation error. |
2 | Copyright (C) 2012-2016 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 <float.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 | |
28 | static inline void |
29 | add_split (long double *hi, long double *lo, long double x, long double y) |
30 | { |
31 | /* Apply Dekker's algorithm. */ |
32 | *hi = x + y; |
33 | *lo = (x - *hi) + y; |
34 | } |
35 | |
36 | /* Calculate X * Y exactly and store the result in *HI + *LO. It is |
37 | given that the values are small enough that no overflow occurs and |
38 | large enough (or zero) that no underflow occurs. */ |
39 | |
40 | static inline void |
41 | mul_split (long double *hi, long double *lo, long double x, long double y) |
42 | { |
43 | #ifdef __FP_FAST_FMAL |
44 | /* Fast built-in fused multiply-add. */ |
45 | *hi = x * y; |
46 | *lo = __builtin_fmal (x, y, -*hi); |
47 | #elif defined FP_FAST_FMAL |
48 | /* Fast library fused multiply-add, compiler before GCC 4.6. */ |
49 | *hi = x * y; |
50 | *lo = __fmal (x, y, -*hi); |
51 | #else |
52 | /* Apply Dekker's algorithm. */ |
53 | *hi = x * y; |
54 | # define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1) |
55 | long double x1 = x * C; |
56 | long double y1 = y * C; |
57 | # undef C |
58 | x1 = (x - x1) + x1; |
59 | y1 = (y - y1) + y1; |
60 | long double x2 = x - x1; |
61 | long double y2 = y - y1; |
62 | *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2; |
63 | #endif |
64 | } |
65 | |
66 | /* Compare absolute values of floating-point values pointed to by P |
67 | and Q for qsort. */ |
68 | |
69 | static int |
70 | compare (const void *p, const void *q) |
71 | { |
72 | long double pld = fabsl (*(const long double *) p); |
73 | long double qld = fabsl (*(const long double *) q); |
74 | if (pld < qld) |
75 | return -1; |
76 | else if (pld == qld) |
77 | return 0; |
78 | else |
79 | return 1; |
80 | } |
81 | |
82 | /* Return X^2 + Y^2 - 1, computed without large cancellation error. |
83 | It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >= |
84 | 0.5. */ |
85 | |
86 | long double |
87 | __x2y2m1l (long double x, long double y) |
88 | { |
89 | long double vals[5]; |
90 | SET_RESTORE_ROUNDL (FE_TONEAREST); |
91 | mul_split (&vals[1], &vals[0], x, x); |
92 | mul_split (&vals[3], &vals[2], y, y); |
93 | vals[4] = -1.0L; |
94 | qsort (vals, 5, sizeof (long double), compare); |
95 | /* Add up the values so that each element of VALS has absolute value |
96 | at most equal to the last set bit of the next nonzero |
97 | element. */ |
98 | for (size_t i = 0; i <= 3; i++) |
99 | { |
100 | add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]); |
101 | qsort (vals + i + 1, 4 - i, sizeof (long double), compare); |
102 | } |
103 | /* Now any error from this addition will be small. */ |
104 | return vals[4] + vals[3] + vals[2] + vals[1] + vals[0]; |
105 | } |
106 | |