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d032e0d2 | 1 | /* Compute x^2 + y^2 - 1, without large cancellation error. |
f7a9f785 | 2 | Copyright (C) 2012-2016 Free Software Foundation, Inc. |
d032e0d2 JM |
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. | |
a5721ebc JM |
83 | It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >= |
84 | 0.5. */ | |
d032e0d2 JM |
85 | |
86 | long double | |
87 | __x2y2m1l (long double x, long double y) | |
88 | { | |
a5721ebc | 89 | long double vals[5]; |
d032e0d2 JM |
90 | SET_RESTORE_ROUNDL (FE_TONEAREST); |
91 | mul_split (&vals[1], &vals[0], x, x); | |
92 | mul_split (&vals[3], &vals[2], y, y); | |
a5721ebc JM |
93 | vals[4] = -1.0L; |
94 | qsort (vals, 5, sizeof (long double), compare); | |
d032e0d2 JM |
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. */ | |
a5721ebc | 98 | for (size_t i = 0; i <= 3; i++) |
d032e0d2 JM |
99 | { |
100 | add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]); | |
a5721ebc | 101 | qsort (vals + i + 1, 4 - i, sizeof (long double), compare); |
d032e0d2 JM |
102 | } |
103 | /* Now any error from this addition will be small. */ | |
a5721ebc | 104 | return vals[4] + vals[3] + vals[2] + vals[1] + vals[0]; |
d032e0d2 | 105 | } |