1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
|
/* Compute x^2 + y^2 - 1, without large cancellation error.
Copyright (C) 2012-2016 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <math.h>
#include <math_private.h>
#include <float.h>
#include <stdlib.h>
/* Calculate X + Y exactly and store the result in *HI + *LO. It is
given that |X| >= |Y| and the values are small enough that no
overflow occurs. */
static inline void
add_split (long double *hi, long double *lo, long double x, long double y)
{
/* Apply Dekker's algorithm. */
*hi = x + y;
*lo = (x - *hi) + y;
}
/* Calculate X * Y exactly and store the result in *HI + *LO. It is
given that the values are small enough that no overflow occurs and
large enough (or zero) that no underflow occurs. */
static inline void
mul_split (long double *hi, long double *lo, long double x, long double y)
{
#ifdef __FP_FAST_FMAL
/* Fast built-in fused multiply-add. */
*hi = x * y;
*lo = __builtin_fmal (x, y, -*hi);
#elif defined FP_FAST_FMAL
/* Fast library fused multiply-add, compiler before GCC 4.6. */
*hi = x * y;
*lo = __fmal (x, y, -*hi);
#else
/* Apply Dekker's algorithm. */
*hi = x * y;
# define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
long double x1 = x * C;
long double y1 = y * C;
# undef C
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
long double x2 = x - x1;
long double y2 = y - y1;
*lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
#endif
}
/* Compare absolute values of floating-point values pointed to by P
and Q for qsort. */
static int
compare (const void *p, const void *q)
{
long double pld = fabsl (*(const long double *) p);
long double qld = fabsl (*(const long double *) q);
if (pld < qld)
return -1;
else if (pld == qld)
return 0;
else
return 1;
}
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
0.5. */
long double
__x2y2m1l (long double x, long double y)
{
long double vals[5];
SET_RESTORE_ROUNDL (FE_TONEAREST);
mul_split (&vals[1], &vals[0], x, x);
mul_split (&vals[3], &vals[2], y, y);
vals[4] = -1.0L;
qsort (vals, 5, sizeof (long double), compare);
/* Add up the values so that each element of VALS has absolute value
at most equal to the last set bit of the next nonzero
element. */
for (size_t i = 0; i <= 3; i++)
{
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
qsort (vals + i + 1, 4 - i, sizeof (long double), compare);
}
/* Now any error from this addition will be small. */
return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
}
|
