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
|
/* Compute x^2 + y^2 - 1, without large cancellation error.
Copyright (C) 2012-2017 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 <mul_split.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 (double *hi, double *lo, double x, double y)
{
/* Apply Dekker's algorithm. */
*hi = x + y;
*lo = (x - *hi) + y;
}
/* Compare absolute values of floating-point values pointed to by P
and Q for qsort. */
static int
compare (const void *p, const void *q)
{
double pd = fabs (*(const double *) p);
double qd = fabs (*(const double *) q);
if (pd < qd)
return -1;
else if (pd == qd)
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)
{
double vals[13];
SET_RESTORE_ROUND (FE_TONEAREST);
union ibm_extended_long_double xu, yu;
xu.ld = x;
yu.ld = y;
if (fabs (xu.d[1].d) < 0x1p-500)
xu.d[1].d = 0.0;
if (fabs (yu.d[1].d) < 0x1p-500)
yu.d[1].d = 0.0;
mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d);
mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d);
vals[2] *= 2.0;
vals[3] *= 2.0;
mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d);
mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d);
mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d);
vals[8] *= 2.0;
vals[9] *= 2.0;
mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
vals[12] = -1.0;
qsort (vals, 13, sizeof (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 <= 11; i++)
{
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
qsort (vals + i + 1, 12 - i, sizeof (double), compare);
}
/* Now any error from this addition will be small. */
long double retval = (long double) vals[12];
for (size_t i = 11; i != (size_t) -1; i--)
retval += (long double) vals[i];
return retval;
}
|
