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
106
107
108
109
110
111
112
113
114
115
|
/* Compute full X * Y for double type.
Copyright (C) 2013-2022 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
<https://www.gnu.org/licenses/>. */
#ifndef _MUL_SPLIT_H
#define _MUL_SPLIT_H
#include <float.h>
/* 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 void
mul_split (double *hi, double *lo, double x, double y)
{
#ifdef __FP_FAST_FMA
/* Fast built-in fused multiply-add. */
*hi = x * y;
*lo = __builtin_fma (x, y, -*hi);
#else
/* Apply Dekker's algorithm. */
*hi = x * y;
# define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
double x1 = x * C;
double y1 = y * C;
# undef C
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
double x2 = x - x1;
double y2 = y - y1;
*lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
#endif
}
/* Add a + b exactly, such that *hi + *lo = a + b.
Assumes |a| >= |b| and rounding to nearest. */
static inline void
fast_two_sum (double *hi, double *lo, double a, double b)
{
double e;
*hi = a + b;
e = *hi - a; /* exact */
*lo = b - e; /* exact */
/* Now *hi + *lo = a + b exactly. */
}
/* Multiplication of two floating-point expansions: *hi + *lo is an
approximation of (h1+l1)*(h2+l2), assuming |l1| <= 1/2*ulp(h1)
and |l2| <= 1/2*ulp(h2) and rounding to nearest. */
static inline void
mul_expansion (double *hi, double *lo, double h1, double l1,
double h2, double l2)
{
double r, e;
mul_split (hi, lo, h1, h2);
r = h1 * l2 + h2 * l1;
/* Now add r to (hi,lo) using fast two-sum, where we know |r| < |hi|. */
fast_two_sum (hi, &e, *hi, r);
*lo -= e;
}
/* Calculate X / Y and store the approximate result in *HI + *LO. It is
assumed that Y is not zero, that no overflow nor underflow occurs, and
rounding is to nearest. */
static inline void
div_split (double *hi, double *lo, double x, double y)
{
double a, b;
*hi = x / y;
mul_split (&a, &b, *hi, y);
/* a + b = hi*y, which should be near x. */
a = x - a; /* huge cancellation */
a = a - b;
/* Now x ~ hi*y + a thus x/y ~ hi + a/y. */
*lo = a / y;
}
/* Division of two floating-point expansions: *hi + *lo is an
approximation of (h1+l1)/(h2+l2), assuming |l1| <= 1/2*ulp(h1)
and |l2| <= 1/2*ulp(h2), h2+l2 is not zero, and rounding to nearest. */
static inline void
div_expansion (double *hi, double *lo, double h1, double l1,
double h2, double l2)
{
double r, e;
div_split (hi, lo, h1, h2);
/* (h1+l1)/(h2+l2) ~ h1/h2 + (l1*h2 - l2*h1)/h2^2 */
r = (l1 * h2 - l2 * h1) / (h2 * h2);
/* Now add r to (hi,lo) using fast two-sum, where we know |r| < |hi|. */
fast_two_sum (hi, &e, *hi, r);
*lo += e;
/* Renormalize since |lo| might be larger than 0.5 ulp(hi). */
fast_two_sum (hi, lo, *hi, *lo);
}
#endif /* _MUL_SPLIT_H */
|
