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
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
|
/* mpn_divexact_by3c -- mpn exact division by 3.
Copyright 2000-2003, 2008 Free Software Foundation, Inc.
This file is part of the GNU MP Library.
The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of either:
* the GNU Lesser General Public License as published by the Free
Software Foundation; either version 3 of the License, or (at your
option) any later version.
or
* the GNU General Public License as published by the Free Software
Foundation; either version 2 of the License, or (at your option) any
later version.
or both in parallel, as here.
The GNU MP 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 General Public License
for more details.
You should have received copies of the GNU General Public License and the
GNU Lesser General Public License along with the GNU MP Library. If not,
see https://www.gnu.org/licenses/. */
#include "gmp-impl.h"
#if DIVEXACT_BY3_METHOD == 0
mp_limb_t
mpn_divexact_by3c (mp_ptr rp, mp_srcptr up, mp_size_t un, mp_limb_t c)
{
mp_limb_t r;
r = mpn_bdiv_dbm1c (rp, up, un, GMP_NUMB_MASK / 3, GMP_NUMB_MASK / 3 * c);
/* Possible bdiv_dbm1 return values are C * (GMP_NUMB_MASK / 3), 0 <= C < 3.
We want to return C. We compute the remainder mod 4 and notice that the
inverse of (2^(2k)-1)/3 mod 4 is 1. */
return r & 3;
}
#endif
#if DIVEXACT_BY3_METHOD == 1
/* The algorithm here is basically the same as mpn_divexact_1, as described
in the manual. Namely at each step q = (src[i]-c)*inverse, and new c =
borrow(src[i]-c) + high(divisor*q). But because the divisor is just 3,
high(divisor*q) can be determined with two comparisons instead of a
multiply.
The "c += ..."s add the high limb of 3*l to c. That high limb will be 0,
1 or 2. Doing two separate "+="s seems to give better code on gcc (as of
2.95.2 at least).
It will be noted that the new c is formed by adding three values each 0
or 1. But the total is only 0, 1 or 2. When the subtraction src[i]-c
causes a borrow, that leaves a limb value of either 0xFF...FF or
0xFF...FE. The multiply by MODLIMB_INVERSE_3 gives 0x55...55 or
0xAA...AA respectively, and in those cases high(3*q) is only 0 or 1
respectively, hence a total of no more than 2.
Alternatives:
This implementation has each multiply on the dependent chain, due to
"l=s-c". See below for alternative code which avoids that. */
mp_limb_t
mpn_divexact_by3c (mp_ptr restrict rp, mp_srcptr restrict up, mp_size_t un, mp_limb_t c)
{
mp_limb_t l, q, s;
mp_size_t i;
ASSERT (un >= 1);
ASSERT (c == 0 || c == 1 || c == 2);
ASSERT (MPN_SAME_OR_SEPARATE_P (rp, up, un));
i = 0;
do
{
s = up[i];
SUBC_LIMB (c, l, s, c);
q = (l * MODLIMB_INVERSE_3) & GMP_NUMB_MASK;
rp[i] = q;
c += (q >= GMP_NUMB_CEIL_MAX_DIV3);
c += (q >= GMP_NUMB_CEIL_2MAX_DIV3);
}
while (++i < un);
ASSERT (c == 0 || c == 1 || c == 2);
return c;
}
#endif
#if DIVEXACT_BY3_METHOD == 2
/* The following alternative code re-arranges the quotient calculation from
(src[i]-c)*inverse to instead
q = src[i]*inverse - c*inverse
thereby allowing src[i]*inverse to be scheduled back as far as desired,
making full use of multiplier throughput and leaving just some carry
handing on the dependent chain.
The carry handling consists of determining the c for the next iteration.
This is the same as described above, namely look for any borrow from
src[i]-c, and at the high of 3*q.
high(3*q) is done with two comparisons as above (in c2 and c3). The
borrow from src[i]-c is incorporated into those by noting that if there's
a carry then then we have src[i]-c == 0xFF..FF or 0xFF..FE, in turn
giving q = 0x55..55 or 0xAA..AA. Adding 1 to either of those q values is
enough to make high(3*q) come out 1 bigger, as required.
l = -c*inverse is calculated at the same time as c, since for most chips
it can be more conveniently derived from separate c1/c2/c3 values than
from a combined c equal to 0, 1 or 2.
The net effect is that with good pipelining this loop should be able to
run at perhaps 4 cycles/limb, depending on available execute resources
etc.
Usage:
This code is not used by default, since we really can't rely on the
compiler generating a good software pipeline, nor on such an approach
even being worthwhile on all CPUs.
Itanium is one chip where this algorithm helps though, see
mpn/ia64/diveby3.asm. */
mp_limb_t
mpn_divexact_by3c (mp_ptr restrict rp, mp_srcptr restrict up, mp_size_t un, mp_limb_t cy)
{
mp_limb_t s, sm, cl, q, qx, c2, c3;
mp_size_t i;
ASSERT (un >= 1);
ASSERT (cy == 0 || cy == 1 || cy == 2);
ASSERT (MPN_SAME_OR_SEPARATE_P (rp, up, un));
cl = cy == 0 ? 0 : cy == 1 ? -MODLIMB_INVERSE_3 : -2*MODLIMB_INVERSE_3;
for (i = 0; i < un; i++)
{
s = up[i];
sm = (s * MODLIMB_INVERSE_3) & GMP_NUMB_MASK;
q = (cl + sm) & GMP_NUMB_MASK;
rp[i] = q;
qx = q + (s < cy);
c2 = qx >= GMP_NUMB_CEIL_MAX_DIV3;
c3 = qx >= GMP_NUMB_CEIL_2MAX_DIV3 ;
cy = c2 + c3;
cl = (-c2 & -MODLIMB_INVERSE_3) + (-c3 & -MODLIMB_INVERSE_3);
}
return cy;
}
#endif
|
