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DEFINITIONS
This source file includes following definitions.
- __muldi3
- __lmulq
1 /*-
2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * This software was developed by the Computer Systems Engineering group
6 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
7 * contributed to Berkeley.
8 *
9 * Redistribution and use in source and binary forms, with or without
10 * modification, are permitted provided that the following conditions
11 * are met:
12 * 1. Redistributions of source code must retain the above copyright
13 * notice, this list of conditions and the following disclaimer.
14 * 2. Redistributions in binary form must reproduce the above copyright
15 * notice, this list of conditions and the following disclaimer in the
16 * documentation and/or other materials provided with the distribution.
17 * 3. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
20 *
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
31 * SUCH DAMAGE.
32 *
33 * From:
34 * @(#)muldi3.c 8.1 (Berkeley) 6/4/93
35 * NetBSD: muldi3.c,v 1.1 2005/12/20 19:28:51 christos Exp
36 */
37
38 #include "longlong.h"
39
40 /*
41 * Multiply two long longs.
42 *
43 * Our algorithm is based on the following. Split incoming long long
44 * values u and v (where u,v >= 0) into
45 *
46 * u = 2^n u1 * u0 (n = number of bits in `unsigned int', usu. 32)
47 *
48 * and
49 *
50 * v = 2^n v1 * v0
51 *
52 * Then
53 *
54 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
55 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
56 *
57 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
58 * and add 2^n u0 v0 to the last term and subtract it from the middle.
59 * This gives:
60 *
61 * uv = (2^2n + 2^n) (u1 v1) +
62 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
63 * (2^n + 1) (u0 v0)
64 *
65 * Factoring the middle a bit gives us:
66 *
67 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
68 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
69 * (2^n + 1) (u0 v0) [u0v0 = low]
70 *
71 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
72 * in just half the precision of the original. (Note that either or both
73 * of (u1 - u0) or (v0 - v1) may be negative.)
74 *
75 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
76 *
77 * Since C does not give us a `int * int = long long' operator, we split
78 * our input long longs into two ints, then split the two ints into two
79 * shorts. We can then calculate `short * short = int' in native
80 * arithmetic.
81 *
82 * Our product should, strictly speaking, be a `long long long', with
83 * 128 bits, but we are going to discard the upper 64. In other words,
84 * we are not interested in uv, but rather in (uv mod 2^2n). This
85 * makes some of the terms above vanish, and we get:
86 *
87 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
88 *
89 * or
90 *
91 * (2^n)(high + mid + low) + low
92 *
93 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
94 * of 2^n in either one will also vanish. Only `low' need be computed
95 * mod 2^2n, and only because of the final term above.
96 */
97 static long long __lmulq(unsigned int, unsigned int);
98
99 long long
100 __muldi3(long long a, long long b)
101 {
102 union uu u, v, low, prod;
103 unsigned int high, mid, udiff, vdiff;
104 int negall, negmid;
105 #define u1 u.ui[H]
106 #define u0 u.ui[L]
107 #define v1 v.ui[H]
108 #define v0 v.ui[L]
109
110 /*
111 * Get u and v such that u, v >= 0. When this is finished,
112 * u1, u0, v1, and v0 will be directly accessible through the
113 * int fields.
114 */
115 if (a >= 0)
116 u.ll = a, negall = 0;
117 else
118 u.ll = -a, negall = 1;
119 if (b >= 0)
120 v.ll = b;
121 else
122 v.ll = -b, negall ^= 1;
123
124 if (u1 == 0 && v1 == 0) {
125 /*
126 * An (I hope) important optimization occurs when u1 and v1
127 * are both 0. This should be common since most numbers
128 * are small. Here the product is just u0*v0.
129 */
130 prod.ll = __lmulq(u0, v0);
131 } else {
132 /*
133 * Compute the three intermediate products, remembering
134 * whether the middle term is negative. We can discard
135 * any upper bits in high and mid, so we can use native
136 * unsigned int * unsigned int => unsigned int arithmetic.
137 */
138 low.ll = __lmulq(u0, v0);
139
140 if (u1 >= u0)
141 negmid = 0, udiff = u1 - u0;
142 else
143 negmid = 1, udiff = u0 - u1;
144 if (v0 >= v1)
145 vdiff = v0 - v1;
146 else
147 vdiff = v1 - v0, negmid ^= 1;
148 mid = udiff * vdiff;
149
150 high = u1 * v1;
151
152 /*
153 * Assemble the final product.
154 */
155 prod.ui[H] = high + (negmid ? -mid : mid) + low.ui[L] +
156 low.ui[H];
157 prod.ui[L] = low.ui[L];
158 }
159 return (negall ? -prod.ll : prod.ll);
160 #undef u1
161 #undef u0
162 #undef v1
163 #undef v0
164 }
165
166 /*
167 * Multiply two 2N-bit ints to produce a 4N-bit long long, where N is
168 * half the number of bits in an int (whatever that is---the code
169 * below does not care as long as the header file does its part of the
170 * bargain---but typically N==16).
171 *
172 * We use the same algorithm from Knuth, but this time the modulo refinement
173 * does not apply. On the other hand, since N is half the size of an int,
174 * we can get away with native multiplication---none of our input terms
175 * exceeds (UINT_MAX >> 1).
176 *
177 * Note that, for unsigned int l, the quad-precision (long long) result
178 *
179 * l << N
180 *
181 * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
182 */
183 static long long
184 __lmulq(unsigned int u, unsigned int v)
185 {
186 unsigned int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
187 unsigned int prodh, prodl, was;
188 union uu prod;
189 int neg;
190
191 u1 = HHALF(u);
192 u0 = LHALF(u);
193 v1 = HHALF(v);
194 v0 = LHALF(v);
195
196 low = u0 * v0;
197
198 /* This is the same small-number optimization as before. */
199 if (u1 == 0 && v1 == 0)
200 return (low);
201
202 if (u1 >= u0)
203 udiff = u1 - u0, neg = 0;
204 else
205 udiff = u0 - u1, neg = 1;
206 if (v0 >= v1)
207 vdiff = v0 - v1;
208 else
209 vdiff = v1 - v0, neg ^= 1;
210 mid = udiff * vdiff;
211
212 high = u1 * v1;
213
214 /* prod = (high << 2N) + (high << N); */
215 prodh = high + HHALF(high);
216 prodl = LHUP(high);
217
218 /* if (neg) prod -= mid << N; else prod += mid << N; */
219 if (neg) {
220 was = prodl;
221 prodl -= LHUP(mid);
222 prodh -= HHALF(mid) + (prodl > was);
223 } else {
224 was = prodl;
225 prodl += LHUP(mid);
226 prodh += HHALF(mid) + (prodl < was);
227 }
228
229 /* prod += low << N */
230 was = prodl;
231 prodl += LHUP(low);
232 prodh += HHALF(low) + (prodl < was);
233 /* ... + low; */
234 if ((prodl += low) < low)
235 prodh++;
236
237 /* return 4N-bit product */
238 prod.ui[H] = prodh;
239 prod.ui[L] = prodl;
240 return (prod.ll);
241 }