1 /* $OpenBSD: muldi3.c,v 1.5 2005/08/08 08:05:35 espie Exp $ */
2 /*-
3 * Copyright (c) 1992, 1993
4 * The Regents of the University of California. All rights reserved.
5 *
6 * This software was developed by the Computer Systems Engineering group
7 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
8 * contributed to Berkeley.
9 *
10 * Redistribution and use in source and binary forms, with or without
11 * modification, are permitted provided that the following conditions
12 * are met:
13 * 1. Redistributions of source code must retain the above copyright
14 * notice, this list of conditions and the following disclaimer.
15 * 2. Redistributions in binary form must reproduce the above copyright
16 * notice, this list of conditions and the following disclaimer in the
17 * documentation and/or other materials provided with the distribution.
18 * 3. Neither the name of the University nor the names of its contributors
19 * may be used to endorse or promote products derived from this software
20 * without specific prior written permission.
21 *
22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
32 * SUCH DAMAGE.
33 */
34
35 #include "quad.h"
36
37 /*
38 * Multiply two quads.
39 *
40 * Our algorithm is based on the following. Split incoming quad values
41 * u and v (where u,v >= 0) into
42 *
43 * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32)
44 *
45 * and
46 *
47 * v = 2^n v1 * v0
48 *
49 * Then
50 *
51 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
52 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
53 *
54 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
55 * and add 2^n u0 v0 to the last term and subtract it from the middle.
56 * This gives:
57 *
58 * uv = (2^2n + 2^n) (u1 v1) +
59 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
60 * (2^n + 1) (u0 v0)
61 *
62 * Factoring the middle a bit gives us:
63 *
64 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
65 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
66 * (2^n + 1) (u0 v0) [u0v0 = low]
67 *
68 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
69 * in just half the precision of the original. (Note that either or both
70 * of (u1 - u0) or (v0 - v1) may be negative.)
71 *
72 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
73 *
74 * Since C does not give us a `int * int = quad' operator, we split
75 * our input quads into two ints, then split the two ints into two
76 * shorts. We can then calculate `short * short = int' in native
77 * arithmetic.
78 *
79 * Our product should, strictly speaking, be a `long quad', with 128
80 * bits, but we are going to discard the upper 64. In other words,
81 * we are not interested in uv, but rather in (uv mod 2^2n). This
82 * makes some of the terms above vanish, and we get:
83 *
84 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
85 *
86 * or
87 *
88 * (2^n)(high + mid + low) + low
89 *
90 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
91 * of 2^n in either one will also vanish. Only `low' need be computed
92 * mod 2^2n, and only because of the final term above.
93 */
94 static quad_t __lmulq(u_int, u_int);
95
96 quad_t
__muldi3(quad_t a,quad_t b)97 __muldi3(quad_t a, quad_t b)
98 {
99 union uu u, v, low, prod;
100 u_int high, mid, udiff, vdiff;
101 int negall, negmid;
102 #define u1 u.ul[H]
103 #define u0 u.ul[L]
104 #define v1 v.ul[H]
105 #define v0 v.ul[L]
106
107 /*
108 * Get u and v such that u, v >= 0. When this is finished,
109 * u1, u0, v1, and v0 will be directly accessible through the
110 * int fields.
111 */
112 if (a >= 0)
113 u.q = a, negall = 0;
114 else
115 u.q = -a, negall = 1;
116 if (b >= 0)
117 v.q = b;
118 else
119 v.q = -b, negall ^= 1;
120
121 if (u1 == 0 && v1 == 0) {
122 /*
123 * An (I hope) important optimization occurs when u1 and v1
124 * are both 0. This should be common since most numbers
125 * are small. Here the product is just u0*v0.
126 */
127 prod.q = __lmulq(u0, v0);
128 } else {
129 /*
130 * Compute the three intermediate products, remembering
131 * whether the middle term is negative. We can discard
132 * any upper bits in high and mid, so we can use native
133 * u_int * u_int => u_int arithmetic.
134 */
135 low.q = __lmulq(u0, v0);
136
137 if (u1 >= u0)
138 negmid = 0, udiff = u1 - u0;
139 else
140 negmid = 1, udiff = u0 - u1;
141 if (v0 >= v1)
142 vdiff = v0 - v1;
143 else
144 vdiff = v1 - v0, negmid ^= 1;
145 mid = udiff * vdiff;
146
147 high = u1 * v1;
148
149 /*
150 * Assemble the final product.
151 */
152 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
153 low.ul[H];
154 prod.ul[L] = low.ul[L];
155 }
156 return (negall ? -prod.q : prod.q);
157 #undef u1
158 #undef u0
159 #undef v1
160 #undef v0
161 }
162
163 /*
164 * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
165 * the number of bits in an int (whatever that is---the code below
166 * does not care as long as quad.h does its part of the bargain---but
167 * typically N==16).
168 *
169 * We use the same algorithm from Knuth, but this time the modulo refinement
170 * does not apply. On the other hand, since N is half the size of an int,
171 * we can get away with native multiplication---none of our input terms
172 * exceeds (UINT_MAX >> 1).
173 *
174 * Note that, for u_int l, the quad-precision result
175 *
176 * l << N
177 *
178 * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
179 */
180 static quad_t
__lmulq(u_int u,u_int v)181 __lmulq(u_int u, u_int v)
182 {
183 u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
184 u_int prodh, prodl, was;
185 union uu prod;
186 int neg;
187
188 u1 = HHALF(u);
189 u0 = LHALF(u);
190 v1 = HHALF(v);
191 v0 = LHALF(v);
192
193 low = u0 * v0;
194
195 /* This is the same small-number optimization as before. */
196 if (u1 == 0 && v1 == 0)
197 return (low);
198
199 if (u1 >= u0)
200 udiff = u1 - u0, neg = 0;
201 else
202 udiff = u0 - u1, neg = 1;
203 if (v0 >= v1)
204 vdiff = v0 - v1;
205 else
206 vdiff = v1 - v0, neg ^= 1;
207 mid = udiff * vdiff;
208
209 high = u1 * v1;
210
211 /* prod = (high << 2N) + (high << N); */
212 prodh = high + HHALF(high);
213 prodl = LHUP(high);
214
215 /* if (neg) prod -= mid << N; else prod += mid << N; */
216 if (neg) {
217 was = prodl;
218 prodl -= LHUP(mid);
219 prodh -= HHALF(mid) + (prodl > was);
220 } else {
221 was = prodl;
222 prodl += LHUP(mid);
223 prodh += HHALF(mid) + (prodl < was);
224 }
225
226 /* prod += low << N */
227 was = prodl;
228 prodl += LHUP(low);
229 prodh += HHALF(low) + (prodl < was);
230 /* ... + low; */
231 if ((prodl += low) < low)
232 prodh++;
233
234 /* return 4N-bit product */
235 prod.ul[H] = prodh;
236 prod.ul[L] = prodl;
237 return (prod.q);
238 }
239