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
34 #if defined(LIBC_SCCS) && !defined(lint)
35 static char rcsid[] = "$OpenBSD: muldi3.c,v 1.7 2004/11/28 07:23:41 mickey Exp $";
36 #endif /* LIBC_SCCS and not lint */
37
38 #include "quad.h"
39
40 /*
41 * Multiply two quads.
42 *
43 * Our algorithm is based on the following. Split incoming quad values
44 * u and v (where u,v >= 0) into
45 *
46 * u = 2^n u1 * u0 (n = number of bits in `u_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 = quad' operator, we split
78 * our input quads 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 quad', with 128
83 * 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 quad_t __lmulq(u_int, u_int);
98
99 quad_t
100 __muldi3(a, b)
101 quad_t a, b;
102 {
103 union uu u, v, low, prod;
104 u_int high, mid, udiff, vdiff;
105 int negall, negmid;
106 #define u1 u.ul[H]
107 #define u0 u.ul[L]
108 #define v1 v.ul[H]
109 #define v0 v.ul[L]
110
111 /*
112 * Get u and v such that u, v >= 0. When this is finished,
113 * u1, u0, v1, and v0 will be directly accessible through the
114 * int fields.
115 */
116 if (a >= 0)
117 u.q = a, negall = 0;
118 else
119 u.q = -a, negall = 1;
120 if (b >= 0)
121 v.q = b;
122 else
123 v.q = -b, negall ^= 1;
124
125 if (u1 == 0 && v1 == 0) {
126 /*
127 * An (I hope) important optimization occurs when u1 and v1
128 * are both 0. This should be common since most numbers
129 * are small. Here the product is just u0*v0.
130 */
131 prod.q = __lmulq(u0, v0);
132 } else {
133 /*
134 * Compute the three intermediate products, remembering
135 * whether the middle term is negative. We can discard
136 * any upper bits in high and mid, so we can use native
137 * u_int * u_int => u_int arithmetic.
138 */
139 low.q = __lmulq(u0, v0);
140
141 if (u1 >= u0)
142 negmid = 0, udiff = u1 - u0;
143 else
144 negmid = 1, udiff = u0 - u1;
145 if (v0 >= v1)
146 vdiff = v0 - v1;
147 else
148 vdiff = v1 - v0, negmid ^= 1;
149 mid = udiff * vdiff;
150
151 high = u1 * v1;
152
153 /*
154 * Assemble the final product.
155 */
156 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
157 low.ul[H];
158 prod.ul[L] = low.ul[L];
159 }
160 return (negall ? -prod.q : prod.q);
161 #undef u1
162 #undef u0
163 #undef v1
164 #undef v0
165 }
166
167 /*
168 * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
169 * the number of bits in an int (whatever that is---the code below
170 * does not care as long as quad.h does its part of the bargain---but
171 * typically N==16).
172 *
173 * We use the same algorithm from Knuth, but this time the modulo refinement
174 * does not apply. On the other hand, since N is half the size of an int,
175 * we can get away with native multiplication---none of our input terms
176 * exceeds (UINT_MAX >> 1).
177 *
178 * Note that, for u_int l, the quad-precision result
179 *
180 * l << N
181 *
182 * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
183 */
184 static quad_t
185 __lmulq(u_int u, u_int v)
186 {
187 u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
188 u_int prodh, prodl, was;
189 union uu prod;
190 int neg;
191
192 u1 = HHALF(u);
193 u0 = LHALF(u);
194 v1 = HHALF(v);
195 v0 = LHALF(v);
196
197 low = u0 * v0;
198
199 /* This is the same small-number optimization as before. */
200 if (u1 == 0 && v1 == 0)
201 return (low);
202
203 if (u1 >= u0)
204 udiff = u1 - u0, neg = 0;
205 else
206 udiff = u0 - u1, neg = 1;
207 if (v0 >= v1)
208 vdiff = v0 - v1;
209 else
210 vdiff = v1 - v0, neg ^= 1;
211 mid = udiff * vdiff;
212
213 high = u1 * v1;
214
215 /* prod = (high << 2N) + (high << N); */
216 prodh = high + HHALF(high);
217 prodl = LHUP(high);
218
219 /* if (neg) prod -= mid << N; else prod += mid << N; */
220 if (neg) {
221 was = prodl;
222 prodl -= LHUP(mid);
223 prodh -= HHALF(mid) + (prodl > was);
224 } else {
225 was = prodl;
226 prodl += LHUP(mid);
227 prodh += HHALF(mid) + (prodl < was);
228 }
229
230 /* prod += low << N */
231 was = prodl;
232 prodl += LHUP(low);
233 prodh += HHALF(low) + (prodl < was);
234 /* ... + low; */
235 if ((prodl += low) < low)
236 prodh++;
237
238 /* return 4N-bit product */
239 prod.ul[H] = prodh;
240 prod.ul[L] = prodl;
241 return (prod.q);
242 }