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 #include "quad.h"
35
36 /*
37 * Multiply two quads.
38 *
39 * Our algorithm is based on the following. Split incoming quad values
40 * u and v (where u,v >= 0) into
41 *
42 * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32)
43 *
44 * and
45 *
46 * v = 2^n v1 * v0
47 *
48 * Then
49 *
50 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
51 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
52 *
53 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
54 * and add 2^n u0 v0 to the last term and subtract it from the middle.
55 * This gives:
56 *
57 * uv = (2^2n + 2^n) (u1 v1) +
58 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
59 * (2^n + 1) (u0 v0)
60 *
61 * Factoring the middle a bit gives us:
62 *
63 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
64 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
65 * (2^n + 1) (u0 v0) [u0v0 = low]
66 *
67 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
68 * in just half the precision of the original. (Note that either or both
69 * of (u1 - u0) or (v0 - v1) may be negative.)
70 *
71 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
72 *
73 * Since C does not give us a `int * int = quad' operator, we split
74 * our input quads into two ints, then split the two ints into two
75 * shorts. We can then calculate `short * short = int' in native
76 * arithmetic.
77 *
78 * Our product should, strictly speaking, be a `long quad', with 128
79 * bits, but we are going to discard the upper 64. In other words,
80 * we are not interested in uv, but rather in (uv mod 2^2n). This
81 * makes some of the terms above vanish, and we get:
82 *
83 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
84 *
85 * or
86 *
87 * (2^n)(high + mid + low) + low
88 *
89 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
90 * of 2^n in either one will also vanish. Only `low' need be computed
91 * mod 2^2n, and only because of the final term above.
92 */
93 static quad_t __lmulq(u_int, u_int);
94
95 quad_t
__muldi3(quad_t a,quad_t b)96 __muldi3(quad_t a, quad_t b)
97 {
98 union uu u, v, low, prod;
99 u_int high, mid, udiff, vdiff;
100 int negall, negmid;
101 #define u1 u.ul[H]
102 #define u0 u.ul[L]
103 #define v1 v.ul[H]
104 #define v0 v.ul[L]
105
106 /*
107 * Get u and v such that u, v >= 0. When this is finished,
108 * u1, u0, v1, and v0 will be directly accessible through the
109 * int fields.
110 */
111 if (a >= 0)
112 u.q = a, negall = 0;
113 else
114 u.q = -a, negall = 1;
115 if (b >= 0)
116 v.q = b;
117 else
118 v.q = -b, negall ^= 1;
119
120 if (u1 == 0 && v1 == 0) {
121 /*
122 * An (I hope) important optimization occurs when u1 and v1
123 * are both 0. This should be common since most numbers
124 * are small. Here the product is just u0*v0.
125 */
126 prod.q = __lmulq(u0, v0);
127 } else {
128 /*
129 * Compute the three intermediate products, remembering
130 * whether the middle term is negative. We can discard
131 * any upper bits in high and mid, so we can use native
132 * u_int * u_int => u_int arithmetic.
133 */
134 low.q = __lmulq(u0, v0);
135
136 if (u1 >= u0)
137 negmid = 0, udiff = u1 - u0;
138 else
139 negmid = 1, udiff = u0 - u1;
140 if (v0 >= v1)
141 vdiff = v0 - v1;
142 else
143 vdiff = v1 - v0, negmid ^= 1;
144 mid = udiff * vdiff;
145
146 high = u1 * v1;
147
148 /*
149 * Assemble the final product.
150 */
151 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
152 low.ul[H];
153 prod.ul[L] = low.ul[L];
154 }
155 return (negall ? -prod.q : prod.q);
156 #undef u1
157 #undef u0
158 #undef v1
159 #undef v0
160 }
161
162 /*
163 * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
164 * the number of bits in an int (whatever that is---the code below
165 * does not care as long as quad.h does its part of the bargain---but
166 * typically N==16).
167 *
168 * We use the same algorithm from Knuth, but this time the modulo refinement
169 * does not apply. On the other hand, since N is half the size of an int,
170 * we can get away with native multiplication---none of our input terms
171 * exceeds (UINT_MAX >> 1).
172 *
173 * Note that, for u_int l, the quad-precision result
174 *
175 * l << N
176 *
177 * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
178 */
179 static quad_t
__lmulq(u_int u,u_int v)180 __lmulq(u_int u, u_int v)
181 {
182 u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
183 u_int prodh, prodl, was;
184 union uu prod;
185 int neg;
186
187 u1 = HHALF(u);
188 u0 = LHALF(u);
189 v1 = HHALF(v);
190 v0 = LHALF(v);
191
192 low = u0 * v0;
193
194 /* This is the same small-number optimization as before. */
195 if (u1 == 0 && v1 == 0)
196 return (low);
197
198 if (u1 >= u0)
199 udiff = u1 - u0, neg = 0;
200 else
201 udiff = u0 - u1, neg = 1;
202 if (v0 >= v1)
203 vdiff = v0 - v1;
204 else
205 vdiff = v1 - v0, neg ^= 1;
206 mid = udiff * vdiff;
207
208 high = u1 * v1;
209
210 /* prod = (high << 2N) + (high << N); */
211 prodh = high + HHALF(high);
212 prodl = LHUP(high);
213
214 /* if (neg) prod -= mid << N; else prod += mid << N; */
215 if (neg) {
216 was = prodl;
217 prodl -= LHUP(mid);
218 prodh -= HHALF(mid) + (prodl > was);
219 } else {
220 was = prodl;
221 prodl += LHUP(mid);
222 prodh += HHALF(mid) + (prodl < was);
223 }
224
225 /* prod += low << N */
226 was = prodl;
227 prodl += LHUP(low);
228 prodh += HHALF(low) + (prodl < was);
229 /* ... + low; */
230 if ((prodl += low) < low)
231 prodh++;
232
233 /* return 4N-bit product */
234 prod.ul[H] = prodh;
235 prod.ul[L] = prodl;
236 return (prod.q);
237 }
238