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