1 /* $NetBSD: muldi3.c,v 1.3 2012/08/06 02:31:54 matt Exp $ */
2
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 <sys/cdefs.h>
37 #if defined(LIBC_SCCS) && !defined(lint)
38 #if 0
39 static char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93";
40 #else
41 __RCSID("$NetBSD: muldi3.c,v 1.3 2012/08/06 02:31:54 matt Exp $");
42 #endif
43 #endif /* LIBC_SCCS and not lint */
44
45 #include "quad.h"
46
47 ARM_EABI_ALIAS(__aeabi_lmul, __muldi3) /* no semicolon */
48
49 /*
50 * Multiply two quads.
51 *
52 * Our algorithm is based on the following. Split incoming quad values
53 * u and v (where u,v >= 0) into
54 *
55 * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32)
56 *
57 * and
58 *
59 * v = 2^n v1 * v0
60 *
61 * Then
62 *
63 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
64 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
65 *
66 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
67 * and add 2^n u0 v0 to the last term and subtract it from the middle.
68 * This gives:
69 *
70 * uv = (2^2n + 2^n) (u1 v1) +
71 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
72 * (2^n + 1) (u0 v0)
73 *
74 * Factoring the middle a bit gives us:
75 *
76 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
77 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
78 * (2^n + 1) (u0 v0) [u0v0 = low]
79 *
80 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
81 * in just half the precision of the original. (Note that either or both
82 * of (u1 - u0) or (v0 - v1) may be negative.)
83 *
84 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
85 *
86 * Since C does not give us a `int * int = quad' operator, we split
87 * our input quads into two ints, then split the two ints into two
88 * shorts. We can then calculate `short * short = int' in native
89 * arithmetic.
90 *
91 * Our product should, strictly speaking, be a `long quad', with 128
92 * bits, but we are going to discard the upper 64. In other words,
93 * we are not interested in uv, but rather in (uv mod 2^2n). This
94 * makes some of the terms above vanish, and we get:
95 *
96 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
97 *
98 * or
99 *
100 * (2^n)(high + mid + low) + low
101 *
102 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
103 * of 2^n in either one will also vanish. Only `low' need be computed
104 * mod 2^2n, and only because of the final term above.
105 */
106 static quad_t __lmulq(u_int, u_int);
107
108 quad_t
__muldi3(quad_t a,quad_t b)109 __muldi3(quad_t a, quad_t b)
110 {
111 union uu u, v, low, prod;
112 u_int high, mid, udiff, vdiff;
113 int negall, negmid;
114 #define u1 u.ul[H]
115 #define u0 u.ul[L]
116 #define v1 v.ul[H]
117 #define v0 v.ul[L]
118
119 /*
120 * Get u and v such that u, v >= 0. When this is finished,
121 * u1, u0, v1, and v0 will be directly accessible through the
122 * int fields.
123 */
124 if (a >= 0)
125 u.q = a, negall = 0;
126 else
127 u.q = -a, negall = 1;
128 if (b >= 0)
129 v.q = b;
130 else
131 v.q = -b, negall ^= 1;
132
133 if (u1 == 0 && v1 == 0) {
134 /*
135 * An (I hope) important optimization occurs when u1 and v1
136 * are both 0. This should be common since most numbers
137 * are small. Here the product is just u0*v0.
138 */
139 prod.q = __lmulq(u0, v0);
140 } else {
141 /*
142 * Compute the three intermediate products, remembering
143 * whether the middle term is negative. We can discard
144 * any upper bits in high and mid, so we can use native
145 * u_int * u_int => u_int arithmetic.
146 */
147 low.q = __lmulq(u0, v0);
148
149 if (u1 >= u0)
150 negmid = 0, udiff = u1 - u0;
151 else
152 negmid = 1, udiff = u0 - u1;
153 if (v0 >= v1)
154 vdiff = v0 - v1;
155 else
156 vdiff = v1 - v0, negmid ^= 1;
157 mid = udiff * vdiff;
158
159 high = u1 * v1;
160
161 /*
162 * Assemble the final product.
163 */
164 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
165 low.ul[H];
166 prod.ul[L] = low.ul[L];
167 }
168 return (negall ? -prod.q : prod.q);
169 #undef u1
170 #undef u0
171 #undef v1
172 #undef v0
173 }
174
175 /*
176 * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
177 * the number of bits in an int (whatever that is---the code below
178 * does not care as long as quad.h does its part of the bargain---but
179 * typically N==16).
180 *
181 * We use the same algorithm from Knuth, but this time the modulo refinement
182 * does not apply. On the other hand, since N is half the size of an int,
183 * we can get away with native multiplication---none of our input terms
184 * exceeds (UINT_MAX >> 1).
185 *
186 * Note that, for u_int l, the quad-precision result
187 *
188 * l << N
189 *
190 * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
191 */
192 static quad_t
__lmulq(u_int u,u_int v)193 __lmulq(u_int u, u_int v)
194 {
195 u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
196 u_int prodh, prodl, was;
197 union uu prod;
198 int neg;
199
200 u1 = HHALF(u);
201 u0 = LHALF(u);
202 v1 = HHALF(v);
203 v0 = LHALF(v);
204
205 low = u0 * v0;
206
207 /* This is the same small-number optimization as before. */
208 if (u1 == 0 && v1 == 0)
209 return (low);
210
211 if (u1 >= u0)
212 udiff = u1 - u0, neg = 0;
213 else
214 udiff = u0 - u1, neg = 1;
215 if (v0 >= v1)
216 vdiff = v0 - v1;
217 else
218 vdiff = v1 - v0, neg ^= 1;
219 mid = udiff * vdiff;
220
221 high = u1 * v1;
222
223 /* prod = (high << 2N) + (high << N); */
224 prodh = high + HHALF(high);
225 prodl = LHUP(high);
226
227 /* if (neg) prod -= mid << N; else prod += mid << N; */
228 if (neg) {
229 was = prodl;
230 prodl -= LHUP(mid);
231 prodh -= HHALF(mid) + (prodl > was);
232 } else {
233 was = prodl;
234 prodl += LHUP(mid);
235 prodh += HHALF(mid) + (prodl < was);
236 }
237
238 /* prod += low << N */
239 was = prodl;
240 prodl += LHUP(low);
241 prodh += HHALF(low) + (prodl < was);
242 /* ... + low; */
243 if ((prodl += low) < low)
244 prodh++;
245
246 /* return 4N-bit product */
247 prod.ul[H] = prodh;
248 prod.ul[L] = prodl;
249 return (prod.q);
250 }
251