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. All advertising materials mentioning features or use of this software
18 * must display the following acknowledgement:
19 * This product includes software developed by the University of
20 * California, Berkeley and its contributors.
21 * 4. Neither the name of the University nor the names of its contributors
22 * may be used to endorse or promote products derived from this software
23 * without specific prior written permission.
24 *
25 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
26 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
27 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
28 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
29 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
30 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
31 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
32 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
33 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
34 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 * SUCH DAMAGE.
36 */
37
38 #include "quadint.h"
39
40 #pragma weak __muldi3 = ___muldi3
41
42 /*
43 * Multiply two quads.
44 *
45 * Our algorithm is based on the following. Split incoming quad values
46 * u and v (where u,v >= 0) into
47 *
48 * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32)
49 *
50 * and
51 *
52 * v = 2^n v1 * v0
53 *
54 * Then
55 *
56 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
57 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
58 *
59 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
60 * and add 2^n u0 v0 to the last term and subtract it from the middle.
61 * This gives:
62 *
63 * uv = (2^2n + 2^n) (u1 v1) +
64 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
65 * (2^n + 1) (u0 v0)
66 *
67 * Factoring the middle a bit gives us:
68 *
69 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
70 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
71 * (2^n + 1) (u0 v0) [u0v0 = low]
72 *
73 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
74 * in just half the precision of the original. (Note that either or both
75 * of (u1 - u0) or (v0 - v1) may be negative.)
76 *
77 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
78 *
79 * Since C does not give us a `long * long = quad' operator, we split
80 * our input quads into two longs, then split the two longs into two
81 * shorts. We can then calculate `short * short = long' in native
82 * arithmetic.
83 *
84 * Our product should, strictly speaking, be a `long quad', with 128
85 * bits, but we are going to discard the upper 64. In other words,
86 * we are not interested in uv, but rather in (uv mod 2^2n). This
87 * makes some of the terms above vanish, and we get:
88 *
89 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
90 *
91 * or
92 *
93 * (2^n)(high + mid + low) + low
94 *
95 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
96 * of 2^n in either one will also vanish. Only `low' need be computed
97 * mod 2^2n, and only because of the final term above.
98 */
99 static longlong_t __lmulq(ulong_t, ulong_t);
100
101 longlong_t
___muldi3(longlong_t a,longlong_t b)102 ___muldi3(longlong_t a, longlong_t b)
103 {
104 union uu u, v, low, prod;
105 ulong_t high, mid, udiff, vdiff;
106 int negall, negmid;
107 #define u1 u.ul[H]
108 #define u0 u.ul[L]
109 #define v1 v.ul[H]
110 #define v0 v.ul[L]
111
112 /*
113 * Get u and v such that u, v >= 0. When this is finished,
114 * u1, u0, v1, and v0 will be directly accessible through the
115 * longword fields.
116 */
117 if (a >= 0)
118 u.q = a, negall = 0;
119 else
120 u.q = -a, negall = 1;
121 if (b >= 0)
122 v.q = b;
123 else
124 v.q = -b, negall ^= 1;
125
126 if (u1 == 0 && v1 == 0) {
127 /*
128 * An (I hope) important optimization occurs when u1 and v1
129 * are both 0. This should be common since most numbers
130 * are small. Here the product is just u0*v0.
131 */
132 prod.q = __lmulq(u0, v0);
133 } else {
134 /*
135 * Compute the three intermediate products, remembering
136 * whether the middle term is negative. We can discard
137 * any upper bits in high and mid, so we can use native
138 * ulong_t * ulong_t => ulong_t arithmetic.
139 */
140 low.q = __lmulq(u0, v0);
141
142 if (u1 >= u0)
143 negmid = 0, udiff = u1 - u0;
144 else
145 negmid = 1, udiff = u0 - u1;
146 if (v0 >= v1)
147 vdiff = v0 - v1;
148 else
149 vdiff = v1 - v0, negmid ^= 1;
150 mid = udiff * vdiff;
151
152 high = u1 * v1;
153
154 /*
155 * Assemble the final product.
156 */
157 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
158 low.ul[H];
159 prod.ul[L] = low.ul[L];
160 }
161 return (negall ? -prod.q : prod.q);
162 #undef u1
163 #undef u0
164 #undef v1
165 #undef v0
166 }
167
168 /*
169 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half
170 * the number of bits in a long (whatever that is---the code below
171 * does not care as long as quad.h does its part of the bargain---but
172 * typically N==16).
173 *
174 * We use the same algorithm from Knuth, but this time the modulo refinement
175 * does not apply. On the other hand, since N is half the size of a long,
176 * we can get away with native multiplication---none of our input terms
177 * exceeds (ULONG_MAX >> 1).
178 *
179 * Note that, for ulong_t l, the quad-precision result
180 *
181 * l << N
182 *
183 * splits into high and low longs as HHALF(l) and LHUP(l) respectively.
184 */
185 static longlong_t
__lmulq(ulong_t u,ulong_t v)186 __lmulq(ulong_t u, ulong_t v)
187 {
188 ulong_t u1, u0, v1, v0, udiff, vdiff, high, mid, low;
189 ulong_t prodh, prodl, was;
190 union uu prod;
191 int neg;
192
193 u1 = HHALF(u);
194 u0 = LHALF(u);
195 v1 = HHALF(v);
196 v0 = LHALF(v);
197
198 low = u0 * v0;
199
200 /* This is the same small-number optimization as before. */
201 if (u1 == 0 && v1 == 0)
202 return (low);
203
204 if (u1 >= u0)
205 udiff = u1 - u0, neg = 0;
206 else
207 udiff = u0 - u1, neg = 1;
208 if (v0 >= v1)
209 vdiff = v0 - v1;
210 else
211 vdiff = v1 - v0, neg ^= 1;
212 mid = udiff * vdiff;
213
214 high = u1 * v1;
215
216 /* prod = (high << 2N) + (high << N); */
217 prodh = high + HHALF(high);
218 prodl = LHUP(high);
219
220 /* if (neg) prod -= mid << N; else prod += mid << N; */
221 if (neg) {
222 was = prodl;
223 prodl -= LHUP(mid);
224 prodh -= HHALF(mid) + (prodl > was);
225 } else {
226 was = prodl;
227 prodl += LHUP(mid);
228 prodh += HHALF(mid) + (prodl < was);
229 }
230
231 /* prod += low << N */
232 was = prodl;
233 prodl += LHUP(low);
234 prodh += HHALF(low) + (prodl < was);
235 /* ... + low; */
236 if ((prodl += low) < low)
237 prodh++;
238
239 /* return 4N-bit product */
240 prod.ul[H] = prodh;
241 prod.ul[L] = prodl;
242 return (prod.q);
243 }
244