Annotation of sys/lib/libkern/muldi3.c, Revision 1.1.1.1
1.1 nbrk 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: }
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