xref: /freebsd/lib/libc/quad/muldi3.c (revision d8a0fe102c0cfdfcd5b818f850eff09d8536c9bc)
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 #if defined(LIBC_SCCS) && !defined(lint)
37 static char sccsid[] = "@(#)muldi3.c	8.1 (Berkeley) 6/4/93";
38 #endif /* LIBC_SCCS and not lint */
39 #include <sys/cdefs.h>
40 __FBSDID("$FreeBSD$");
41 
42 #include "quad.h"
43 
44 /*
45  * Multiply two quads.
46  *
47  * Our algorithm is based on the following.  Split incoming quad values
48  * u and v (where u,v >= 0) into
49  *
50  *	u = 2^n u1  *  u0	(n = number of bits in `u_long', usu. 32)
51  *
52  * and
53  *
54  *	v = 2^n v1  *  v0
55  *
56  * Then
57  *
58  *	uv = 2^2n u1 v1  +  2^n u1 v0  +  2^n v1 u0  +  u0 v0
59  *	   = 2^2n u1 v1  +     2^n (u1 v0 + v1 u0)   +  u0 v0
60  *
61  * Now add 2^n u1 v1 to the first term and subtract it from the middle,
62  * and add 2^n u0 v0 to the last term and subtract it from the middle.
63  * This gives:
64  *
65  *	uv = (2^2n + 2^n) (u1 v1)  +
66  *	         (2^n)    (u1 v0 - u1 v1 + u0 v1 - u0 v0)  +
67  *	       (2^n + 1)  (u0 v0)
68  *
69  * Factoring the middle a bit gives us:
70  *
71  *	uv = (2^2n + 2^n) (u1 v1)  +			[u1v1 = high]
72  *		 (2^n)    (u1 - u0) (v0 - v1)  +	[(u1-u0)... = mid]
73  *	       (2^n + 1)  (u0 v0)			[u0v0 = low]
74  *
75  * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
76  * in just half the precision of the original.  (Note that either or both
77  * of (u1 - u0) or (v0 - v1) may be negative.)
78  *
79  * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
80  *
81  * Since C does not give us a `long * long = quad' operator, we split
82  * our input quads into two longs, then split the two longs into two
83  * shorts.  We can then calculate `short * short = long' in native
84  * arithmetic.
85  *
86  * Our product should, strictly speaking, be a `long quad', with 128
87  * bits, but we are going to discard the upper 64.  In other words,
88  * we are not interested in uv, but rather in (uv mod 2^2n).  This
89  * makes some of the terms above vanish, and we get:
90  *
91  *	(2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
92  *
93  * or
94  *
95  *	(2^n)(high + mid + low) + low
96  *
97  * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
98  * of 2^n in either one will also vanish.  Only `low' need be computed
99  * mod 2^2n, and only because of the final term above.
100  */
101 static quad_t __lmulq(u_long, u_long);
102 
103 quad_t
104 __muldi3(a, b)
105 	quad_t a, b;
106 {
107 	union uu u, v, low, prod;
108 	u_long high, mid, udiff, vdiff;
109 	int negall, negmid;
110 #define	u1	u.ul[H]
111 #define	u0	u.ul[L]
112 #define	v1	v.ul[H]
113 #define	v0	v.ul[L]
114 
115 	/*
116 	 * Get u and v such that u, v >= 0.  When this is finished,
117 	 * u1, u0, v1, and v0 will be directly accessible through the
118 	 * longword fields.
119 	 */
120 	if (a >= 0)
121 		u.q = a, negall = 0;
122 	else
123 		u.q = -a, negall = 1;
124 	if (b >= 0)
125 		v.q = b;
126 	else
127 		v.q = -b, negall ^= 1;
128 
129 	if (u1 == 0 && v1 == 0) {
130 		/*
131 		 * An (I hope) important optimization occurs when u1 and v1
132 		 * are both 0.  This should be common since most numbers
133 		 * are small.  Here the product is just u0*v0.
134 		 */
135 		prod.q = __lmulq(u0, v0);
136 	} else {
137 		/*
138 		 * Compute the three intermediate products, remembering
139 		 * whether the middle term is negative.  We can discard
140 		 * any upper bits in high and mid, so we can use native
141 		 * u_long * u_long => u_long arithmetic.
142 		 */
143 		low.q = __lmulq(u0, v0);
144 
145 		if (u1 >= u0)
146 			negmid = 0, udiff = u1 - u0;
147 		else
148 			negmid = 1, udiff = u0 - u1;
149 		if (v0 >= v1)
150 			vdiff = v0 - v1;
151 		else
152 			vdiff = v1 - v0, negmid ^= 1;
153 		mid = udiff * vdiff;
154 
155 		high = u1 * v1;
156 
157 		/*
158 		 * Assemble the final product.
159 		 */
160 		prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
161 		    low.ul[H];
162 		prod.ul[L] = low.ul[L];
163 	}
164 	return (negall ? -prod.q : prod.q);
165 #undef u1
166 #undef u0
167 #undef v1
168 #undef v0
169 }
170 
171 /*
172  * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half
173  * the number of bits in a long (whatever that is---the code below
174  * does not care as long as quad.h does its part of the bargain---but
175  * typically N==16).
176  *
177  * We use the same algorithm from Knuth, but this time the modulo refinement
178  * does not apply.  On the other hand, since N is half the size of a long,
179  * we can get away with native multiplication---none of our input terms
180  * exceeds (ULONG_MAX >> 1).
181  *
182  * Note that, for u_long l, the quad-precision result
183  *
184  *	l << N
185  *
186  * splits into high and low longs as HHALF(l) and LHUP(l) respectively.
187  */
188 static quad_t
189 __lmulq(u_long u, u_long v)
190 {
191 	u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low;
192 	u_long prodh, prodl, was;
193 	union uu prod;
194 	int neg;
195 
196 	u1 = HHALF(u);
197 	u0 = LHALF(u);
198 	v1 = HHALF(v);
199 	v0 = LHALF(v);
200 
201 	low = u0 * v0;
202 
203 	/* This is the same small-number optimization as before. */
204 	if (u1 == 0 && v1 == 0)
205 		return (low);
206 
207 	if (u1 >= u0)
208 		udiff = u1 - u0, neg = 0;
209 	else
210 		udiff = u0 - u1, neg = 1;
211 	if (v0 >= v1)
212 		vdiff = v0 - v1;
213 	else
214 		vdiff = v1 - v0, neg ^= 1;
215 	mid = udiff * vdiff;
216 
217 	high = u1 * v1;
218 
219 	/* prod = (high << 2N) + (high << N); */
220 	prodh = high + HHALF(high);
221 	prodl = LHUP(high);
222 
223 	/* if (neg) prod -= mid << N; else prod += mid << N; */
224 	if (neg) {
225 		was = prodl;
226 		prodl -= LHUP(mid);
227 		prodh -= HHALF(mid) + (prodl > was);
228 	} else {
229 		was = prodl;
230 		prodl += LHUP(mid);
231 		prodh += HHALF(mid) + (prodl < was);
232 	}
233 
234 	/* prod += low << N */
235 	was = prodl;
236 	prodl += LHUP(low);
237 	prodh += HHALF(low) + (prodl < was);
238 	/* ... + low; */
239 	if ((prodl += low) < low)
240 		prodh++;
241 
242 	/* return 4N-bit product */
243 	prod.ul[H] = prodh;
244 	prod.ul[L] = prodl;
245 	return (prod.q);
246 }
247