1 /* $NetBSD: muldi3.c,v 1.8 2003/08/07 16:32:09 agc Exp $ */ 2 3 /*- 4 * SPDX-License-Identifier: BSD-3-Clause 5 * 6 * Copyright (c) 1992, 1993 7 * The Regents of the University of California. All rights reserved. 8 * 9 * This software was developed by the Computer Systems Engineering group 10 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and 11 * contributed to Berkeley. 12 * 13 * Redistribution and use in source and binary forms, with or without 14 * modification, are permitted provided that the following conditions 15 * are met: 16 * 1. Redistributions of source code must retain the above copyright 17 * notice, this list of conditions and the following disclaimer. 18 * 2. Redistributions in binary form must reproduce the above copyright 19 * notice, this list of conditions and the following disclaimer in the 20 * documentation and/or other materials provided with the distribution. 21 * 3. 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 <sys/cdefs.h> 39 #if defined(LIBC_SCCS) && !defined(lint) 40 #if 0 41 static char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93"; 42 #else 43 #endif 44 #endif /* LIBC_SCCS and not lint */ 45 46 #include <libkern/quad.h> 47 48 /* 49 * Multiply two quads. 50 * 51 * Our algorithm is based on the following. Split incoming quad values 52 * u and v (where u,v >= 0) into 53 * 54 * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32) 55 * 56 * and 57 * 58 * v = 2^n v1 * v0 59 * 60 * Then 61 * 62 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 63 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 64 * 65 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 66 * and add 2^n u0 v0 to the last term and subtract it from the middle. 67 * This gives: 68 * 69 * uv = (2^2n + 2^n) (u1 v1) + 70 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 71 * (2^n + 1) (u0 v0) 72 * 73 * Factoring the middle a bit gives us: 74 * 75 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 76 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 77 * (2^n + 1) (u0 v0) [u0v0 = low] 78 * 79 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 80 * in just half the precision of the original. (Note that either or both 81 * of (u1 - u0) or (v0 - v1) may be negative.) 82 * 83 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 84 * 85 * Since C does not give us a `int * int = quad' operator, we split 86 * our input quads into two ints, then split the two ints into two 87 * shorts. We can then calculate `short * short = int' in native 88 * arithmetic. 89 * 90 * Our product should, strictly speaking, be a `long quad', with 128 91 * bits, but we are going to discard the upper 64. In other words, 92 * we are not interested in uv, but rather in (uv mod 2^2n). This 93 * makes some of the terms above vanish, and we get: 94 * 95 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 96 * 97 * or 98 * 99 * (2^n)(high + mid + low) + low 100 * 101 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 102 * of 2^n in either one will also vanish. Only `low' need be computed 103 * mod 2^2n, and only because of the final term above. 104 */ 105 static quad_t __lmulq(u_int, u_int); 106 107 quad_t __muldi3(quad_t, quad_t); 108 quad_t 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 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