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 <libkern/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 __muldi3(quad_t, quad_t); 100 quad_t 101 __muldi3(quad_t a, quad_t 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 } 243