muldi3.c revision 330897
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 41static char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93"; 42#else 43__FBSDID("$FreeBSD: stable/11/sys/libkern/arm/muldi3.c 330897 2018-03-14 03:19:51Z eadler $"); 44#endif 45#endif /* LIBC_SCCS and not lint */ 46 47#include <libkern/quad.h> 48 49/* 50 * Multiply two quads. 51 * 52 * Our algorithm is based on the following. Split incoming quad values 53 * u and v (where u,v >= 0) into 54 * 55 * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32) 56 * 57 * and 58 * 59 * v = 2^n v1 * v0 60 * 61 * Then 62 * 63 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 64 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 65 * 66 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 67 * and add 2^n u0 v0 to the last term and subtract it from the middle. 68 * This gives: 69 * 70 * uv = (2^2n + 2^n) (u1 v1) + 71 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 72 * (2^n + 1) (u0 v0) 73 * 74 * Factoring the middle a bit gives us: 75 * 76 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 77 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 78 * (2^n + 1) (u0 v0) [u0v0 = low] 79 * 80 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 81 * in just half the precision of the original. (Note that either or both 82 * of (u1 - u0) or (v0 - v1) may be negative.) 83 * 84 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 85 * 86 * Since C does not give us a `int * int = quad' operator, we split 87 * our input quads into two ints, then split the two ints into two 88 * shorts. We can then calculate `short * short = int' in native 89 * arithmetic. 90 * 91 * Our product should, strictly speaking, be a `long quad', with 128 92 * bits, but we are going to discard the upper 64. In other words, 93 * we are not interested in uv, but rather in (uv mod 2^2n). This 94 * makes some of the terms above vanish, and we get: 95 * 96 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 97 * 98 * or 99 * 100 * (2^n)(high + mid + low) + low 101 * 102 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 103 * of 2^n in either one will also vanish. Only `low' need be computed 104 * mod 2^2n, and only because of the final term above. 105 */ 106static quad_t __lmulq(u_int, u_int); 107 108quad_t __muldi3(quad_t, quad_t); 109quad_t 110__muldi3(quad_t a, quad_t b) 111{ 112 union uu u, v, low, prod; 113 u_int high, mid, udiff, vdiff; 114 int negall, negmid; 115#define u1 u.ul[H] 116#define u0 u.ul[L] 117#define v1 v.ul[H] 118#define v0 v.ul[L] 119 120 /* 121 * Get u and v such that u, v >= 0. When this is finished, 122 * u1, u0, v1, and v0 will be directly accessible through the 123 * int fields. 124 */ 125 if (a >= 0) 126 u.q = a, negall = 0; 127 else 128 u.q = -a, negall = 1; 129 if (b >= 0) 130 v.q = b; 131 else 132 v.q = -b, negall ^= 1; 133 134 if (u1 == 0 && v1 == 0) { 135 /* 136 * An (I hope) important optimization occurs when u1 and v1 137 * are both 0. This should be common since most numbers 138 * are small. Here the product is just u0*v0. 139 */ 140 prod.q = __lmulq(u0, v0); 141 } else { 142 /* 143 * Compute the three intermediate products, remembering 144 * whether the middle term is negative. We can discard 145 * any upper bits in high and mid, so we can use native 146 * u_int * u_int => u_int arithmetic. 147 */ 148 low.q = __lmulq(u0, v0); 149 150 if (u1 >= u0) 151 negmid = 0, udiff = u1 - u0; 152 else 153 negmid = 1, udiff = u0 - u1; 154 if (v0 >= v1) 155 vdiff = v0 - v1; 156 else 157 vdiff = v1 - v0, negmid ^= 1; 158 mid = udiff * vdiff; 159 160 high = u1 * v1; 161 162 /* 163 * Assemble the final product. 164 */ 165 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 166 low.ul[H]; 167 prod.ul[L] = low.ul[L]; 168 } 169 return (negall ? -prod.q : prod.q); 170#undef u1 171#undef u0 172#undef v1 173#undef v0 174} 175 176/* 177 * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half 178 * the number of bits in an int (whatever that is---the code below 179 * does not care as long as quad.h does its part of the bargain---but 180 * typically N==16). 181 * 182 * We use the same algorithm from Knuth, but this time the modulo refinement 183 * does not apply. On the other hand, since N is half the size of an int, 184 * we can get away with native multiplication---none of our input terms 185 * exceeds (UINT_MAX >> 1). 186 * 187 * Note that, for u_int l, the quad-precision result 188 * 189 * l << N 190 * 191 * splits into high and low ints as HHALF(l) and LHUP(l) respectively. 192 */ 193static quad_t 194__lmulq(u_int u, u_int v) 195{ 196 u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low; 197 u_int prodh, prodl, was; 198 union uu prod; 199 int neg; 200 201 u1 = HHALF(u); 202 u0 = LHALF(u); 203 v1 = HHALF(v); 204 v0 = LHALF(v); 205 206 low = u0 * v0; 207 208 /* This is the same small-number optimization as before. */ 209 if (u1 == 0 && v1 == 0) 210 return (low); 211 212 if (u1 >= u0) 213 udiff = u1 - u0, neg = 0; 214 else 215 udiff = u0 - u1, neg = 1; 216 if (v0 >= v1) 217 vdiff = v0 - v1; 218 else 219 vdiff = v1 - v0, neg ^= 1; 220 mid = udiff * vdiff; 221 222 high = u1 * v1; 223 224 /* prod = (high << 2N) + (high << N); */ 225 prodh = high + HHALF(high); 226 prodl = LHUP(high); 227 228 /* if (neg) prod -= mid << N; else prod += mid << N; */ 229 if (neg) { 230 was = prodl; 231 prodl -= LHUP(mid); 232 prodh -= HHALF(mid) + (prodl > was); 233 } else { 234 was = prodl; 235 prodl += LHUP(mid); 236 prodh += HHALF(mid) + (prodl < was); 237 } 238 239 /* prod += low << N */ 240 was = prodl; 241 prodl += LHUP(low); 242 prodh += HHALF(low) + (prodl < was); 243 /* ... + low; */ 244 if ((prodl += low) < low) 245 prodh++; 246 247 /* return 4N-bit product */ 248 prod.ul[H] = prodh; 249 prod.ul[L] = prodl; 250 return (prod.q); 251} 252