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 sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93"; 36 #endif /* LIBC_SCCS and not lint */ 37 #include <sys/cdefs.h> 38 __FBSDID("$FreeBSD$"); 39 40 #include "quad.h" 41 42 /* 43 * Multiply two quads. 44 * 45 * Our algorithm is based on the following. Split incoming quad values 46 * u and v (where u,v >= 0) into 47 * 48 * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32) 49 * 50 * and 51 * 52 * v = 2^n v1 * v0 53 * 54 * Then 55 * 56 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 57 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 58 * 59 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 60 * and add 2^n u0 v0 to the last term and subtract it from the middle. 61 * This gives: 62 * 63 * uv = (2^2n + 2^n) (u1 v1) + 64 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 65 * (2^n + 1) (u0 v0) 66 * 67 * Factoring the middle a bit gives us: 68 * 69 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 70 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 71 * (2^n + 1) (u0 v0) [u0v0 = low] 72 * 73 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 74 * in just half the precision of the original. (Note that either or both 75 * of (u1 - u0) or (v0 - v1) may be negative.) 76 * 77 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 78 * 79 * Since C does not give us a `long * long = quad' operator, we split 80 * our input quads into two longs, then split the two longs into two 81 * shorts. We can then calculate `short * short = long' in native 82 * arithmetic. 83 * 84 * Our product should, strictly speaking, be a `long quad', with 128 85 * bits, but we are going to discard the upper 64. In other words, 86 * we are not interested in uv, but rather in (uv mod 2^2n). This 87 * makes some of the terms above vanish, and we get: 88 * 89 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 90 * 91 * or 92 * 93 * (2^n)(high + mid + low) + low 94 * 95 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 96 * of 2^n in either one will also vanish. Only `low' need be computed 97 * mod 2^2n, and only because of the final term above. 98 */ 99 static quad_t __lmulq(u_long, u_long); 100 101 quad_t 102 __muldi3(a, b) 103 quad_t a, b; 104 { 105 union uu u, v, low, prod; 106 u_long high, mid, udiff, vdiff; 107 int negall, negmid; 108 #define u1 u.ul[H] 109 #define u0 u.ul[L] 110 #define v1 v.ul[H] 111 #define v0 v.ul[L] 112 113 /* 114 * Get u and v such that u, v >= 0. When this is finished, 115 * u1, u0, v1, and v0 will be directly accessible through the 116 * longword fields. 117 */ 118 if (a >= 0) 119 u.q = a, negall = 0; 120 else 121 u.q = -a, negall = 1; 122 if (b >= 0) 123 v.q = b; 124 else 125 v.q = -b, negall ^= 1; 126 127 if (u1 == 0 && v1 == 0) { 128 /* 129 * An (I hope) important optimization occurs when u1 and v1 130 * are both 0. This should be common since most numbers 131 * are small. Here the product is just u0*v0. 132 */ 133 prod.q = __lmulq(u0, v0); 134 } else { 135 /* 136 * Compute the three intermediate products, remembering 137 * whether the middle term is negative. We can discard 138 * any upper bits in high and mid, so we can use native 139 * u_long * u_long => u_long arithmetic. 140 */ 141 low.q = __lmulq(u0, v0); 142 143 if (u1 >= u0) 144 negmid = 0, udiff = u1 - u0; 145 else 146 negmid = 1, udiff = u0 - u1; 147 if (v0 >= v1) 148 vdiff = v0 - v1; 149 else 150 vdiff = v1 - v0, negmid ^= 1; 151 mid = udiff * vdiff; 152 153 high = u1 * v1; 154 155 /* 156 * Assemble the final product. 157 */ 158 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 159 low.ul[H]; 160 prod.ul[L] = low.ul[L]; 161 } 162 return (negall ? -prod.q : prod.q); 163 #undef u1 164 #undef u0 165 #undef v1 166 #undef v0 167 } 168 169 /* 170 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half 171 * the number of bits in a long (whatever that is---the code below 172 * does not care as long as quad.h does its part of the bargain---but 173 * typically N==16). 174 * 175 * We use the same algorithm from Knuth, but this time the modulo refinement 176 * does not apply. On the other hand, since N is half the size of a long, 177 * we can get away with native multiplication---none of our input terms 178 * exceeds (ULONG_MAX >> 1). 179 * 180 * Note that, for u_long l, the quad-precision result 181 * 182 * l << N 183 * 184 * splits into high and low longs as HHALF(l) and LHUP(l) respectively. 185 */ 186 static quad_t 187 __lmulq(u_long u, u_long v) 188 { 189 u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low; 190 u_long prodh, prodl, was; 191 union uu prod; 192 int neg; 193 194 u1 = HHALF(u); 195 u0 = LHALF(u); 196 v1 = HHALF(v); 197 v0 = LHALF(v); 198 199 low = u0 * v0; 200 201 /* This is the same small-number optimization as before. */ 202 if (u1 == 0 && v1 == 0) 203 return (low); 204 205 if (u1 >= u0) 206 udiff = u1 - u0, neg = 0; 207 else 208 udiff = u0 - u1, neg = 1; 209 if (v0 >= v1) 210 vdiff = v0 - v1; 211 else 212 vdiff = v1 - v0, neg ^= 1; 213 mid = udiff * vdiff; 214 215 high = u1 * v1; 216 217 /* prod = (high << 2N) + (high << N); */ 218 prodh = high + HHALF(high); 219 prodl = LHUP(high); 220 221 /* if (neg) prod -= mid << N; else prod += mid << N; */ 222 if (neg) { 223 was = prodl; 224 prodl -= LHUP(mid); 225 prodh -= HHALF(mid) + (prodl > was); 226 } else { 227 was = prodl; 228 prodl += LHUP(mid); 229 prodh += HHALF(mid) + (prodl < was); 230 } 231 232 /* prod += low << N */ 233 was = prodl; 234 prodl += LHUP(low); 235 prodh += HHALF(low) + (prodl < was); 236 /* ... + low; */ 237 if ((prodl += low) < low) 238 prodh++; 239 240 /* return 4N-bit product */ 241 prod.ul[H] = prodh; 242 prod.ul[L] = prodl; 243 return (prod.q); 244 } 245