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 #include "quad.h" 35 36 /* 37 * Multiply two quads. 38 * 39 * Our algorithm is based on the following. Split incoming quad values 40 * u and v (where u,v >= 0) into 41 * 42 * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32) 43 * 44 * and 45 * 46 * v = 2^n v1 * v0 47 * 48 * Then 49 * 50 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 51 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 52 * 53 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 54 * and add 2^n u0 v0 to the last term and subtract it from the middle. 55 * This gives: 56 * 57 * uv = (2^2n + 2^n) (u1 v1) + 58 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 59 * (2^n + 1) (u0 v0) 60 * 61 * Factoring the middle a bit gives us: 62 * 63 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 64 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 65 * (2^n + 1) (u0 v0) [u0v0 = low] 66 * 67 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 68 * in just half the precision of the original. (Note that either or both 69 * of (u1 - u0) or (v0 - v1) may be negative.) 70 * 71 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 72 * 73 * Since C does not give us a `int * int = quad' operator, we split 74 * our input quads into two ints, then split the two ints into two 75 * shorts. We can then calculate `short * short = int' in native 76 * arithmetic. 77 * 78 * Our product should, strictly speaking, be a `long quad', with 128 79 * bits, but we are going to discard the upper 64. In other words, 80 * we are not interested in uv, but rather in (uv mod 2^2n). This 81 * makes some of the terms above vanish, and we get: 82 * 83 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 84 * 85 * or 86 * 87 * (2^n)(high + mid + low) + low 88 * 89 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 90 * of 2^n in either one will also vanish. Only `low' need be computed 91 * mod 2^2n, and only because of the final term above. 92 */ 93 static quad_t __lmulq(u_int, u_int); 94 95 quad_t 96 __muldi3(a, b) 97 quad_t a, b; 98 { 99 union uu u, v, low, prod; 100 u_int high, mid, udiff, vdiff; 101 int negall, negmid; 102 #define u1 u.ul[H] 103 #define u0 u.ul[L] 104 #define v1 v.ul[H] 105 #define v0 v.ul[L] 106 107 /* 108 * Get u and v such that u, v >= 0. When this is finished, 109 * u1, u0, v1, and v0 will be directly accessible through the 110 * int fields. 111 */ 112 if (a >= 0) 113 u.q = a, negall = 0; 114 else 115 u.q = -a, negall = 1; 116 if (b >= 0) 117 v.q = b; 118 else 119 v.q = -b, negall ^= 1; 120 121 if (u1 == 0 && v1 == 0) { 122 /* 123 * An (I hope) important optimization occurs when u1 and v1 124 * are both 0. This should be common since most numbers 125 * are small. Here the product is just u0*v0. 126 */ 127 prod.q = __lmulq(u0, v0); 128 } else { 129 /* 130 * Compute the three intermediate products, remembering 131 * whether the middle term is negative. We can discard 132 * any upper bits in high and mid, so we can use native 133 * u_int * u_int => u_int arithmetic. 134 */ 135 low.q = __lmulq(u0, v0); 136 137 if (u1 >= u0) 138 negmid = 0, udiff = u1 - u0; 139 else 140 negmid = 1, udiff = u0 - u1; 141 if (v0 >= v1) 142 vdiff = v0 - v1; 143 else 144 vdiff = v1 - v0, negmid ^= 1; 145 mid = udiff * vdiff; 146 147 high = u1 * v1; 148 149 /* 150 * Assemble the final product. 151 */ 152 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 153 low.ul[H]; 154 prod.ul[L] = low.ul[L]; 155 } 156 return (negall ? -prod.q : prod.q); 157 #undef u1 158 #undef u0 159 #undef v1 160 #undef v0 161 } 162 163 /* 164 * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half 165 * the number of bits in an int (whatever that is---the code below 166 * does not care as long as quad.h does its part of the bargain---but 167 * typically N==16). 168 * 169 * We use the same algorithm from Knuth, but this time the modulo refinement 170 * does not apply. On the other hand, since N is half the size of an int, 171 * we can get away with native multiplication---none of our input terms 172 * exceeds (UINT_MAX >> 1). 173 * 174 * Note that, for u_int l, the quad-precision result 175 * 176 * l << N 177 * 178 * splits into high and low ints as HHALF(l) and LHUP(l) respectively. 179 */ 180 static quad_t 181 __lmulq(u_int u, u_int v) 182 { 183 u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low; 184 u_int prodh, prodl, was; 185 union uu prod; 186 int neg; 187 188 u1 = HHALF(u); 189 u0 = LHALF(u); 190 v1 = HHALF(v); 191 v0 = LHALF(v); 192 193 low = u0 * v0; 194 195 /* This is the same small-number optimization as before. */ 196 if (u1 == 0 && v1 == 0) 197 return (low); 198 199 if (u1 >= u0) 200 udiff = u1 - u0, neg = 0; 201 else 202 udiff = u0 - u1, neg = 1; 203 if (v0 >= v1) 204 vdiff = v0 - v1; 205 else 206 vdiff = v1 - v0, neg ^= 1; 207 mid = udiff * vdiff; 208 209 high = u1 * v1; 210 211 /* prod = (high << 2N) + (high << N); */ 212 prodh = high + HHALF(high); 213 prodl = LHUP(high); 214 215 /* if (neg) prod -= mid << N; else prod += mid << N; */ 216 if (neg) { 217 was = prodl; 218 prodl -= LHUP(mid); 219 prodh -= HHALF(mid) + (prodl > was); 220 } else { 221 was = prodl; 222 prodl += LHUP(mid); 223 prodh += HHALF(mid) + (prodl < was); 224 } 225 226 /* prod += low << N */ 227 was = prodl; 228 prodl += LHUP(low); 229 prodh += HHALF(low) + (prodl < was); 230 /* ... + low; */ 231 if ((prodl += low) < low) 232 prodh++; 233 234 /* return 4N-bit product */ 235 prod.ul[H] = prodh; 236 prod.ul[L] = prodl; 237 return (prod.q); 238 } 239