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. All advertising materials mentioning features or use of this software 18 * must display the following acknowledgement: 19 * This product includes software developed by the University of 20 * California, Berkeley and its contributors. 21 * 4. 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 #pragma ident "%Z%%M% %I% %E% SMI" 39 40 #include "quadint.h" 41 42 #pragma weak __muldi3 = ___muldi3 43 44 /* 45 * Multiply two quads. 46 * 47 * Our algorithm is based on the following. Split incoming quad values 48 * u and v (where u,v >= 0) into 49 * 50 * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32) 51 * 52 * and 53 * 54 * v = 2^n v1 * v0 55 * 56 * Then 57 * 58 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 59 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 60 * 61 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 62 * and add 2^n u0 v0 to the last term and subtract it from the middle. 63 * This gives: 64 * 65 * uv = (2^2n + 2^n) (u1 v1) + 66 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 67 * (2^n + 1) (u0 v0) 68 * 69 * Factoring the middle a bit gives us: 70 * 71 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 72 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 73 * (2^n + 1) (u0 v0) [u0v0 = low] 74 * 75 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 76 * in just half the precision of the original. (Note that either or both 77 * of (u1 - u0) or (v0 - v1) may be negative.) 78 * 79 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 80 * 81 * Since C does not give us a `long * long = quad' operator, we split 82 * our input quads into two longs, then split the two longs into two 83 * shorts. We can then calculate `short * short = long' in native 84 * arithmetic. 85 * 86 * Our product should, strictly speaking, be a `long quad', with 128 87 * bits, but we are going to discard the upper 64. In other words, 88 * we are not interested in uv, but rather in (uv mod 2^2n). This 89 * makes some of the terms above vanish, and we get: 90 * 91 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 92 * 93 * or 94 * 95 * (2^n)(high + mid + low) + low 96 * 97 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 98 * of 2^n in either one will also vanish. Only `low' need be computed 99 * mod 2^2n, and only because of the final term above. 100 */ 101 static longlong_t __lmulq(ulong_t, ulong_t); 102 103 longlong_t 104 ___muldi3(longlong_t a, longlong_t b) 105 { 106 union uu u, v, low, prod; 107 ulong_t high, mid, udiff, vdiff; 108 int negall, negmid; 109 #define u1 u.ul[H] 110 #define u0 u.ul[L] 111 #define v1 v.ul[H] 112 #define v0 v.ul[L] 113 114 /* 115 * Get u and v such that u, v >= 0. When this is finished, 116 * u1, u0, v1, and v0 will be directly accessible through the 117 * longword fields. 118 */ 119 if (a >= 0) 120 u.q = a, negall = 0; 121 else 122 u.q = -a, negall = 1; 123 if (b >= 0) 124 v.q = b; 125 else 126 v.q = -b, negall ^= 1; 127 128 if (u1 == 0 && v1 == 0) { 129 /* 130 * An (I hope) important optimization occurs when u1 and v1 131 * are both 0. This should be common since most numbers 132 * are small. Here the product is just u0*v0. 133 */ 134 prod.q = __lmulq(u0, v0); 135 } else { 136 /* 137 * Compute the three intermediate products, remembering 138 * whether the middle term is negative. We can discard 139 * any upper bits in high and mid, so we can use native 140 * ulong_t * ulong_t => ulong_t arithmetic. 141 */ 142 low.q = __lmulq(u0, v0); 143 144 if (u1 >= u0) 145 negmid = 0, udiff = u1 - u0; 146 else 147 negmid = 1, udiff = u0 - u1; 148 if (v0 >= v1) 149 vdiff = v0 - v1; 150 else 151 vdiff = v1 - v0, negmid ^= 1; 152 mid = udiff * vdiff; 153 154 high = u1 * v1; 155 156 /* 157 * Assemble the final product. 158 */ 159 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 160 low.ul[H]; 161 prod.ul[L] = low.ul[L]; 162 } 163 return (negall ? -prod.q : prod.q); 164 #undef u1 165 #undef u0 166 #undef v1 167 #undef v0 168 } 169 170 /* 171 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half 172 * the number of bits in a long (whatever that is---the code below 173 * does not care as long as quad.h does its part of the bargain---but 174 * typically N==16). 175 * 176 * We use the same algorithm from Knuth, but this time the modulo refinement 177 * does not apply. On the other hand, since N is half the size of a long, 178 * we can get away with native multiplication---none of our input terms 179 * exceeds (ULONG_MAX >> 1). 180 * 181 * Note that, for ulong_t l, the quad-precision result 182 * 183 * l << N 184 * 185 * splits into high and low longs as HHALF(l) and LHUP(l) respectively. 186 */ 187 static longlong_t 188 __lmulq(ulong_t u, ulong_t v) 189 { 190 ulong_t u1, u0, v1, v0, udiff, vdiff, high, mid, low; 191 ulong_t prodh, prodl, was; 192 union uu prod; 193 int neg; 194 195 u1 = HHALF(u); 196 u0 = LHALF(u); 197 v1 = HHALF(v); 198 v0 = LHALF(v); 199 200 low = u0 * v0; 201 202 /* This is the same small-number optimization as before. */ 203 if (u1 == 0 && v1 == 0) 204 return (low); 205 206 if (u1 >= u0) 207 udiff = u1 - u0, neg = 0; 208 else 209 udiff = u0 - u1, neg = 1; 210 if (v0 >= v1) 211 vdiff = v0 - v1; 212 else 213 vdiff = v1 - v0, neg ^= 1; 214 mid = udiff * vdiff; 215 216 high = u1 * v1; 217 218 /* prod = (high << 2N) + (high << N); */ 219 prodh = high + HHALF(high); 220 prodl = LHUP(high); 221 222 /* if (neg) prod -= mid << N; else prod += mid << N; */ 223 if (neg) { 224 was = prodl; 225 prodl -= LHUP(mid); 226 prodh -= HHALF(mid) + (prodl > was); 227 } else { 228 was = prodl; 229 prodl += LHUP(mid); 230 prodh += HHALF(mid) + (prodl < was); 231 } 232 233 /* prod += low << N */ 234 was = prodl; 235 prodl += LHUP(low); 236 prodh += HHALF(low) + (prodl < was); 237 /* ... + low; */ 238 if ((prodl += low) < low) 239 prodh++; 240 241 /* return 4N-bit product */ 242 prod.ul[H] = prodh; 243 prod.ul[L] = prodl; 244 return (prod.q); 245 } 246