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 #include "quadint.h" 39 40 #pragma weak __muldi3 = ___muldi3 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 longlong_t __lmulq(ulong_t, ulong_t); 100 101 longlong_t 102 ___muldi3(longlong_t a, longlong_t b) 103 { 104 union uu u, v, low, prod; 105 ulong_t high, mid, udiff, vdiff; 106 int negall, negmid; 107 #define u1 u.ul[H] 108 #define u0 u.ul[L] 109 #define v1 v.ul[H] 110 #define v0 v.ul[L] 111 112 /* 113 * Get u and v such that u, v >= 0. When this is finished, 114 * u1, u0, v1, and v0 will be directly accessible through the 115 * longword fields. 116 */ 117 if (a >= 0) 118 u.q = a, negall = 0; 119 else 120 u.q = -a, negall = 1; 121 if (b >= 0) 122 v.q = b; 123 else 124 v.q = -b, negall ^= 1; 125 126 if (u1 == 0 && v1 == 0) { 127 /* 128 * An (I hope) important optimization occurs when u1 and v1 129 * are both 0. This should be common since most numbers 130 * are small. Here the product is just u0*v0. 131 */ 132 prod.q = __lmulq(u0, v0); 133 } else { 134 /* 135 * Compute the three intermediate products, remembering 136 * whether the middle term is negative. We can discard 137 * any upper bits in high and mid, so we can use native 138 * ulong_t * ulong_t => ulong_t arithmetic. 139 */ 140 low.q = __lmulq(u0, v0); 141 142 if (u1 >= u0) 143 negmid = 0, udiff = u1 - u0; 144 else 145 negmid = 1, udiff = u0 - u1; 146 if (v0 >= v1) 147 vdiff = v0 - v1; 148 else 149 vdiff = v1 - v0, negmid ^= 1; 150 mid = udiff * vdiff; 151 152 high = u1 * v1; 153 154 /* 155 * Assemble the final product. 156 */ 157 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 158 low.ul[H]; 159 prod.ul[L] = low.ul[L]; 160 } 161 return (negall ? -prod.q : prod.q); 162 #undef u1 163 #undef u0 164 #undef v1 165 #undef v0 166 } 167 168 /* 169 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half 170 * the number of bits in a long (whatever that is---the code below 171 * does not care as long as quad.h does its part of the bargain---but 172 * typically N==16). 173 * 174 * We use the same algorithm from Knuth, but this time the modulo refinement 175 * does not apply. On the other hand, since N is half the size of a long, 176 * we can get away with native multiplication---none of our input terms 177 * exceeds (ULONG_MAX >> 1). 178 * 179 * Note that, for ulong_t l, the quad-precision result 180 * 181 * l << N 182 * 183 * splits into high and low longs as HHALF(l) and LHUP(l) respectively. 184 */ 185 static longlong_t 186 __lmulq(ulong_t u, ulong_t v) 187 { 188 ulong_t u1, u0, v1, v0, udiff, vdiff, high, mid, low; 189 ulong_t prodh, prodl, was; 190 union uu prod; 191 int neg; 192 193 u1 = HHALF(u); 194 u0 = LHALF(u); 195 v1 = HHALF(v); 196 v0 = LHALF(v); 197 198 low = u0 * v0; 199 200 /* This is the same small-number optimization as before. */ 201 if (u1 == 0 && v1 == 0) 202 return (low); 203 204 if (u1 >= u0) 205 udiff = u1 - u0, neg = 0; 206 else 207 udiff = u0 - u1, neg = 1; 208 if (v0 >= v1) 209 vdiff = v0 - v1; 210 else 211 vdiff = v1 - v0, neg ^= 1; 212 mid = udiff * vdiff; 213 214 high = u1 * v1; 215 216 /* prod = (high << 2N) + (high << N); */ 217 prodh = high + HHALF(high); 218 prodl = LHUP(high); 219 220 /* if (neg) prod -= mid << N; else prod += mid << N; */ 221 if (neg) { 222 was = prodl; 223 prodl -= LHUP(mid); 224 prodh -= HHALF(mid) + (prodl > was); 225 } else { 226 was = prodl; 227 prodl += LHUP(mid); 228 prodh += HHALF(mid) + (prodl < was); 229 } 230 231 /* prod += low << N */ 232 was = prodl; 233 prodl += LHUP(low); 234 prodh += HHALF(low) + (prodl < was); 235 /* ... + low; */ 236 if ((prodl += low) < low) 237 prodh++; 238 239 /* return 4N-bit product */ 240 prod.ul[H] = prodh; 241 prod.ul[L] = prodl; 242 return (prod.q); 243 } 244