1 /*- 2 * Copyright (c) 1992 The Regents of the University of California. 3 * 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 * %sccs.include.redist.c% 10 */ 11 12 #if defined(LIBC_SCCS) && !defined(lint) 13 static char sccsid[] = "@(#)muldi3.c 5.7 (Berkeley) 06/02/92"; 14 #endif /* LIBC_SCCS and not lint */ 15 16 #include "quad.h" 17 18 /* 19 * Multiply two quads. 20 * 21 * Our algorithm is based on the following. Split incoming quad values 22 * u and v (where u,v >= 0) into 23 * 24 * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32) 25 * 26 * and 27 * 28 * v = 2^n v1 * v0 29 * 30 * Then 31 * 32 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 33 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 34 * 35 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 36 * and add 2^n u0 v0 to the last term and subtract it from the middle. 37 * This gives: 38 * 39 * uv = (2^2n + 2^n) (u1 v1) + 40 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 41 * (2^n + 1) (u0 v0) 42 * 43 * Factoring the middle a bit gives us: 44 * 45 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 46 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 47 * (2^n + 1) (u0 v0) [u0v0 = low] 48 * 49 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 50 * in just half the precision of the original. (Note that either or both 51 * of (u1 - u0) or (v0 - v1) may be negative.) 52 * 53 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 54 * 55 * Since C does not give us a `long * long = quad' operator, we split 56 * our input quads into two longs, then split the two longs into two 57 * shorts. We can then calculate `short * short = long' in native 58 * arithmetic. 59 * 60 * Our product should, strictly speaking, be a `long quad', with 128 61 * bits, but we are going to discard the upper 64. In other words, 62 * we are not interested in uv, but rather in (uv mod 2^2n). This 63 * makes some of the terms above vanish, and we get: 64 * 65 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 66 * 67 * or 68 * 69 * (2^n)(high + mid + low) + low 70 * 71 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 72 * of 2^n in either one will also vanish. Only `low' need be computed 73 * mod 2^2n, and only because of the final term above. 74 */ 75 static quad __lmulq(u_long, u_long); 76 77 quad 78 __muldi3(quad a, quad b) 79 { 80 union uu u, v, low, prod; 81 register u_long high, mid, udiff, vdiff; 82 register int negall, negmid; 83 #define u1 u.ul[H] 84 #define u0 u.ul[L] 85 #define v1 v.ul[H] 86 #define v0 v.ul[L] 87 88 /* 89 * Get u and v such that u, v >= 0. When this is finished, 90 * u1, u0, v1, and v0 will be directly accessible through the 91 * longword fields. 92 */ 93 if (a >= 0) 94 u.q = a, negall = 0; 95 else 96 u.q = -a, negall = 1; 97 if (b >= 0) 98 v.q = b; 99 else 100 v.q = -b, negall ^= 1; 101 102 if (u1 == 0 && v1 == 0) { 103 /* 104 * An (I hope) important optimization occurs when u1 and v1 105 * are both 0. This should be common since most numbers 106 * are small. Here the product is just u0*v0. 107 */ 108 prod.q = __lmulq(u0, v0); 109 } else { 110 /* 111 * Compute the three intermediate products, remembering 112 * whether the middle term is negative. We can discard 113 * any upper bits in high and mid, so we can use native 114 * u_long * u_long => u_long arithmetic. 115 */ 116 low.q = __lmulq(u0, v0); 117 118 if (u1 >= u0) 119 negmid = 0, udiff = u1 - u0; 120 else 121 negmid = 1, udiff = u0 - u1; 122 if (v0 >= v1) 123 vdiff = v0 - v1; 124 else 125 vdiff = v1 - v0, negmid ^= 1; 126 mid = udiff * vdiff; 127 128 high = u1 * v1; 129 130 /* 131 * Assemble the final product. 132 */ 133 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 134 low.ul[H]; 135 prod.ul[L] = low.ul[L]; 136 } 137 return (negall ? -prod.q : prod.q); 138 #undef u1 139 #undef u0 140 #undef v1 141 #undef v0 142 } 143 144 /* 145 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half 146 * the number of bits in a long (whatever that is---the code below 147 * does not care as long as quad.h does its part of the bargain---but 148 * typically N==16). 149 * 150 * We use the same algorithm from Knuth, but this time the modulo refinement 151 * does not apply. On the other hand, since N is half the size of a long, 152 * we can get away with native multiplication---none of our input terms 153 * exceeds (ULONG_MAX >> 1). 154 * 155 * Note that, for u_long l, the quad-precision result 156 * 157 * l << N 158 * 159 * splits into high and low longs as HHALF(l) and LHUP(l) respectively. 160 */ 161 static quad 162 __lmulq(u_long u, u_long v) 163 { 164 u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low; 165 u_long prodh, prodl, was; 166 union uu prod; 167 int neg; 168 169 u1 = HHALF(u); 170 u0 = LHALF(u); 171 v1 = HHALF(v); 172 v0 = LHALF(v); 173 174 low = u0 * v0; 175 176 /* This is the same small-number optimization as before. */ 177 if (u1 == 0 && v1 == 0) 178 return (low); 179 180 if (u1 >= u0) 181 udiff = u1 - u0, neg = 0; 182 else 183 udiff = u0 - u1, neg = 1; 184 if (v0 >= v1) 185 vdiff = v0 - v1; 186 else 187 vdiff = v1 - v0, neg ^= 1; 188 mid = udiff * vdiff; 189 190 high = u1 * v1; 191 192 /* prod = (high << 2N) + (high << N); */ 193 prodh = high + HHALF(high); 194 prodl = LHUP(high); 195 196 /* if (neg) prod -= mid << N; else prod += mid << N; */ 197 if (neg) { 198 was = prodl; 199 prodl -= LHUP(mid); 200 prodh -= HHALF(mid) + (prodl > was); 201 } else { 202 was = prodl; 203 prodl += LHUP(mid); 204 prodh += HHALF(mid) + (prodl < was); 205 } 206 207 /* prod += low << N */ 208 was = prodl; 209 prodl += LHUP(low); 210 prodh += HHALF(low) + (prodl < was); 211 /* ... + low; */ 212 if ((prodl += low) < low) 213 prodh++; 214 215 /* return 4N-bit product */ 216 prod.ul[H] = prodh; 217 prod.ul[L] = prodl; 218 return (prod.q); 219 } 220