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 /* 41 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed), 42 * section 4.3.1, pp. 257--259. 43 */ 44 45 #include "quadint.h" 46 47 #define B (1 << HALF_BITS) /* digit base */ 48 49 /* Combine two `digits' to make a single two-digit number. */ 50 #define COMBINE(a, b) (((ulong_t)(a) << HALF_BITS) | (b)) 51 52 /* select a type for digits in base B: use unsigned short if they fit */ 53 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff 54 typedef unsigned short digit; 55 #else 56 typedef ulong_t digit; 57 #endif 58 59 /* 60 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that 61 * `fall out' the left (there never will be any such anyway). 62 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS. 63 */ 64 static void 65 shl(digit *p, int len, int sh) 66 { 67 int i; 68 69 for (i = 0; i < len; i++) 70 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh)); 71 p[i] = LHALF(p[i] << sh); 72 } 73 74 /* 75 * ___qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v. 76 * 77 * We do this in base 2-sup-HALF_BITS, so that all intermediate products 78 * fit within ulong_t. As a consequence, the maximum length dividend and 79 * divisor are 4 `digits' in this base (they are shorter if they have 80 * leading zeros). 81 */ 82 u_longlong_t 83 ___qdivrem(u_longlong_t uq, u_longlong_t vq, u_longlong_t *arq) 84 { 85 union uu tmp; 86 digit *u, *v, *q; 87 digit v1, v2; 88 ulong_t qhat, rhat, t; 89 int m, n, d, j, i; 90 digit uspace[5], vspace[5], qspace[5]; 91 92 /* 93 * Take care of special cases: divide by zero, and u < v. 94 */ 95 if (vq == 0) { 96 /* divide by zero. */ 97 static volatile const unsigned int zero = 0; 98 99 tmp.ul[H] = tmp.ul[L] = 1 / zero; 100 if (arq) 101 *arq = uq; 102 return (tmp.q); 103 } 104 if (uq < vq) { 105 if (arq) 106 *arq = uq; 107 return (0); 108 } 109 u = &uspace[0]; 110 v = &vspace[0]; 111 q = &qspace[0]; 112 113 /* 114 * Break dividend and divisor into digits in base B, then 115 * count leading zeros to determine m and n. When done, we 116 * will have: 117 * u = (u[1]u[2]...u[m+n]) sub B 118 * v = (v[1]v[2]...v[n]) sub B 119 * v[1] != 0 120 * 1 < n <= 4 (if n = 1, we use a different division algorithm) 121 * m >= 0 (otherwise u < v, which we already checked) 122 * m + n = 4 123 * and thus 124 * m = 4 - n <= 2 125 */ 126 tmp.uq = uq; 127 u[0] = 0; 128 u[1] = HHALF(tmp.ul[H]); 129 u[2] = LHALF(tmp.ul[H]); 130 u[3] = HHALF(tmp.ul[L]); 131 u[4] = LHALF(tmp.ul[L]); 132 tmp.uq = vq; 133 v[1] = HHALF(tmp.ul[H]); 134 v[2] = LHALF(tmp.ul[H]); 135 v[3] = HHALF(tmp.ul[L]); 136 v[4] = LHALF(tmp.ul[L]); 137 for (n = 4; v[1] == 0; v++) { 138 if (--n == 1) { 139 ulong_t rbj; /* r*B+u[j] (not root boy jim) */ 140 digit q1, q2, q3, q4; 141 142 /* 143 * Change of plan, per exercise 16. 144 * r = 0; 145 * for j = 1..4: 146 * q[j] = floor((r*B + u[j]) / v), 147 * r = (r*B + u[j]) % v; 148 * We unroll this completely here. 149 */ 150 t = v[2]; /* nonzero, by definition */ 151 q1 = u[1] / t; 152 rbj = COMBINE(u[1] % t, u[2]); 153 q2 = rbj / t; 154 rbj = COMBINE(rbj % t, u[3]); 155 q3 = rbj / t; 156 rbj = COMBINE(rbj % t, u[4]); 157 q4 = rbj / t; 158 if (arq) 159 *arq = rbj % t; 160 tmp.ul[H] = COMBINE(q1, q2); 161 tmp.ul[L] = COMBINE(q3, q4); 162 return (tmp.q); 163 } 164 } 165 166 /* 167 * By adjusting q once we determine m, we can guarantee that 168 * there is a complete four-digit quotient at &qspace[1] when 169 * we finally stop. 170 */ 171 for (m = 4 - n; u[1] == 0; u++) 172 m--; 173 for (i = 4 - m; --i >= 0; ) 174 q[i] = 0; 175 q += 4 - m; 176 177 /* 178 * Here we run Program D, translated from MIX to C and acquiring 179 * a few minor changes. 180 * 181 * D1: choose multiplier 1 << d to ensure v[1] >= B/2. 182 */ 183 d = 0; 184 for (t = v[1]; t < B / 2; t <<= 1) 185 d++; 186 if (d > 0) { 187 shl(&u[0], m + n, d); /* u <<= d */ 188 shl(&v[1], n - 1, d); /* v <<= d */ 189 } 190 /* 191 * D2: j = 0. 192 */ 193 j = 0; 194 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */ 195 v2 = v[2]; /* for D3 */ 196 do { 197 digit uj0, uj1, uj2; 198 199 /* 200 * D3: Calculate qhat (\^q, in TeX notation). 201 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and 202 * let rhat = (u[j]*B + u[j+1]) mod v[1]. 203 * While rhat < B and v[2]*qhat > rhat*B+u[j+2], 204 * decrement qhat and increase rhat correspondingly. 205 * Note that if rhat >= B, v[2]*qhat < rhat*B. 206 */ 207 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */ 208 uj1 = u[j + 1]; /* for D3 only */ 209 uj2 = u[j + 2]; /* for D3 only */ 210 if (uj0 == v1) { 211 qhat = B; 212 rhat = uj1; 213 goto qhat_too_big; 214 } else { 215 ulong_t n = COMBINE(uj0, uj1); 216 qhat = n / v1; 217 rhat = n % v1; 218 } 219 while (v2 * qhat > COMBINE(rhat, uj2)) { 220 qhat_too_big: 221 qhat--; 222 if ((rhat += v1) >= B) 223 break; 224 } 225 /* 226 * D4: Multiply and subtract. 227 * The variable `t' holds any borrows across the loop. 228 * We split this up so that we do not require v[0] = 0, 229 * and to eliminate a final special case. 230 */ 231 for (t = 0, i = n; i > 0; i--) { 232 t = u[i + j] - v[i] * qhat - t; 233 u[i + j] = LHALF(t); 234 t = (B - HHALF(t)) & (B - 1); 235 } 236 t = u[j] - t; 237 u[j] = LHALF(t); 238 /* 239 * D5: test remainder. 240 * There is a borrow if and only if HHALF(t) is nonzero; 241 * in that (rare) case, qhat was too large (by exactly 1). 242 * Fix it by adding v[1..n] to u[j..j+n]. 243 */ 244 if (HHALF(t)) { 245 qhat--; 246 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */ 247 t += u[i + j] + v[i]; 248 u[i + j] = LHALF(t); 249 t = HHALF(t); 250 } 251 u[j] = LHALF(u[j] + t); 252 } 253 q[j] = (digit)qhat; 254 } while (++j <= m); /* D7: loop on j. */ 255 256 /* 257 * If caller wants the remainder, we have to calculate it as 258 * u[m..m+n] >> d (this is at most n digits and thus fits in 259 * u[m+1..m+n], but we may need more source digits). 260 */ 261 if (arq) { 262 if (d) { 263 for (i = m + n; i > m; --i) 264 u[i] = (u[i] >> d) | 265 LHALF(u[i - 1] << (HALF_BITS - d)); 266 u[i] = 0; 267 } 268 tmp.ul[H] = COMBINE(uspace[1], uspace[2]); 269 tmp.ul[L] = COMBINE(uspace[3], uspace[4]); 270 *arq = tmp.q; 271 } 272 273 tmp.ul[H] = COMBINE(qspace[1], qspace[2]); 274 tmp.ul[L] = COMBINE(qspace[3], qspace[4]); 275 return (tmp.q); 276 } 277