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 * 4. 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 #if defined(LIBC_SCCS) && !defined(lint) 35 static char sccsid[] = "@(#)qdivrem.c 8.1 (Berkeley) 6/4/93"; 36 #endif /* LIBC_SCCS and not lint */ 37 #include <sys/cdefs.h> 38 __FBSDID("$FreeBSD$"); 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 "quad.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) (((u_long)(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 u_long 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 u_long. 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_quad_t 83 __qdivrem(uq, vq, arq) 84 u_quad_t uq, vq, *arq; 85 { 86 union uu tmp; 87 digit *u, *v, *q; 88 digit v1, v2; 89 u_long qhat, rhat, t; 90 int m, n, d, j, i; 91 digit uspace[5], vspace[5], qspace[5]; 92 93 /* 94 * Take care of special cases: divide by zero, and u < v. 95 */ 96 if (vq == 0) { 97 /* divide by zero. */ 98 static volatile const unsigned int zero = 0; 99 100 tmp.ul[H] = tmp.ul[L] = 1 / zero; 101 if (arq) 102 *arq = uq; 103 return (tmp.q); 104 } 105 if (uq < vq) { 106 if (arq) 107 *arq = uq; 108 return (0); 109 } 110 u = &uspace[0]; 111 v = &vspace[0]; 112 q = &qspace[0]; 113 114 /* 115 * Break dividend and divisor into digits in base B, then 116 * count leading zeros to determine m and n. When done, we 117 * will have: 118 * u = (u[1]u[2]...u[m+n]) sub B 119 * v = (v[1]v[2]...v[n]) sub B 120 * v[1] != 0 121 * 1 < n <= 4 (if n = 1, we use a different division algorithm) 122 * m >= 0 (otherwise u < v, which we already checked) 123 * m + n = 4 124 * and thus 125 * m = 4 - n <= 2 126 */ 127 tmp.uq = uq; 128 u[0] = 0; 129 u[1] = HHALF(tmp.ul[H]); 130 u[2] = LHALF(tmp.ul[H]); 131 u[3] = HHALF(tmp.ul[L]); 132 u[4] = LHALF(tmp.ul[L]); 133 tmp.uq = vq; 134 v[1] = HHALF(tmp.ul[H]); 135 v[2] = LHALF(tmp.ul[H]); 136 v[3] = HHALF(tmp.ul[L]); 137 v[4] = LHALF(tmp.ul[L]); 138 for (n = 4; v[1] == 0; v++) { 139 if (--n == 1) { 140 u_long rbj; /* r*B+u[j] (not root boy jim) */ 141 digit q1, q2, q3, q4; 142 143 /* 144 * Change of plan, per exercise 16. 145 * r = 0; 146 * for j = 1..4: 147 * q[j] = floor((r*B + u[j]) / v), 148 * r = (r*B + u[j]) % v; 149 * We unroll this completely here. 150 */ 151 t = v[2]; /* nonzero, by definition */ 152 q1 = u[1] / t; 153 rbj = COMBINE(u[1] % t, u[2]); 154 q2 = rbj / t; 155 rbj = COMBINE(rbj % t, u[3]); 156 q3 = rbj / t; 157 rbj = COMBINE(rbj % t, u[4]); 158 q4 = rbj / t; 159 if (arq) 160 *arq = rbj % t; 161 tmp.ul[H] = COMBINE(q1, q2); 162 tmp.ul[L] = COMBINE(q3, q4); 163 return (tmp.q); 164 } 165 } 166 167 /* 168 * By adjusting q once we determine m, we can guarantee that 169 * there is a complete four-digit quotient at &qspace[1] when 170 * we finally stop. 171 */ 172 for (m = 4 - n; u[1] == 0; u++) 173 m--; 174 for (i = 4 - m; --i >= 0;) 175 q[i] = 0; 176 q += 4 - m; 177 178 /* 179 * Here we run Program D, translated from MIX to C and acquiring 180 * a few minor changes. 181 * 182 * D1: choose multiplier 1 << d to ensure v[1] >= B/2. 183 */ 184 d = 0; 185 for (t = v[1]; t < B / 2; t <<= 1) 186 d++; 187 if (d > 0) { 188 shl(&u[0], m + n, d); /* u <<= d */ 189 shl(&v[1], n - 1, d); /* v <<= d */ 190 } 191 /* 192 * D2: j = 0. 193 */ 194 j = 0; 195 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */ 196 v2 = v[2]; /* for D3 */ 197 do { 198 digit uj0, uj1, uj2; 199 200 /* 201 * D3: Calculate qhat (\^q, in TeX notation). 202 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and 203 * let rhat = (u[j]*B + u[j+1]) mod v[1]. 204 * While rhat < B and v[2]*qhat > rhat*B+u[j+2], 205 * decrement qhat and increase rhat correspondingly. 206 * Note that if rhat >= B, v[2]*qhat < rhat*B. 207 */ 208 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */ 209 uj1 = u[j + 1]; /* for D3 only */ 210 uj2 = u[j + 2]; /* for D3 only */ 211 if (uj0 == v1) { 212 qhat = B; 213 rhat = uj1; 214 goto qhat_too_big; 215 } else { 216 u_long n = COMBINE(uj0, uj1); 217 qhat = n / v1; 218 rhat = n % v1; 219 } 220 while (v2 * qhat > COMBINE(rhat, uj2)) { 221 qhat_too_big: 222 qhat--; 223 if ((rhat += v1) >= B) 224 break; 225 } 226 /* 227 * D4: Multiply and subtract. 228 * The variable `t' holds any borrows across the loop. 229 * We split this up so that we do not require v[0] = 0, 230 * and to eliminate a final special case. 231 */ 232 for (t = 0, i = n; i > 0; i--) { 233 t = u[i + j] - v[i] * qhat - t; 234 u[i + j] = LHALF(t); 235 t = (B - HHALF(t)) & (B - 1); 236 } 237 t = u[j] - t; 238 u[j] = LHALF(t); 239 /* 240 * D5: test remainder. 241 * There is a borrow if and only if HHALF(t) is nonzero; 242 * in that (rare) case, qhat was too large (by exactly 1). 243 * Fix it by adding v[1..n] to u[j..j+n]. 244 */ 245 if (HHALF(t)) { 246 qhat--; 247 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */ 248 t += u[i + j] + v[i]; 249 u[i + j] = LHALF(t); 250 t = HHALF(t); 251 } 252 u[j] = LHALF(u[j] + t); 253 } 254 q[j] = qhat; 255 } while (++j <= m); /* D7: loop on j. */ 256 257 /* 258 * If caller wants the remainder, we have to calculate it as 259 * u[m..m+n] >> d (this is at most n digits and thus fits in 260 * u[m+1..m+n], but we may need more source digits). 261 */ 262 if (arq) { 263 if (d) { 264 for (i = m + n; i > m; --i) 265 u[i] = (u[i] >> d) | 266 LHALF(u[i - 1] << (HALF_BITS - d)); 267 u[i] = 0; 268 } 269 tmp.ul[H] = COMBINE(uspace[1], uspace[2]); 270 tmp.ul[L] = COMBINE(uspace[3], uspace[4]); 271 *arq = tmp.q; 272 } 273 274 tmp.ul[H] = COMBINE(qspace[1], qspace[2]); 275 tmp.ul[L] = COMBINE(qspace[3], qspace[4]); 276 return (tmp.q); 277 } 278