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 #if defined(LIBC_SCCS) && !defined(lint) 39 static char sccsid[] = "@(#)qdivrem.c 8.1 (Berkeley) 6/4/93"; 40 #endif /* LIBC_SCCS and not lint */ 41 #include <sys/cdefs.h> 42 __FBSDID("$FreeBSD$"); 43 44 /* 45 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed), 46 * section 4.3.1, pp. 257--259. 47 */ 48 49 #include "quad.h" 50 51 #define B (1 << HALF_BITS) /* digit base */ 52 53 /* Combine two `digits' to make a single two-digit number. */ 54 #define COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b)) 55 56 /* select a type for digits in base B: use unsigned short if they fit */ 57 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff 58 typedef unsigned short digit; 59 #else 60 typedef u_long digit; 61 #endif 62 63 /* 64 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that 65 * `fall out' the left (there never will be any such anyway). 66 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS. 67 */ 68 static void 69 shl(digit *p, int len, int sh) 70 { 71 int i; 72 73 for (i = 0; i < len; i++) 74 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh)); 75 p[i] = LHALF(p[i] << sh); 76 } 77 78 /* 79 * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v. 80 * 81 * We do this in base 2-sup-HALF_BITS, so that all intermediate products 82 * fit within u_long. As a consequence, the maximum length dividend and 83 * divisor are 4 `digits' in this base (they are shorter if they have 84 * leading zeros). 85 */ 86 u_quad_t 87 __qdivrem(uq, vq, arq) 88 u_quad_t uq, vq, *arq; 89 { 90 union uu tmp; 91 digit *u, *v, *q; 92 digit v1, v2; 93 u_long qhat, rhat, t; 94 int m, n, d, j, i; 95 digit uspace[5], vspace[5], qspace[5]; 96 97 /* 98 * Take care of special cases: divide by zero, and u < v. 99 */ 100 if (vq == 0) { 101 /* divide by zero. */ 102 static volatile const unsigned int zero = 0; 103 104 tmp.ul[H] = tmp.ul[L] = 1 / zero; 105 if (arq) 106 *arq = uq; 107 return (tmp.q); 108 } 109 if (uq < vq) { 110 if (arq) 111 *arq = uq; 112 return (0); 113 } 114 u = &uspace[0]; 115 v = &vspace[0]; 116 q = &qspace[0]; 117 118 /* 119 * Break dividend and divisor into digits in base B, then 120 * count leading zeros to determine m and n. When done, we 121 * will have: 122 * u = (u[1]u[2]...u[m+n]) sub B 123 * v = (v[1]v[2]...v[n]) sub B 124 * v[1] != 0 125 * 1 < n <= 4 (if n = 1, we use a different division algorithm) 126 * m >= 0 (otherwise u < v, which we already checked) 127 * m + n = 4 128 * and thus 129 * m = 4 - n <= 2 130 */ 131 tmp.uq = uq; 132 u[0] = 0; 133 u[1] = HHALF(tmp.ul[H]); 134 u[2] = LHALF(tmp.ul[H]); 135 u[3] = HHALF(tmp.ul[L]); 136 u[4] = LHALF(tmp.ul[L]); 137 tmp.uq = vq; 138 v[1] = HHALF(tmp.ul[H]); 139 v[2] = LHALF(tmp.ul[H]); 140 v[3] = HHALF(tmp.ul[L]); 141 v[4] = LHALF(tmp.ul[L]); 142 for (n = 4; v[1] == 0; v++) { 143 if (--n == 1) { 144 u_long rbj; /* r*B+u[j] (not root boy jim) */ 145 digit q1, q2, q3, q4; 146 147 /* 148 * Change of plan, per exercise 16. 149 * r = 0; 150 * for j = 1..4: 151 * q[j] = floor((r*B + u[j]) / v), 152 * r = (r*B + u[j]) % v; 153 * We unroll this completely here. 154 */ 155 t = v[2]; /* nonzero, by definition */ 156 q1 = u[1] / t; 157 rbj = COMBINE(u[1] % t, u[2]); 158 q2 = rbj / t; 159 rbj = COMBINE(rbj % t, u[3]); 160 q3 = rbj / t; 161 rbj = COMBINE(rbj % t, u[4]); 162 q4 = rbj / t; 163 if (arq) 164 *arq = rbj % t; 165 tmp.ul[H] = COMBINE(q1, q2); 166 tmp.ul[L] = COMBINE(q3, q4); 167 return (tmp.q); 168 } 169 } 170 171 /* 172 * By adjusting q once we determine m, we can guarantee that 173 * there is a complete four-digit quotient at &qspace[1] when 174 * we finally stop. 175 */ 176 for (m = 4 - n; u[1] == 0; u++) 177 m--; 178 for (i = 4 - m; --i >= 0;) 179 q[i] = 0; 180 q += 4 - m; 181 182 /* 183 * Here we run Program D, translated from MIX to C and acquiring 184 * a few minor changes. 185 * 186 * D1: choose multiplier 1 << d to ensure v[1] >= B/2. 187 */ 188 d = 0; 189 for (t = v[1]; t < B / 2; t <<= 1) 190 d++; 191 if (d > 0) { 192 shl(&u[0], m + n, d); /* u <<= d */ 193 shl(&v[1], n - 1, d); /* v <<= d */ 194 } 195 /* 196 * D2: j = 0. 197 */ 198 j = 0; 199 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */ 200 v2 = v[2]; /* for D3 */ 201 do { 202 digit uj0, uj1, uj2; 203 204 /* 205 * D3: Calculate qhat (\^q, in TeX notation). 206 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and 207 * let rhat = (u[j]*B + u[j+1]) mod v[1]. 208 * While rhat < B and v[2]*qhat > rhat*B+u[j+2], 209 * decrement qhat and increase rhat correspondingly. 210 * Note that if rhat >= B, v[2]*qhat < rhat*B. 211 */ 212 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */ 213 uj1 = u[j + 1]; /* for D3 only */ 214 uj2 = u[j + 2]; /* for D3 only */ 215 if (uj0 == v1) { 216 qhat = B; 217 rhat = uj1; 218 goto qhat_too_big; 219 } else { 220 u_long n = COMBINE(uj0, uj1); 221 qhat = n / v1; 222 rhat = n % v1; 223 } 224 while (v2 * qhat > COMBINE(rhat, uj2)) { 225 qhat_too_big: 226 qhat--; 227 if ((rhat += v1) >= B) 228 break; 229 } 230 /* 231 * D4: Multiply and subtract. 232 * The variable `t' holds any borrows across the loop. 233 * We split this up so that we do not require v[0] = 0, 234 * and to eliminate a final special case. 235 */ 236 for (t = 0, i = n; i > 0; i--) { 237 t = u[i + j] - v[i] * qhat - t; 238 u[i + j] = LHALF(t); 239 t = (B - HHALF(t)) & (B - 1); 240 } 241 t = u[j] - t; 242 u[j] = LHALF(t); 243 /* 244 * D5: test remainder. 245 * There is a borrow if and only if HHALF(t) is nonzero; 246 * in that (rare) case, qhat was too large (by exactly 1). 247 * Fix it by adding v[1..n] to u[j..j+n]. 248 */ 249 if (HHALF(t)) { 250 qhat--; 251 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */ 252 t += u[i + j] + v[i]; 253 u[i + j] = LHALF(t); 254 t = HHALF(t); 255 } 256 u[j] = LHALF(u[j] + t); 257 } 258 q[j] = qhat; 259 } while (++j <= m); /* D7: loop on j. */ 260 261 /* 262 * If caller wants the remainder, we have to calculate it as 263 * u[m..m+n] >> d (this is at most n digits and thus fits in 264 * u[m+1..m+n], but we may need more source digits). 265 */ 266 if (arq) { 267 if (d) { 268 for (i = m + n; i > m; --i) 269 u[i] = (u[i] >> d) | 270 LHALF(u[i - 1] << (HALF_BITS - d)); 271 u[i] = 0; 272 } 273 tmp.ul[H] = COMBINE(uspace[1], uspace[2]); 274 tmp.ul[L] = COMBINE(uspace[3], uspace[4]); 275 *arq = tmp.q; 276 } 277 278 tmp.ul[H] = COMBINE(qspace[1], qspace[2]); 279 tmp.ul[L] = COMBINE(qspace[3], qspace[4]); 280 return (tmp.q); 281 } 282