1 /*- 2 * SPDX-License-Identifier: BSD-3-Clause 3 * 4 * Copyright (c) 1992, 1993 5 * The Regents of the University of California. All rights reserved. 6 * 7 * This software was developed by the Computer Systems Engineering group 8 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and 9 * contributed to Berkeley. 10 * 11 * Redistribution and use in source and binary forms, with or without 12 * modification, are permitted provided that the following conditions 13 * are met: 14 * 1. Redistributions of source code must retain the above copyright 15 * notice, this list of conditions and the following disclaimer. 16 * 2. Redistributions in binary form must reproduce the above copyright 17 * notice, this list of conditions and the following disclaimer in the 18 * documentation and/or other materials provided with the distribution. 19 * 3. Neither the name of the University nor the names of its contributors 20 * may be used to endorse or promote products derived from this software 21 * without specific prior written permission. 22 * 23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 26 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 33 * SUCH DAMAGE. 34 */ 35 36 #if defined(LIBC_SCCS) && !defined(lint) 37 static char sccsid[] = "@(#)qdivrem.c 8.1 (Berkeley) 6/4/93"; 38 #endif /* LIBC_SCCS and not lint */ 39 #include <sys/cdefs.h> 40 __FBSDID("$FreeBSD$"); 41 42 /* 43 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed), 44 * section 4.3.1, pp. 257--259. 45 */ 46 47 #include "quad.h" 48 49 #define B (1L << HALF_BITS) /* digit base */ 50 51 /* Combine two `digits' to make a single two-digit number. */ 52 #define COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b)) 53 54 /* select a type for digits in base B: use unsigned short if they fit */ 55 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff 56 typedef unsigned short digit; 57 #else 58 typedef u_long digit; 59 #endif 60 61 /* 62 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that 63 * `fall out' the left (there never will be any such anyway). 64 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS. 65 */ 66 static void 67 shl(digit *p, int len, int sh) 68 { 69 int i; 70 71 for (i = 0; i < len; i++) 72 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh)); 73 p[i] = LHALF(p[i] << sh); 74 } 75 76 /* 77 * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v. 78 * 79 * We do this in base 2-sup-HALF_BITS, so that all intermediate products 80 * fit within u_long. As a consequence, the maximum length dividend and 81 * divisor are 4 `digits' in this base (they are shorter if they have 82 * leading zeros). 83 */ 84 u_quad_t 85 __qdivrem(u_quad_t uq, u_quad_t vq, u_quad_t *arq) 86 { 87 union uu tmp; 88 digit *u, *v, *q; 89 digit v1, v2; 90 u_long qhat, rhat, t; 91 int m, n, d, j, i; 92 digit uspace[5], vspace[5], qspace[5]; 93 94 /* 95 * Take care of special cases: divide by zero, and u < v. 96 */ 97 if (__predict_false(vq == 0)) { 98 /* divide by zero. */ 99 static volatile const unsigned int zero = 0; 100 101 tmp.ul[H] = tmp.ul[L] = 1 / zero; 102 if (arq) 103 *arq = uq; 104 return (tmp.q); 105 } 106 if (uq < vq) { 107 if (arq) 108 *arq = uq; 109 return (0); 110 } 111 u = &uspace[0]; 112 v = &vspace[0]; 113 q = &qspace[0]; 114 115 /* 116 * Break dividend and divisor into digits in base B, then 117 * count leading zeros to determine m and n. When done, we 118 * will have: 119 * u = (u[1]u[2]...u[m+n]) sub B 120 * v = (v[1]v[2]...v[n]) sub B 121 * v[1] != 0 122 * 1 < n <= 4 (if n = 1, we use a different division algorithm) 123 * m >= 0 (otherwise u < v, which we already checked) 124 * m + n = 4 125 * and thus 126 * m = 4 - n <= 2 127 */ 128 tmp.uq = uq; 129 u[0] = 0; 130 u[1] = HHALF(tmp.ul[H]); 131 u[2] = LHALF(tmp.ul[H]); 132 u[3] = HHALF(tmp.ul[L]); 133 u[4] = LHALF(tmp.ul[L]); 134 tmp.uq = vq; 135 v[1] = HHALF(tmp.ul[H]); 136 v[2] = LHALF(tmp.ul[H]); 137 v[3] = HHALF(tmp.ul[L]); 138 v[4] = LHALF(tmp.ul[L]); 139 for (n = 4; v[1] == 0; v++) { 140 if (--n == 1) { 141 u_long rbj; /* r*B+u[j] (not root boy jim) */ 142 digit q1, q2, q3, q4; 143 144 /* 145 * Change of plan, per exercise 16. 146 * r = 0; 147 * for j = 1..4: 148 * q[j] = floor((r*B + u[j]) / v), 149 * r = (r*B + u[j]) % v; 150 * We unroll this completely here. 151 */ 152 t = v[2]; /* nonzero, by definition */ 153 q1 = u[1] / t; 154 rbj = COMBINE(u[1] % t, u[2]); 155 q2 = rbj / t; 156 rbj = COMBINE(rbj % t, u[3]); 157 q3 = rbj / t; 158 rbj = COMBINE(rbj % t, u[4]); 159 q4 = rbj / t; 160 if (arq) 161 *arq = rbj % t; 162 tmp.ul[H] = COMBINE(q1, q2); 163 tmp.ul[L] = COMBINE(q3, q4); 164 return (tmp.q); 165 } 166 } 167 168 /* 169 * By adjusting q once we determine m, we can guarantee that 170 * there is a complete four-digit quotient at &qspace[1] when 171 * we finally stop. 172 */ 173 for (m = 4 - n; u[1] == 0; u++) 174 m--; 175 for (i = 4 - m; --i >= 0;) 176 q[i] = 0; 177 q += 4 - m; 178 179 /* 180 * Here we run Program D, translated from MIX to C and acquiring 181 * a few minor changes. 182 * 183 * D1: choose multiplier 1 << d to ensure v[1] >= B/2. 184 */ 185 d = 0; 186 for (t = v[1]; t < B / 2; t <<= 1) 187 d++; 188 if (d > 0) { 189 shl(&u[0], m + n, d); /* u <<= d */ 190 shl(&v[1], n - 1, d); /* v <<= d */ 191 } 192 /* 193 * D2: j = 0. 194 */ 195 j = 0; 196 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */ 197 v2 = v[2]; /* for D3 */ 198 do { 199 digit uj0, uj1, uj2; 200 201 /* 202 * D3: Calculate qhat (\^q, in TeX notation). 203 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and 204 * let rhat = (u[j]*B + u[j+1]) mod v[1]. 205 * While rhat < B and v[2]*qhat > rhat*B+u[j+2], 206 * decrement qhat and increase rhat correspondingly. 207 * Note that if rhat >= B, v[2]*qhat < rhat*B. 208 */ 209 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */ 210 uj1 = u[j + 1]; /* for D3 only */ 211 uj2 = u[j + 2]; /* for D3 only */ 212 if (uj0 == v1) { 213 qhat = B; 214 rhat = uj1; 215 goto qhat_too_big; 216 } else { 217 u_long n = COMBINE(uj0, uj1); 218 qhat = n / v1; 219 rhat = n % v1; 220 } 221 while (v2 * qhat > COMBINE(rhat, uj2)) { 222 qhat_too_big: 223 qhat--; 224 if ((rhat += v1) >= B) 225 break; 226 } 227 /* 228 * D4: Multiply and subtract. 229 * The variable `t' holds any borrows across the loop. 230 * We split this up so that we do not require v[0] = 0, 231 * and to eliminate a final special case. 232 */ 233 for (t = 0, i = n; i > 0; i--) { 234 t = u[i + j] - v[i] * qhat - t; 235 u[i + j] = LHALF(t); 236 t = (B - HHALF(t)) & (B - 1); 237 } 238 t = u[j] - t; 239 u[j] = LHALF(t); 240 /* 241 * D5: test remainder. 242 * There is a borrow if and only if HHALF(t) is nonzero; 243 * in that (rare) case, qhat was too large (by exactly 1). 244 * Fix it by adding v[1..n] to u[j..j+n]. 245 */ 246 if (HHALF(t)) { 247 qhat--; 248 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */ 249 t += u[i + j] + v[i]; 250 u[i + j] = LHALF(t); 251 t = HHALF(t); 252 } 253 u[j] = LHALF(u[j] + t); 254 } 255 q[j] = qhat; 256 } while (++j <= m); /* D7: loop on j. */ 257 258 /* 259 * If caller wants the remainder, we have to calculate it as 260 * u[m..m+n] >> d (this is at most n digits and thus fits in 261 * u[m+1..m+n], but we may need more source digits). 262 */ 263 if (arq) { 264 if (d) { 265 for (i = m + n; i > m; --i) 266 u[i] = (u[i] >> d) | 267 LHALF(u[i - 1] << (HALF_BITS - d)); 268 u[i] = 0; 269 } 270 tmp.ul[H] = COMBINE(uspace[1], uspace[2]); 271 tmp.ul[L] = COMBINE(uspace[3], uspace[4]); 272 *arq = tmp.q; 273 } 274 275 tmp.ul[H] = COMBINE(qspace[1], qspace[2]); 276 tmp.ul[L] = COMBINE(qspace[3], qspace[4]); 277 return (tmp.q); 278 } 279