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. 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 * $FreeBSD: src/lib/libstand/qdivrem.c,v 1.2 1999/08/28 00:05:33 peter Exp $ 34 * From: Id: qdivrem.c,v 1.7 1997/11/07 09:20:40 phk Exp 35 */ 36 37 /* 38 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed), 39 * section 4.3.1, pp. 257--259. 40 */ 41 42 #include "quad.h" 43 #include <sys/endian.h> /* _QUAD_HIGHWORD */ 44 45 #define B (1 << HALF_BITS) /* digit base */ 46 47 /* 48 * Define high and low longwords. 49 */ 50 #define H _QUAD_HIGHWORD 51 #define L _QUAD_LOWWORD 52 53 /* Combine two `digits' to make a single two-digit number. */ 54 #define COMBINE(a, b) (((u_int)(a) << HALF_BITS) | (b)) 55 56 _Static_assert(sizeof(int) / 2 == sizeof(short), 57 "Bitwise functions in libstand are broken on this architecture"); 58 59 /* select a type for digits in base B: use unsigned short if they fit */ 60 typedef unsigned short digit; 61 62 /* 63 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that 64 * `fall out' the left (there never will be any such anyway). 65 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS. 66 */ 67 static void 68 shl(digit *p, int len, int sh) 69 { 70 int i; 71 72 for (i = 0; i < len; i++) 73 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh)); 74 p[i] = LHALF(p[i] << sh); 75 } 76 77 /* 78 * __udivmoddi4(u, v, rem) returns u/v and, optionally, sets *rem to u%v. 79 * 80 * We do this in base 2-sup-HALF_BITS, so that all intermediate products 81 * fit within u_int. As a consequence, the maximum length dividend and 82 * divisor are 4 `digits' in this base (they are shorter if they have 83 * leading zeros). 84 */ 85 u_quad_t 86 __udivmoddi4(u_quad_t uq, u_quad_t vq, u_quad_t *arq) 87 { 88 union uu tmp; 89 digit *u, *v, *q; 90 digit v1, v2; 91 u_int qhat, rhat, t; 92 int m, n, d, j, i; 93 digit uspace[5], vspace[5], qspace[5]; 94 95 /* 96 * Take care of special cases: divide by zero, and u < v. 97 */ 98 if (vq == 0) { 99 /* divide by zero. */ 100 static volatile const unsigned int zero = 0; 101 102 tmp.ul[H] = tmp.ul[L] = 1 / zero; 103 if (arq) 104 *arq = uq; 105 return (tmp.q); 106 } 107 if (uq < vq) { 108 if (arq) 109 *arq = uq; 110 return (0); 111 } 112 u = &uspace[0]; 113 v = &vspace[0]; 114 q = &qspace[0]; 115 116 /* 117 * Break dividend and divisor into digits in base B, then 118 * count leading zeros to determine m and n. When done, we 119 * will have: 120 * u = (u[1]u[2]...u[m+n]) sub B 121 * v = (v[1]v[2]...v[n]) sub B 122 * v[1] != 0 123 * 1 < n <= 4 (if n = 1, we use a different division algorithm) 124 * m >= 0 (otherwise u < v, which we already checked) 125 * m + n = 4 126 * and thus 127 * m = 4 - n <= 2 128 */ 129 tmp.uq = uq; 130 u[0] = 0; 131 u[1] = HHALF(tmp.ul[H]); 132 u[2] = LHALF(tmp.ul[H]); 133 u[3] = HHALF(tmp.ul[L]); 134 u[4] = LHALF(tmp.ul[L]); 135 tmp.uq = vq; 136 v[1] = HHALF(tmp.ul[H]); 137 v[2] = LHALF(tmp.ul[H]); 138 v[3] = HHALF(tmp.ul[L]); 139 v[4] = LHALF(tmp.ul[L]); 140 for (n = 4; v[1] == 0; v++) { 141 if (--n == 1) { 142 u_int rbj; /* r*B+u[j] (not root boy jim) */ 143 digit q1, q2, q3, q4; 144 145 /* 146 * Change of plan, per exercise 16. 147 * r = 0; 148 * for j = 1..4: 149 * q[j] = floor((r*B + u[j]) / v), 150 * r = (r*B + u[j]) % v; 151 * We unroll this completely here. 152 */ 153 t = v[2]; /* nonzero, by definition */ 154 q1 = u[1] / t; 155 rbj = COMBINE(u[1] % t, u[2]); 156 q2 = rbj / t; 157 rbj = COMBINE(rbj % t, u[3]); 158 q3 = rbj / t; 159 rbj = COMBINE(rbj % t, u[4]); 160 q4 = rbj / t; 161 if (arq) 162 *arq = rbj % t; 163 tmp.ul[H] = COMBINE(q1, q2); 164 tmp.ul[L] = COMBINE(q3, q4); 165 return (tmp.q); 166 } 167 } 168 169 /* 170 * By adjusting q once we determine m, we can guarantee that 171 * there is a complete four-digit quotient at &qspace[1] when 172 * we finally stop. 173 */ 174 for (m = 4 - n; u[1] == 0; u++) 175 m--; 176 for (i = 4 - m; --i >= 0;) 177 q[i] = 0; 178 q += 4 - m; 179 180 /* 181 * Here we run Program D, translated from MIX to C and acquiring 182 * a few minor changes. 183 * 184 * D1: choose multiplier 1 << d to ensure v[1] >= B/2. 185 */ 186 d = 0; 187 for (t = v[1]; t < B / 2; t <<= 1) 188 d++; 189 if (d > 0) { 190 shl(&u[0], m + n, d); /* u <<= d */ 191 shl(&v[1], n - 1, d); /* v <<= d */ 192 } 193 /* 194 * D2: j = 0. 195 */ 196 j = 0; 197 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */ 198 v2 = v[2]; /* for D3 */ 199 do { 200 digit uj0, uj1, uj2; 201 202 /* 203 * D3: Calculate qhat (\^q, in TeX notation). 204 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and 205 * let rhat = (u[j]*B + u[j+1]) mod v[1]. 206 * While rhat < B and v[2]*qhat > rhat*B+u[j+2], 207 * decrement qhat and increase rhat correspondingly. 208 * Note that if rhat >= B, v[2]*qhat < rhat*B. 209 */ 210 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */ 211 uj1 = u[j + 1]; /* for D3 only */ 212 uj2 = u[j + 2]; /* for D3 only */ 213 if (uj0 == v1) { 214 qhat = B; 215 rhat = uj1; 216 goto qhat_too_big; 217 } else { 218 u_int nn = COMBINE(uj0, uj1); 219 qhat = nn / v1; 220 rhat = nn % v1; 221 } 222 while (v2 * qhat > COMBINE(rhat, uj2)) { 223 qhat_too_big: 224 qhat--; 225 if ((rhat += v1) >= B) 226 break; 227 } 228 /* 229 * D4: Multiply and subtract. 230 * The variable `t' holds any borrows across the loop. 231 * We split this up so that we do not require v[0] = 0, 232 * and to eliminate a final special case. 233 */ 234 for (t = 0, i = n; i > 0; i--) { 235 t = u[i + j] - v[i] * qhat - t; 236 u[i + j] = LHALF(t); 237 t = (B - HHALF(t)) & (B - 1); 238 } 239 t = u[j] - t; 240 u[j] = LHALF(t); 241 /* 242 * D5: test remainder. 243 * There is a borrow if and only if HHALF(t) is nonzero; 244 * in that (rare) case, qhat was too large (by exactly 1). 245 * Fix it by adding v[1..n] to u[j..j+n]. 246 */ 247 if (HHALF(t)) { 248 qhat--; 249 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */ 250 t += u[i + j] + v[i]; 251 u[i + j] = LHALF(t); 252 t = HHALF(t); 253 } 254 u[j] = LHALF(u[j] + t); 255 } 256 q[j] = qhat; 257 } while (++j <= m); /* D7: loop on j. */ 258 259 /* 260 * If caller wants the remainder, we have to calculate it as 261 * u[m..m+n] >> d (this is at most n digits and thus fits in 262 * u[m+1..m+n], but we may need more source digits). 263 */ 264 if (arq) { 265 if (d) { 266 for (i = m + n; i > m; --i) 267 u[i] = (u[i] >> d) | 268 LHALF(u[i - 1] << (HALF_BITS - d)); 269 u[i] = 0; 270 } 271 tmp.ul[H] = COMBINE(uspace[1], uspace[2]); 272 tmp.ul[L] = COMBINE(uspace[3], uspace[4]); 273 *arq = tmp.q; 274 } 275 276 tmp.ul[H] = COMBINE(qspace[1], qspace[2]); 277 tmp.ul[L] = COMBINE(qspace[3], qspace[4]); 278 return (tmp.q); 279 } 280 281 /* 282 * Divide two unsigned quads. 283 */ 284 285 u_quad_t 286 __udivdi3(u_quad_t a, u_quad_t b) 287 { 288 289 return (__udivmoddi4(a, b, (u_quad_t *)0)); 290 } 291 292 /* 293 * Return remainder after dividing two unsigned quads. 294 */ 295 u_quad_t 296 __umoddi3(u_quad_t a, u_quad_t b) 297 { 298 u_quad_t r; 299 300 (void)__udivmoddi4(a, b, &r); 301 return (r); 302 } 303