lib/libstand/qdivrem.c

   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  * $FreeBSD: src/lib/libstand/qdivrem.c,v 1.2 1999/08/28 00:05:33 peter Exp $
  38  * $DragonFly: src/lib/libstand/qdivrem.c,v 1.4 2005/12/11 02:27:26 swildner Exp $
  39  *      From: Id: qdivrem.c,v 1.7 1997/11/07 09:20:40 phk Exp
  40  */
  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       (1 << 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 (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 nn = COMBINE(uj0, uj1);
 218                         qhat = nn / v1;
 219                         rhat = nn % 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
 280 /*
 281  * Divide two unsigned quads.
 282  */
 283
 284 u_quad_t
 285 __udivdi3(u_quad_t a, u_quad_t b)
 286 {
 287
 288         return (__qdivrem(a, b, (u_quad_t *)0));
 289 }
 290
 291 /*
 292  * Return remainder after dividing two unsigned quads.
 293  */
 294 u_quad_t
 295 __umoddi3(u_quad_t a, u_quad_t b)
 296 {
 297         u_quad_t r;
 298
 299         (void)__qdivrem(a, b, &r);
 300         return (r);
 301 }