The Design and Implementation of the FreeBSD Operating System, Second Edition
Now available: The Design and Implementation of the FreeBSD Operating System (Second Edition)


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FreeBSD/Linux Kernel Cross Reference
sys/libkern/qdivrem.c

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    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/sys/libkern/qdivrem.c,v 1.8 1999/08/28 00:46:35 peter Exp $
   34  * $DragonFly: src/sys/libkern/qdivrem.c,v 1.4 2004/01/26 11:09:44 joerg 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 <libkern/quad.h>
   43 
   44 #define B       (1 << HALF_BITS)        /* digit base */
   45 
   46 /* Combine two `digits' to make a single two-digit number. */
   47 #define COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b))
   48 
   49 /* select a type for digits in base B: use unsigned short if they fit */
   50 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff
   51 typedef unsigned short digit;
   52 #else
   53 typedef u_long digit;
   54 #endif
   55 
   56 /*
   57  * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
   58  * `fall out' the left (there never will be any such anyway).
   59  * We may assume len >= 0.  NOTE THAT THIS WRITES len+1 DIGITS.
   60  */
   61 static void
   62 shl(digit *p, int len, int sh)
   63 {
   64         int i;
   65 
   66         for (i = 0; i < len; i++)
   67                 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh));
   68         p[i] = LHALF(p[i] << sh);
   69 }
   70 
   71 /*
   72  * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
   73  *
   74  * We do this in base 2-sup-HALF_BITS, so that all intermediate products
   75  * fit within u_long.  As a consequence, the maximum length dividend and
   76  * divisor are 4 `digits' in this base (they are shorter if they have
   77  * leading zeros).
   78  */
   79 u_quad_t
   80 __qdivrem(u_quad_t uq, u_quad_t vq, u_quad_t *arq)
   81 {
   82         union uu tmp;
   83         digit *u, *v, *q;
   84         digit v1, v2;
   85         u_long qhat, rhat, t;
   86         int m, n, d, j, i;
   87         digit uspace[5], vspace[5], qspace[5];
   88 
   89         /*
   90          * Take care of special cases: divide by zero, and u < v.
   91          */
   92         if (vq == 0) {
   93                 /* divide by zero. */
   94                 static volatile const unsigned int zero = 0;
   95 
   96                 tmp.ul[H] = tmp.ul[L] = 1 / zero;
   97                 if (arq)
   98                         *arq = uq;
   99                 return (tmp.q);
  100         }
  101         if (uq < vq) {
  102                 if (arq)
  103                         *arq = uq;
  104                 return (0);
  105         }
  106         u = &uspace[0];
  107         v = &vspace[0];
  108         q = &qspace[0];
  109 
  110         /*
  111          * Break dividend and divisor into digits in base B, then
  112          * count leading zeros to determine m and n.  When done, we
  113          * will have:
  114          *      u = (u[1]u[2]...u[m+n]) sub B
  115          *      v = (v[1]v[2]...v[n]) sub B
  116          *      v[1] != 0
  117          *      1 < n <= 4 (if n = 1, we use a different division algorithm)
  118          *      m >= 0 (otherwise u < v, which we already checked)
  119          *      m + n = 4
  120          * and thus
  121          *      m = 4 - n <= 2
  122          */
  123         tmp.uq = uq;
  124         u[0] = 0;
  125         u[1] = HHALF(tmp.ul[H]);
  126         u[2] = LHALF(tmp.ul[H]);
  127         u[3] = HHALF(tmp.ul[L]);
  128         u[4] = LHALF(tmp.ul[L]);
  129         tmp.uq = vq;
  130         v[1] = HHALF(tmp.ul[H]);
  131         v[2] = LHALF(tmp.ul[H]);
  132         v[3] = HHALF(tmp.ul[L]);
  133         v[4] = LHALF(tmp.ul[L]);
  134         for (n = 4; v[1] == 0; v++) {
  135                 if (--n == 1) {
  136                         u_long rbj;     /* r*B+u[j] (not root boy jim) */
  137                         digit q1, q2, q3, q4;
  138 
  139                         /*
  140                          * Change of plan, per exercise 16.
  141                          *      r = 0;
  142                          *      for j = 1..4:
  143                          *              q[j] = floor((r*B + u[j]) / v),
  144                          *              r = (r*B + u[j]) % v;
  145                          * We unroll this completely here.
  146                          */
  147                         t = v[2];       /* nonzero, by definition */
  148                         q1 = u[1] / t;
  149                         rbj = COMBINE(u[1] % t, u[2]);
  150                         q2 = rbj / t;
  151                         rbj = COMBINE(rbj % t, u[3]);
  152                         q3 = rbj / t;
  153                         rbj = COMBINE(rbj % t, u[4]);
  154                         q4 = rbj / t;
  155                         if (arq)
  156                                 *arq = rbj % t;
  157                         tmp.ul[H] = COMBINE(q1, q2);
  158                         tmp.ul[L] = COMBINE(q3, q4);
  159                         return (tmp.q);
  160                 }
  161         }
  162 
  163         /*
  164          * By adjusting q once we determine m, we can guarantee that
  165          * there is a complete four-digit quotient at &qspace[1] when
  166          * we finally stop.
  167          */
  168         for (m = 4 - n; u[1] == 0; u++)
  169                 m--;
  170         for (i = 4 - m; --i >= 0;)
  171                 q[i] = 0;
  172         q += 4 - m;
  173 
  174         /*
  175          * Here we run Program D, translated from MIX to C and acquiring
  176          * a few minor changes.
  177          *
  178          * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
  179          */
  180         d = 0;
  181         for (t = v[1]; t < B / 2; t <<= 1)
  182                 d++;
  183         if (d > 0) {
  184                 shl(&u[0], m + n, d);           /* u <<= d */
  185                 shl(&v[1], n - 1, d);           /* v <<= d */
  186         }
  187         /*
  188          * D2: j = 0.
  189          */
  190         j = 0;
  191         v1 = v[1];      /* for D3 -- note that v[1..n] are constant */
  192         v2 = v[2];      /* for D3 */
  193         do {
  194                 digit uj0, uj1, uj2;
  195 
  196                 /*
  197                  * D3: Calculate qhat (\^q, in TeX notation).
  198                  * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
  199                  * let rhat = (u[j]*B + u[j+1]) mod v[1].
  200                  * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
  201                  * decrement qhat and increase rhat correspondingly.
  202                  * Note that if rhat >= B, v[2]*qhat < rhat*B.
  203                  */
  204                 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */
  205                 uj1 = u[j + 1]; /* for D3 only */
  206                 uj2 = u[j + 2]; /* for D3 only */
  207                 if (uj0 == v1) {
  208                         qhat = B;
  209                         rhat = uj1;
  210                         goto qhat_too_big;
  211                 } else {
  212                         u_long nn = COMBINE(uj0, uj1);
  213                         qhat = nn / v1;
  214                         rhat = nn % v1;
  215                 }
  216                 while (v2 * qhat > COMBINE(rhat, uj2)) {
  217         qhat_too_big:
  218                         qhat--;
  219                         if ((rhat += v1) >= B)
  220                                 break;
  221                 }
  222                 /*
  223                  * D4: Multiply and subtract.
  224                  * The variable `t' holds any borrows across the loop.
  225                  * We split this up so that we do not require v[0] = 0,
  226                  * and to eliminate a final special case.
  227                  */
  228                 for (t = 0, i = n; i > 0; i--) {
  229                         t = u[i + j] - v[i] * qhat - t;
  230                         u[i + j] = LHALF(t);
  231                         t = (B - HHALF(t)) & (B - 1);
  232                 }
  233                 t = u[j] - t;
  234                 u[j] = LHALF(t);
  235                 /*
  236                  * D5: test remainder.
  237                  * There is a borrow if and only if HHALF(t) is nonzero;
  238                  * in that (rare) case, qhat was too large (by exactly 1).
  239                  * Fix it by adding v[1..n] to u[j..j+n].
  240                  */
  241                 if (HHALF(t)) {
  242                         qhat--;
  243                         for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
  244                                 t += u[i + j] + v[i];
  245                                 u[i + j] = LHALF(t);
  246                                 t = HHALF(t);
  247                         }
  248                         u[j] = LHALF(u[j] + t);
  249                 }
  250                 q[j] = qhat;
  251         } while (++j <= m);             /* D7: loop on j. */
  252 
  253         /*
  254          * If caller wants the remainder, we have to calculate it as
  255          * u[m..m+n] >> d (this is at most n digits and thus fits in
  256          * u[m+1..m+n], but we may need more source digits).
  257          */
  258         if (arq) {
  259                 if (d) {
  260                         for (i = m + n; i > m; --i)
  261                                 u[i] = (u[i] >> d) |
  262                                     LHALF(u[i - 1] << (HALF_BITS - d));
  263                         u[i] = 0;
  264                 }
  265                 tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
  266                 tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
  267                 *arq = tmp.q;
  268         }
  269 
  270         tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
  271         tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
  272         return (tmp.q);
  273 }

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