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. 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 #include <sys/cdefs.h>
   39 __FBSDID("$FreeBSD: releng/5.2/sys/libkern/qdivrem.c 116189 2003-06-11 05:37:42Z obrien $");
   40 
   41 /*
   42  * Multiprecision divide.  This algorithm is from Knuth vol. 2 (2nd ed),
   43  * section 4.3.1, pp. 257--259.
   44  */
   45 
   46 #include <libkern/quad.h>
   47 
   48 #define B       (1 << HALF_BITS)        /* digit base */
   49 
   50 /* Combine two `digits' to make a single two-digit number. */
   51 #define COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b))
   52 
   53 /* select a type for digits in base B: use unsigned short if they fit */
   54 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff
   55 typedef unsigned short digit;
   56 #else
   57 typedef u_long digit;
   58 #endif
   59 
   60 /*
   61  * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
   62  * `fall out' the left (there never will be any such anyway).
   63  * We may assume len >= 0.  NOTE THAT THIS WRITES len+1 DIGITS.
   64  */
   65 static void
   66 shl(register digit *p, register int len, register int sh)
   67 {
   68         register int i;
   69 
   70         for (i = 0; i < len; i++)
   71                 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh));
   72         p[i] = LHALF(p[i] << sh);
   73 }
   74 
   75 /*
   76  * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
   77  *
   78  * We do this in base 2-sup-HALF_BITS, so that all intermediate products
   79  * fit within u_long.  As a consequence, the maximum length dividend and
   80  * divisor are 4 `digits' in this base (they are shorter if they have
   81  * leading zeros).
   82  */
   83 u_quad_t
   84 __qdivrem(uq, vq, arq)
   85         u_quad_t uq, vq, *arq;
   86 {
   87         union uu tmp;
   88         digit *u, *v, *q;
   89         register 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                 register 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 }
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FreeBSD/Linux Kernel Cross Reference sys/libkern/qdivrem.c

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sys/libkern/qdivrem.c
