The Design and Implementation of the FreeBSD Operating System, Second Edition
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FreeBSD/Linux Kernel Cross Reference
sys/libkern/arm/muldi3.c

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    1 /*      $NetBSD: muldi3.c,v 1.8 2003/08/07 16:32:09 agc Exp $   */
    2 
    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 #include <sys/cdefs.h>
   37 #if defined(LIBC_SCCS) && !defined(lint)
   38 #if 0
   39 static char sccsid[] = "@(#)muldi3.c    8.1 (Berkeley) 6/4/93";
   40 #else
   41 __FBSDID("$FreeBSD: releng/10.4/sys/libkern/arm/muldi3.c 129210 2004-05-14 12:28:31Z cognet $");
   42 #endif
   43 #endif /* LIBC_SCCS and not lint */
   44 
   45 #include <libkern/quad.h>
   46 
   47 /*
   48  * Multiply two quads.
   49  *
   50  * Our algorithm is based on the following.  Split incoming quad values
   51  * u and v (where u,v >= 0) into
   52  *
   53  *      u = 2^n u1  *  u0       (n = number of bits in `u_int', usu. 32)
   54  *
   55  * and 
   56  *
   57  *      v = 2^n v1  *  v0
   58  *
   59  * Then
   60  *
   61  *      uv = 2^2n u1 v1  +  2^n u1 v0  +  2^n v1 u0  +  u0 v0
   62  *         = 2^2n u1 v1  +     2^n (u1 v0 + v1 u0)   +  u0 v0
   63  *
   64  * Now add 2^n u1 v1 to the first term and subtract it from the middle,
   65  * and add 2^n u0 v0 to the last term and subtract it from the middle.
   66  * This gives:
   67  *
   68  *      uv = (2^2n + 2^n) (u1 v1)  +
   69  *               (2^n)    (u1 v0 - u1 v1 + u0 v1 - u0 v0)  +
   70  *             (2^n + 1)  (u0 v0)
   71  *
   72  * Factoring the middle a bit gives us:
   73  *
   74  *      uv = (2^2n + 2^n) (u1 v1)  +                    [u1v1 = high]
   75  *               (2^n)    (u1 - u0) (v0 - v1)  +        [(u1-u0)... = mid]
   76  *             (2^n + 1)  (u0 v0)                       [u0v0 = low]
   77  *
   78  * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
   79  * in just half the precision of the original.  (Note that either or both
   80  * of (u1 - u0) or (v0 - v1) may be negative.)
   81  *
   82  * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
   83  *
   84  * Since C does not give us a `int * int = quad' operator, we split
   85  * our input quads into two ints, then split the two ints into two
   86  * shorts.  We can then calculate `short * short = int' in native
   87  * arithmetic.
   88  *
   89  * Our product should, strictly speaking, be a `long quad', with 128
   90  * bits, but we are going to discard the upper 64.  In other words,
   91  * we are not interested in uv, but rather in (uv mod 2^2n).  This
   92  * makes some of the terms above vanish, and we get:
   93  *
   94  *      (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
   95  *
   96  * or
   97  *
   98  *      (2^n)(high + mid + low) + low
   99  *
  100  * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
  101  * of 2^n in either one will also vanish.  Only `low' need be computed
  102  * mod 2^2n, and only because of the final term above.
  103  */
  104 static quad_t __lmulq(u_int, u_int);
  105 
  106 quad_t __muldi3(quad_t, quad_t);
  107 quad_t
  108 __muldi3(quad_t a, quad_t b)
  109 {
  110         union uu u, v, low, prod;
  111         u_int high, mid, udiff, vdiff;
  112         int negall, negmid;
  113 #define u1      u.ul[H]
  114 #define u0      u.ul[L]
  115 #define v1      v.ul[H]
  116 #define v0      v.ul[L]
  117 
  118         /*
  119          * Get u and v such that u, v >= 0.  When this is finished,
  120          * u1, u0, v1, and v0 will be directly accessible through the
  121          * int fields.
  122          */
  123         if (a >= 0)
  124                 u.q = a, negall = 0;
  125         else
  126                 u.q = -a, negall = 1;
  127         if (b >= 0)
  128                 v.q = b;
  129         else
  130                 v.q = -b, negall ^= 1;
  131 
  132         if (u1 == 0 && v1 == 0) {
  133                 /*
  134                  * An (I hope) important optimization occurs when u1 and v1
  135                  * are both 0.  This should be common since most numbers
  136                  * are small.  Here the product is just u0*v0.
  137                  */
  138                 prod.q = __lmulq(u0, v0);
  139         } else {
  140                 /*
  141                  * Compute the three intermediate products, remembering
  142                  * whether the middle term is negative.  We can discard
  143                  * any upper bits in high and mid, so we can use native
  144                  * u_int * u_int => u_int arithmetic.
  145                  */
  146                 low.q = __lmulq(u0, v0);
  147 
  148                 if (u1 >= u0)
  149                         negmid = 0, udiff = u1 - u0;
  150                 else
  151                         negmid = 1, udiff = u0 - u1;
  152                 if (v0 >= v1)
  153                         vdiff = v0 - v1;
  154                 else
  155                         vdiff = v1 - v0, negmid ^= 1;
  156                 mid = udiff * vdiff;
  157 
  158                 high = u1 * v1;
  159 
  160                 /*
  161                  * Assemble the final product.
  162                  */
  163                 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
  164                     low.ul[H];
  165                 prod.ul[L] = low.ul[L];
  166         }
  167         return (negall ? -prod.q : prod.q);
  168 #undef u1
  169 #undef u0
  170 #undef v1
  171 #undef v0
  172 }
  173 
  174 /*
  175  * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
  176  * the number of bits in an int (whatever that is---the code below
  177  * does not care as long as quad.h does its part of the bargain---but
  178  * typically N==16).
  179  *
  180  * We use the same algorithm from Knuth, but this time the modulo refinement
  181  * does not apply.  On the other hand, since N is half the size of an int,
  182  * we can get away with native multiplication---none of our input terms
  183  * exceeds (UINT_MAX >> 1).
  184  *
  185  * Note that, for u_int l, the quad-precision result
  186  *
  187  *      l << N
  188  *
  189  * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
  190  */
  191 static quad_t
  192 __lmulq(u_int u, u_int v)
  193 {
  194         u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
  195         u_int prodh, prodl, was;
  196         union uu prod;
  197         int neg;
  198 
  199         u1 = HHALF(u);
  200         u0 = LHALF(u);
  201         v1 = HHALF(v);
  202         v0 = LHALF(v);
  203 
  204         low = u0 * v0;
  205 
  206         /* This is the same small-number optimization as before. */
  207         if (u1 == 0 && v1 == 0)
  208                 return (low);
  209 
  210         if (u1 >= u0)
  211                 udiff = u1 - u0, neg = 0;
  212         else
  213                 udiff = u0 - u1, neg = 1;
  214         if (v0 >= v1)
  215                 vdiff = v0 - v1;
  216         else
  217                 vdiff = v1 - v0, neg ^= 1;
  218         mid = udiff * vdiff;
  219 
  220         high = u1 * v1;
  221 
  222         /* prod = (high << 2N) + (high << N); */
  223         prodh = high + HHALF(high);
  224         prodl = LHUP(high);
  225 
  226         /* if (neg) prod -= mid << N; else prod += mid << N; */
  227         if (neg) {
  228                 was = prodl;
  229                 prodl -= LHUP(mid);
  230                 prodh -= HHALF(mid) + (prodl > was);
  231         } else {
  232                 was = prodl;
  233                 prodl += LHUP(mid);
  234                 prodh += HHALF(mid) + (prodl < was);
  235         }
  236 
  237         /* prod += low << N */
  238         was = prodl;
  239         prodl += LHUP(low);
  240         prodh += HHALF(low) + (prodl < was);
  241         /* ... + low; */
  242         if ((prodl += low) < low)
  243                 prodh++;
  244 
  245         /* return 4N-bit product */
  246         prod.ul[H] = prodh;
  247         prod.ul[L] = prodl;
  248         return (prod.q);
  249 }

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