The Design and Implementation of the FreeBSD Operating System, Second Edition
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FreeBSD/Linux Kernel Cross Reference
sys/libkern/muldi3.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/muldi3.c,v 1.6 1999/08/28 00:46:34 peter Exp $
   34  * $DragonFly: src/sys/libkern/muldi3.c,v 1.4 2004/01/26 11:09:44 joerg Exp $
   35  */
   36 
   37 #include "quad.h"
   38 
   39 /*
   40  * Multiply two quads.
   41  *
   42  * Our algorithm is based on the following.  Split incoming quad values
   43  * u and v (where u,v >= 0) into
   44  *
   45  *      u = 2^n u1  *  u0       (n = number of bits in `u_long', usu. 32)
   46  *
   47  * and
   48  *
   49  *      v = 2^n v1  *  v0
   50  *
   51  * Then
   52  *
   53  *      uv = 2^2n u1 v1  +  2^n u1 v0  +  2^n v1 u0  +  u0 v0
   54  *         = 2^2n u1 v1  +     2^n (u1 v0 + v1 u0)   +  u0 v0
   55  *
   56  * Now add 2^n u1 v1 to the first term and subtract it from the middle,
   57  * and add 2^n u0 v0 to the last term and subtract it from the middle.
   58  * This gives:
   59  *
   60  *      uv = (2^2n + 2^n) (u1 v1)  +
   61  *               (2^n)    (u1 v0 - u1 v1 + u0 v1 - u0 v0)  +
   62  *             (2^n + 1)  (u0 v0)
   63  *
   64  * Factoring the middle a bit gives us:
   65  *
   66  *      uv = (2^2n + 2^n) (u1 v1)  +                    [u1v1 = high]
   67  *               (2^n)    (u1 - u0) (v0 - v1)  +        [(u1-u0)... = mid]
   68  *             (2^n + 1)  (u0 v0)                       [u0v0 = low]
   69  *
   70  * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
   71  * in just half the precision of the original.  (Note that either or both
   72  * of (u1 - u0) or (v0 - v1) may be negative.)
   73  *
   74  * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
   75  *
   76  * Since C does not give us a `long * long = quad' operator, we split
   77  * our input quads into two longs, then split the two longs into two
   78  * shorts.  We can then calculate `short * short = long' in native
   79  * arithmetic.
   80  *
   81  * Our product should, strictly speaking, be a `long quad', with 128
   82  * bits, but we are going to discard the upper 64.  In other words,
   83  * we are not interested in uv, but rather in (uv mod 2^2n).  This
   84  * makes some of the terms above vanish, and we get:
   85  *
   86  *      (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
   87  *
   88  * or
   89  *
   90  *      (2^n)(high + mid + low) + low
   91  *
   92  * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
   93  * of 2^n in either one will also vanish.  Only `low' need be computed
   94  * mod 2^2n, and only because of the final term above.
   95  */
   96 static quad_t __lmulq(u_long u, u_long v);
   97 
   98 quad_t
   99 __muldi3(quad_t a, quad_t b)
  100 {
  101         union uu u, v, low, prod;
  102         u_long high, mid, udiff, vdiff;
  103         int negall, negmid;
  104 #define u1      u.ul[H]
  105 #define u0      u.ul[L]
  106 #define v1      v.ul[H]
  107 #define v0      v.ul[L]
  108 
  109         /*
  110          * Get u and v such that u, v >= 0.  When this is finished,
  111          * u1, u0, v1, and v0 will be directly accessible through the
  112          * longword fields.
  113          */
  114         if (a >= 0)
  115                 u.q = a, negall = 0;
  116         else
  117                 u.q = -a, negall = 1;
  118         if (b >= 0)
  119                 v.q = b;
  120         else
  121                 v.q = -b, negall ^= 1;
  122 
  123         if (u1 == 0 && v1 == 0) {
  124                 /*
  125                  * An (I hope) important optimization occurs when u1 and v1
  126                  * are both 0.  This should be common since most numbers
  127                  * are small.  Here the product is just u0*v0.
  128                  */
  129                 prod.q = __lmulq(u0, v0);
  130         } else {
  131                 /*
  132                  * Compute the three intermediate products, remembering
  133                  * whether the middle term is negative.  We can discard
  134                  * any upper bits in high and mid, so we can use native
  135                  * u_long * u_long => u_long arithmetic.
  136                  */
  137                 low.q = __lmulq(u0, v0);
  138 
  139                 if (u1 >= u0)
  140                         negmid = 0, udiff = u1 - u0;
  141                 else
  142                         negmid = 1, udiff = u0 - u1;
  143                 if (v0 >= v1)
  144                         vdiff = v0 - v1;
  145                 else
  146                         vdiff = v1 - v0, negmid ^= 1;
  147                 mid = udiff * vdiff;
  148 
  149                 high = u1 * v1;
  150 
  151                 /*
  152                  * Assemble the final product.
  153                  */
  154                 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
  155                     low.ul[H];
  156                 prod.ul[L] = low.ul[L];
  157         }
  158         return (negall ? -prod.q : prod.q);
  159 #undef u1
  160 #undef u0
  161 #undef v1
  162 #undef v0
  163 }
  164 
  165 /*
  166  * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half
  167  * the number of bits in a long (whatever that is---the code below
  168  * does not care as long as quad.h does its part of the bargain---but
  169  * typically N==16).
  170  *
  171  * We use the same algorithm from Knuth, but this time the modulo refinement
  172  * does not apply.  On the other hand, since N is half the size of a long,
  173  * we can get away with native multiplication---none of our input terms
  174  * exceeds (ULONG_MAX >> 1).
  175  *
  176  * Note that, for u_long l, the quad-precision result
  177  *
  178  *      l << N
  179  *
  180  * splits into high and low longs as HHALF(l) and LHUP(l) respectively.
  181  */
  182 static quad_t
  183 __lmulq(u_long u, u_long v)
  184 {
  185         u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low;
  186         u_long prodh, prodl, was;
  187         union uu prod;
  188         int neg;
  189 
  190         u1 = HHALF(u);
  191         u0 = LHALF(u);
  192         v1 = HHALF(v);
  193         v0 = LHALF(v);
  194 
  195         low = u0 * v0;
  196 
  197         /* This is the same small-number optimization as before. */
  198         if (u1 == 0 && v1 == 0)
  199                 return (low);
  200 
  201         if (u1 >= u0)
  202                 udiff = u1 - u0, neg = 0;
  203         else
  204                 udiff = u0 - u1, neg = 1;
  205         if (v0 >= v1)
  206                 vdiff = v0 - v1;
  207         else
  208                 vdiff = v1 - v0, neg ^= 1;
  209         mid = udiff * vdiff;
  210 
  211         high = u1 * v1;
  212 
  213         /* prod = (high << 2N) + (high << N); */
  214         prodh = high + HHALF(high);
  215         prodl = LHUP(high);
  216 
  217         /* if (neg) prod -= mid << N; else prod += mid << N; */
  218         if (neg) {
  219                 was = prodl;
  220                 prodl -= LHUP(mid);
  221                 prodh -= HHALF(mid) + (prodl > was);
  222         } else {
  223                 was = prodl;
  224                 prodl += LHUP(mid);
  225                 prodh += HHALF(mid) + (prodl < was);
  226         }
  227 
  228         /* prod += low << N */
  229         was = prodl;
  230         prodl += LHUP(low);
  231         prodh += HHALF(low) + (prodl < was);
  232         /* ... + low; */
  233         if ((prodl += low) < low)
  234                 prodh++;
  235 
  236         /* return 4N-bit product */
  237         prod.ul[H] = prodh;
  238         prod.ul[L] = prodl;
  239         return (prod.q);
  240 }

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