sys/libkern/muldi3.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/sys/libkern/muldi3.c,v 1.6 1999/08/28 00:46:34 peter Exp $
  38  * $DragonFly: src/sys/libkern/muldi3.c,v 1.4 2004/01/26 11:09:44 joerg Exp $
  39  */
  40
  41 #include "quad.h"
  42
  43 /*
  44  * Multiply two quads.
  45  *
  46  * Our algorithm is based on the following.  Split incoming quad values
  47  * u and v (where u,v >= 0) into
  48  *
  49  *      u = 2^n u1  *  u0       (n = number of bits in `u_long', usu. 32)
  50  *
  51  * and
  52  *
  53  *      v = 2^n v1  *  v0
  54  *
  55  * Then
  56  *
  57  *      uv = 2^2n u1 v1  +  2^n u1 v0  +  2^n v1 u0  +  u0 v0
  58  *         = 2^2n u1 v1  +     2^n (u1 v0 + v1 u0)   +  u0 v0
  59  *
  60  * Now add 2^n u1 v1 to the first term and subtract it from the middle,
  61  * and add 2^n u0 v0 to the last term and subtract it from the middle.
  62  * This gives:
  63  *
  64  *      uv = (2^2n + 2^n) (u1 v1)  +
  65  *               (2^n)    (u1 v0 - u1 v1 + u0 v1 - u0 v0)  +
  66  *             (2^n + 1)  (u0 v0)
  67  *
  68  * Factoring the middle a bit gives us:
  69  *
  70  *      uv = (2^2n + 2^n) (u1 v1)  +                    [u1v1 = high]
  71  *               (2^n)    (u1 - u0) (v0 - v1)  +        [(u1-u0)... = mid]
  72  *             (2^n + 1)  (u0 v0)                       [u0v0 = low]
  73  *
  74  * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
  75  * in just half the precision of the original.  (Note that either or both
  76  * of (u1 - u0) or (v0 - v1) may be negative.)
  77  *
  78  * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
  79  *
  80  * Since C does not give us a `long * long = quad' operator, we split
  81  * our input quads into two longs, then split the two longs into two
  82  * shorts.  We can then calculate `short * short = long' in native
  83  * arithmetic.
  84  *
  85  * Our product should, strictly speaking, be a `long quad', with 128
  86  * bits, but we are going to discard the upper 64.  In other words,
  87  * we are not interested in uv, but rather in (uv mod 2^2n).  This
  88  * makes some of the terms above vanish, and we get:
  89  *
  90  *      (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
  91  *
  92  * or
  93  *
  94  *      (2^n)(high + mid + low) + low
  95  *
  96  * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
  97  * of 2^n in either one will also vanish.  Only `low' need be computed
  98  * mod 2^2n, and only because of the final term above.
  99  */
 100 static quad_t __lmulq(u_long u, u_long v);
 101
 102 quad_t
 103 __muldi3(quad_t a, quad_t b)
 104 {
 105         union uu u, v, low, prod;
 106         u_long high, mid, udiff, vdiff;
 107         int negall, negmid;
 108 #define u1      u.ul[H]
 109 #define u0      u.ul[L]
 110 #define v1      v.ul[H]
 111 #define v0      v.ul[L]
 112
 113         /*
 114          * Get u and v such that u, v >= 0.  When this is finished,
 115          * u1, u0, v1, and v0 will be directly accessible through the
 116          * longword fields.
 117          */
 118         if (a >= 0)
 119                 u.q = a, negall = 0;
 120         else
 121                 u.q = -a, negall = 1;
 122         if (b >= 0)
 123                 v.q = b;
 124         else
 125                 v.q = -b, negall ^= 1;
 126
 127         if (u1 == 0 && v1 == 0) {
 128                 /*
 129                  * An (I hope) important optimization occurs when u1 and v1
 130                  * are both 0.  This should be common since most numbers
 131                  * are small.  Here the product is just u0*v0.
 132                  */
 133                 prod.q = __lmulq(u0, v0);
 134         } else {
 135                 /*
 136                  * Compute the three intermediate products, remembering
 137                  * whether the middle term is negative.  We can discard
 138                  * any upper bits in high and mid, so we can use native
 139                  * u_long * u_long => u_long arithmetic.
 140                  */
 141                 low.q = __lmulq(u0, v0);
 142
 143                 if (u1 >= u0)
 144                         negmid = 0, udiff = u1 - u0;
 145                 else
 146                         negmid = 1, udiff = u0 - u1;
 147                 if (v0 >= v1)
 148                         vdiff = v0 - v1;
 149                 else
 150                         vdiff = v1 - v0, negmid ^= 1;
 151                 mid = udiff * vdiff;
 152
 153                 high = u1 * v1;
 154
 155                 /*
 156                  * Assemble the final product.
 157                  */
 158                 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
 159                     low.ul[H];
 160                 prod.ul[L] = low.ul[L];
 161         }
 162         return (negall ? -prod.q : prod.q);
 163 #undef u1
 164 #undef u0
 165 #undef v1
 166 #undef v0
 167 }
 168
 169 /*
 170  * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half
 171  * the number of bits in a long (whatever that is---the code below
 172  * does not care as long as quad.h does its part of the bargain---but
 173  * typically N==16).
 174  *
 175  * We use the same algorithm from Knuth, but this time the modulo refinement
 176  * does not apply.  On the other hand, since N is half the size of a long,
 177  * we can get away with native multiplication---none of our input terms
 178  * exceeds (ULONG_MAX >> 1).
 179  *
 180  * Note that, for u_long l, the quad-precision result
 181  *
 182  *      l << N
 183  *
 184  * splits into high and low longs as HHALF(l) and LHUP(l) respectively.
 185  */
 186 static quad_t
 187 __lmulq(u_long u, u_long v)
 188 {
 189         u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low;
 190         u_long prodh, prodl, was;
 191         union uu prod;
 192         int neg;
 193
 194         u1 = HHALF(u);
 195         u0 = LHALF(u);
 196         v1 = HHALF(v);
 197         v0 = LHALF(v);
 198
 199         low = u0 * v0;
 200
 201         /* This is the same small-number optimization as before. */
 202         if (u1 == 0 && v1 == 0)
 203                 return (low);
 204
 205         if (u1 >= u0)
 206                 udiff = u1 - u0, neg = 0;
 207         else
 208                 udiff = u0 - u1, neg = 1;
 209         if (v0 >= v1)
 210                 vdiff = v0 - v1;
 211         else
 212                 vdiff = v1 - v0, neg ^= 1;
 213         mid = udiff * vdiff;
 214
 215         high = u1 * v1;
 216
 217         /* prod = (high << 2N) + (high << N); */
 218         prodh = high + HHALF(high);
 219         prodl = LHUP(high);
 220
 221         /* if (neg) prod -= mid << N; else prod += mid << N; */
 222         if (neg) {
 223                 was = prodl;
 224                 prodl -= LHUP(mid);
 225                 prodh -= HHALF(mid) + (prodl > was);
 226         } else {
 227                 was = prodl;
 228                 prodl += LHUP(mid);
 229                 prodh += HHALF(mid) + (prodl < was);
 230         }
 231
 232         /* prod += low << N */
 233         was = prodl;
 234         prodl += LHUP(low);
 235         prodh += HHALF(low) + (prodl < was);
 236         /* ... + low; */
 237         if ((prodl += low) < low)
 238                 prodh++;
 239
 240         /* return 4N-bit product */
 241         prod.ul[H] = prodh;
 242         prod.ul[L] = prodl;
 243         return (prod.q);
 244 }