common/lib/libc/quad/muldi3.c

   1 /*      $NetBSD: muldi3.c,v 1.3 2012/08/06 02:31:54 matt 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 __RCSID("$NetBSD: muldi3.c,v 1.3 2012/08/06 02:31:54 matt Exp $");
  42 #endif
  43 #endif /* LIBC_SCCS and not lint */
  44
  45 #include "quad.h"
  46
  47 ARM_EABI_ALIAS(__aeabi_lmul, __muldi3)  /* no semicolon */
  48
  49 /*
  50  * Multiply two quads.
  51  *
  52  * Our algorithm is based on the following.  Split incoming quad values
  53  * u and v (where u,v >= 0) into
  54  *
  55  *      u = 2^n u1  *  u0       (n = number of bits in `u_int', usu. 32)
  56  *
  57  * and
  58  *
  59  *      v = 2^n v1  *  v0
  60  *
  61  * Then
  62  *
  63  *      uv = 2^2n u1 v1  +  2^n u1 v0  +  2^n v1 u0  +  u0 v0
  64  *         = 2^2n u1 v1  +     2^n (u1 v0 + v1 u0)   +  u0 v0
  65  *
  66  * Now add 2^n u1 v1 to the first term and subtract it from the middle,
  67  * and add 2^n u0 v0 to the last term and subtract it from the middle.
  68  * This gives:
  69  *
  70  *      uv = (2^2n + 2^n) (u1 v1)  +
  71  *               (2^n)    (u1 v0 - u1 v1 + u0 v1 - u0 v0)  +
  72  *             (2^n + 1)  (u0 v0)
  73  *
  74  * Factoring the middle a bit gives us:
  75  *
  76  *      uv = (2^2n + 2^n) (u1 v1)  +                    [u1v1 = high]
  77  *               (2^n)    (u1 - u0) (v0 - v1)  +        [(u1-u0)... = mid]
  78  *             (2^n + 1)  (u0 v0)                       [u0v0 = low]
  79  *
  80  * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
  81  * in just half the precision of the original.  (Note that either or both
  82  * of (u1 - u0) or (v0 - v1) may be negative.)
  83  *
  84  * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
  85  *
  86  * Since C does not give us a `int * int = quad' operator, we split
  87  * our input quads into two ints, then split the two ints into two
  88  * shorts.  We can then calculate `short * short = int' in native
  89  * arithmetic.
  90  *
  91  * Our product should, strictly speaking, be a `long quad', with 128
  92  * bits, but we are going to discard the upper 64.  In other words,
  93  * we are not interested in uv, but rather in (uv mod 2^2n).  This
  94  * makes some of the terms above vanish, and we get:
  95  *
  96  *      (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
  97  *
  98  * or
  99  *
 100  *      (2^n)(high + mid + low) + low
 101  *
 102  * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
 103  * of 2^n in either one will also vanish.  Only `low' need be computed
 104  * mod 2^2n, and only because of the final term above.
 105  */
 106 static quad_t __lmulq(u_int, u_int);
 107
 108 quad_t
 109 __muldi3(quad_t a, quad_t b)
 110 {
 111         union uu u, v, low, prod;
 112         u_int high, mid, udiff, vdiff;
 113         int negall, negmid;
 114 #define u1      u.ul[H]
 115 #define u0      u.ul[L]
 116 #define v1      v.ul[H]
 117 #define v0      v.ul[L]
 118
 119         /*
 120          * Get u and v such that u, v >= 0.  When this is finished,
 121          * u1, u0, v1, and v0 will be directly accessible through the
 122          * int fields.
 123          */
 124         if (a >= 0)
 125                 u.q = a, negall = 0;
 126         else
 127                 u.q = -a, negall = 1;
 128         if (b >= 0)
 129                 v.q = b;
 130         else
 131                 v.q = -b, negall ^= 1;
 132
 133         if (u1 == 0 && v1 == 0) {
 134                 /*
 135                  * An (I hope) important optimization occurs when u1 and v1
 136                  * are both 0.  This should be common since most numbers
 137                  * are small.  Here the product is just u0*v0.
 138                  */
 139                 prod.q = __lmulq(u0, v0);
 140         } else {
 141                 /*
 142                  * Compute the three intermediate products, remembering
 143                  * whether the middle term is negative.  We can discard
 144                  * any upper bits in high and mid, so we can use native
 145                  * u_int * u_int => u_int arithmetic.
 146                  */
 147                 low.q = __lmulq(u0, v0);
 148
 149                 if (u1 >= u0)
 150                         negmid = 0, udiff = u1 - u0;
 151                 else
 152                         negmid = 1, udiff = u0 - u1;
 153                 if (v0 >= v1)
 154                         vdiff = v0 - v1;
 155                 else
 156                         vdiff = v1 - v0, negmid ^= 1;
 157                 mid = udiff * vdiff;
 158
 159                 high = u1 * v1;
 160
 161                 /*
 162                  * Assemble the final product.
 163                  */
 164                 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
 165                     low.ul[H];
 166                 prod.ul[L] = low.ul[L];
 167         }
 168         return (negall ? -prod.q : prod.q);
 169 #undef u1
 170 #undef u0
 171 #undef v1
 172 #undef v0
 173 }
 174
 175 /*
 176  * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
 177  * the number of bits in an int (whatever that is---the code below
 178  * does not care as long as quad.h does its part of the bargain---but
 179  * typically N==16).
 180  *
 181  * We use the same algorithm from Knuth, but this time the modulo refinement
 182  * does not apply.  On the other hand, since N is half the size of an int,
 183  * we can get away with native multiplication---none of our input terms
 184  * exceeds (UINT_MAX >> 1).
 185  *
 186  * Note that, for u_int l, the quad-precision result
 187  *
 188  *      l << N
 189  *
 190  * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
 191  */
 192 static quad_t
 193 __lmulq(u_int u, u_int v)
 194 {
 195         u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
 196         u_int prodh, prodl, was;
 197         union uu prod;
 198         int neg;
 199
 200         u1 = HHALF(u);
 201         u0 = LHALF(u);
 202         v1 = HHALF(v);
 203         v0 = LHALF(v);
 204
 205         low = u0 * v0;
 206
 207         /* This is the same small-number optimization as before. */
 208         if (u1 == 0 && v1 == 0)
 209                 return (low);
 210
 211         if (u1 >= u0)
 212                 udiff = u1 - u0, neg = 0;
 213         else
 214                 udiff = u0 - u1, neg = 1;
 215         if (v0 >= v1)
 216                 vdiff = v0 - v1;
 217         else
 218                 vdiff = v1 - v0, neg ^= 1;
 219         mid = udiff * vdiff;
 220
 221         high = u1 * v1;
 222
 223         /* prod = (high << 2N) + (high << N); */
 224         prodh = high + HHALF(high);
 225         prodl = LHUP(high);
 226
 227         /* if (neg) prod -= mid << N; else prod += mid << N; */
 228         if (neg) {
 229                 was = prodl;
 230                 prodl -= LHUP(mid);
 231                 prodh -= HHALF(mid) + (prodl > was);
 232         } else {
 233                 was = prodl;
 234                 prodl += LHUP(mid);
 235                 prodh += HHALF(mid) + (prodl < was);
 236         }
 237
 238         /* prod += low << N */
 239         was = prodl;
 240         prodl += LHUP(low);
 241         prodh += HHALF(low) + (prodl < was);
 242         /* ... + low; */
 243         if ((prodl += low) < low)
 244                 prodh++;
 245
 246         /* return 4N-bit product */
 247         prod.ul[H] = prodh;
 248         prod.ul[L] = prodl;
 249         return (prod.q);
 250 }