muldi3.c

/*-
 * Copyright (c) 1992, 1993
 *      The Regents of the University of California.  All rights reserved.
 *
 * This software was developed by the Computer Systems Engineering group
 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
 * contributed to Berkeley.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 *
 * From:
 *      @(#)muldi3.c    8.1 (Berkeley) 6/4/93
 *      NetBSD: muldi3.c,v 1.1 2005/12/20 19:28:51 christos Exp
 */

#include "longlong.h"

/*
 * Multiply two long longs.
 *
 * Our algorithm is based on the following.  Split incoming long long
 * values u and v (where u,v >= 0) into
 *
 *      u = 2^n u1  *  u0       (n = number of bits in `unsigned int', usu. 32)
 *
 * and 
 *
 *      v = 2^n v1  *  v0
 *
 * Then
 *
 *      uv = 2^2n u1 v1  +  2^n u1 v0  +  2^n v1 u0  +  u0 v0
 *         = 2^2n u1 v1  +     2^n (u1 v0 + v1 u0)   +  u0 v0
 *
 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
 * and add 2^n u0 v0 to the last term and subtract it from the middle.
 * This gives:
 *
 *      uv = (2^2n + 2^n) (u1 v1)  +
 *               (2^n)    (u1 v0 - u1 v1 + u0 v1 - u0 v0)  +
 *             (2^n + 1)  (u0 v0)
 *
 * Factoring the middle a bit gives us:
 *
 *      uv = (2^2n + 2^n) (u1 v1)  +                    [u1v1 = high]
 *               (2^n)    (u1 - u0) (v0 - v1)  +        [(u1-u0)... = mid]
 *             (2^n + 1)  (u0 v0)                       [u0v0 = low]
 *
 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
 * in just half the precision of the original.  (Note that either or both
 * of (u1 - u0) or (v0 - v1) may be negative.)
 *
 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
 *
 * Since C does not give us a `int * int = long long' operator, we split
 * our input long longs into two ints, then split the two ints into two
 * shorts.  We can then calculate `short * short = int' in native
 * arithmetic.
 *
 * Our product should, strictly speaking, be a `long long long', with
 * 128 bits, but we are going to discard the upper 64.  In other words,
 * we are not interested in uv, but rather in (uv mod 2^2n).  This
 * makes some of the terms above vanish, and we get:
 *
 *      (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
 *
 * or
 *
 *      (2^n)(high + mid + low) + low
 *
 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
 * of 2^n in either one will also vanish.  Only `low' need be computed
 * mod 2^2n, and only because of the final term above.
 */
static long long __lmulq(unsigned int, unsigned int);

long long
__muldi3(long long a, long long b)
{
        union uu u, v, low, prod;
        unsigned int high, mid, udiff, vdiff;
        int negall, negmid;
#define u1      u.ui[H]
#define u0      u.ui[L]
#define v1      v.ui[H]
#define v0      v.ui[L]

        /*
         * Get u and v such that u, v >= 0.  When this is finished,
         * u1, u0, v1, and v0 will be directly accessible through the
         * int fields.
         */
        if (a >= 0)
                u.ll = a, negall = 0;
        else
                u.ll = -a, negall = 1;
        if (b >= 0)
                v.ll = b;
        else
                v.ll = -b, negall ^= 1;

        if (u1 == 0 && v1 == 0) {
                /*
                 * An (I hope) important optimization occurs when u1 and v1
                 * are both 0.  This should be common since most numbers
                 * are small.  Here the product is just u0*v0.
                 */
                prod.ll = __lmulq(u0, v0);
        } else {
                /*
                 * Compute the three intermediate products, remembering
                 * whether the middle term is negative.  We can discard
                 * any upper bits in high and mid, so we can use native
                 * unsigned int * unsigned int => unsigned int arithmetic.
                 */
                low.ll = __lmulq(u0, v0);

                if (u1 >= u0)
                        negmid = 0, udiff = u1 - u0;
                else
                        negmid = 1, udiff = u0 - u1;
                if (v0 >= v1)
                        vdiff = v0 - v1;
                else
                        vdiff = v1 - v0, negmid ^= 1;
                mid = udiff * vdiff;

                high = u1 * v1;

                /*
                 * Assemble the final product.
                 */
                prod.ui[H] = high + (negmid ? -mid : mid) + low.ui[L] +
                    low.ui[H];
                prod.ui[L] = low.ui[L];
        }
        return (negall ? -prod.ll : prod.ll);
#undef u1
#undef u0
#undef v1
#undef v0
}

/*
 * Multiply two 2N-bit ints to produce a 4N-bit long long, where N is
 * half the number of bits in an int (whatever that is---the code
 * below does not care as long as the header file does its part of the
 * bargain---but typically N==16).
 *
 * We use the same algorithm from Knuth, but this time the modulo refinement
 * does not apply.  On the other hand, since N is half the size of an int,
 * we can get away with native multiplication---none of our input terms
 * exceeds (UINT_MAX >> 1).
 *
 * Note that, for unsigned int l, the quad-precision (long long) result
 *
 *      l << N
 *
 * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
 */
static long long
__lmulq(unsigned int u, unsigned int v)
{
        unsigned int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
        unsigned int prodh, prodl, was;
        union uu prod;
        int neg;

        u1 = HHALF(u);
        u0 = LHALF(u);
        v1 = HHALF(v);
        v0 = LHALF(v);

        low = u0 * v0;

        /* This is the same small-number optimization as before. */
        if (u1 == 0 && v1 == 0)
                return (low);

        if (u1 >= u0)
                udiff = u1 - u0, neg = 0;
        else
                udiff = u0 - u1, neg = 1;
        if (v0 >= v1)
                vdiff = v0 - v1;
        else
                vdiff = v1 - v0, neg ^= 1;
        mid = udiff * vdiff;

        high = u1 * v1;

        /* prod = (high << 2N) + (high << N); */
        prodh = high + HHALF(high);
        prodl = LHUP(high);

        /* if (neg) prod -= mid << N; else prod += mid << N; */
        if (neg) {
                was = prodl;
                prodl -= LHUP(mid);
                prodh -= HHALF(mid) + (prodl > was);
        } else {
                was = prodl;
                prodl += LHUP(mid);
                prodh += HHALF(mid) + (prodl < was);
        }

        /* prod += low << N */
        was = prodl;
        prodl += LHUP(low);
        prodh += HHALF(low) + (prodl < was);
        /* ... + low; */
        if ((prodl += low) < low)
                prodh++;

        /* return 4N-bit product */
        prod.ui[H] = prodh;
        prod.ui[L] = prodl;
        return (prod.ll);
}