/* $OpenBSD: muldi3.c,v 1.5 2005/08/08 08:05:35 espie Exp $ */
  /*-
   * 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.
   */
  
  #include "quad.h"
  
  /*
   * Multiply two quads.
   *
   * Our algorithm is based on the following.  Split incoming quad values
   * u and v (where u,v >= 0) into
   *
   *  u = 2^n u1  *  u0   (n = number of bits in `u_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 = quad' operator, we split
   * our input quads 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 quad', 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 quad_t __lmulq(u_int, u_int);
  
  quad_t __muldi3(quad_t a, quad_t b)
  {
      union uu u, v, low, prod;
      u_int high, mid, udiff, vdiff;
      int negall, negmid;
  #define u1  u.ul[H]
  #define u0  u.ul[L]
  #define v1  v.ul[H]
  #define v0  v.ul[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.q = a, negall = 0;
      }
      else {
          u.q = -a, negall = 1;
      }
  
      if (b >= 0) {
          v.q = b;
      }
      else {
          v.q = -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.q = __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
           * u_int * u_int => u_int arithmetic.
           */
          low.q = __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.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + low.ul[H];
          prod.ul[L] = low.ul[L];
      }
  
      return negall ? -prod.q : prod.q;
  #undef u1
  #undef u0
  #undef v1
  #undef v0
  }
  
  /*
   * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
   * the number of bits in an int (whatever that is---the code below
   * does not care as long as quad.h 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 u_int l, the quad-precision result
   *
   *  l << N
   *
   * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
   */
  static quad_t __lmulq(u_int u, u_int v)
  {
      u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
      u_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.ul[H] = prodh;
      prod.ul[L] = prodl;
      return prod.q;
  }