(mpx_udiv): Fix bug in quotient digit estimation.

[u/mdw/catacomb] / mpx.c

/* -*-c-*-
 *
 * $Id: mpx.c,v 1.8 2000/06/25 12:59:02 mdw Exp $
 *
 * Low-level multiprecision arithmetic
 *
 * (c) 1999 Straylight/Edgeware
 */

/*----- Licensing notice --------------------------------------------------* 
 *
 * This file is part of Catacomb.
 *
 * Catacomb is free software; you can redistribute it and/or modify
 * it under the terms of the GNU Library General Public License as
 * published by the Free Software Foundation; either version 2 of the
 * License, or (at your option) any later version.
 * 
 * Catacomb is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Library General Public License for more details.
 * 
 * You should have received a copy of the GNU Library General Public
 * License along with Catacomb; if not, write to the Free
 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
 * MA 02111-1307, USA.
 */

/*----- Revision history --------------------------------------------------* 
 *
 * $Log: mpx.c,v $
 * Revision 1.8  2000/06/25 12:59:02  mdw
 * (mpx_udiv): Fix bug in quotient digit estimation.
 *
 * Revision 1.7  1999/12/22 15:49:07  mdw
 * New function for division by a small integer.
 *
 * Revision 1.6  1999/11/20 22:43:44  mdw
 * Integrate testing for MPX routines.
 *
 * Revision 1.5  1999/11/20 22:23:27  mdw
 * Add function versions of some low-level macros with wider use.
 *
 * Revision 1.4  1999/11/17 18:04:09  mdw
 * Add two's-complement functionality.  Improve mpx_udiv a little by
 * performing the multiplication of the divisor by q with the subtraction
 * from r.
 *
 * Revision 1.3  1999/11/13 01:57:31  mdw
 * Remove stray debugging code.
 *
 * Revision 1.2  1999/11/13 01:50:59  mdw
 * Multiprecision routines finished and tested.
 *
 * Revision 1.1  1999/09/03 08:41:12  mdw
 * Initial import.
 *
 */

/*----- Header files ------------------------------------------------------*/

#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#include <mLib/bits.h>

#include "mptypes.h"
#include "mpx.h"

/*----- Loading and storing -----------------------------------------------*/

/* --- @mpx_storel@ --- *
 *
 * Arguments:   @const mpw *v, *vl@ = base and limit of source vector
 *              @void *pp@ = pointer to octet array
 *              @size_t sz@ = size of octet array
 *
 * Returns:     ---
 *
 * Use:         Stores an MP in an octet array, least significant octet
 *              first.  High-end octets are silently discarded if there
 *              isn't enough space for them.
 */

void mpx_storel(const mpw *v, const mpw *vl, void *pp, size_t sz)
{
  mpw n, w = 0;
  octet *p = pp, *q = p + sz;
  unsigned bits = 0;

  while (p < q) {
    if (bits < 8) {
      if (v >= vl) {
        *p++ = U8(w);
        break;
      }
      n = *v++;
      *p++ = U8(w | n << bits);
      w = n >> (8 - bits);
      bits += MPW_BITS - 8;
    } else {
      *p++ = U8(w);
      w >>= 8;
      bits -= 8;
    }
  }
  memset(p, 0, q - p);
}

/* --- @mpx_loadl@ --- *
 *
 * Arguments:   @mpw *v, *vl@ = base and limit of destination vector
 *              @const void *pp@ = pointer to octet array
 *              @size_t sz@ = size of octet array
 *
 * Returns:     ---
 *
 * Use:         Loads an MP in an octet array, least significant octet
 *              first.  High-end octets are ignored if there isn't enough
 *              space for them.
 */

void mpx_loadl(mpw *v, mpw *vl, const void *pp, size_t sz)
{
  unsigned n;
  mpw w = 0;
  const octet *p = pp, *q = p + sz;
  unsigned bits = 0;

  if (v >= vl)
    return;
  while (p < q) {
    n = U8(*p++);
    w |= n << bits;
    bits += 8;
    if (bits >= MPW_BITS) {
      *v++ = MPW(w);
      w = n >> (MPW_BITS - bits + 8);
      bits -= MPW_BITS;
      if (v >= vl)
        return;
    }
  }
  *v++ = w;
  MPX_ZERO(v, vl);
}

/* --- @mpx_storeb@ --- *
 *
 * Arguments:   @const mpw *v, *vl@ = base and limit of source vector
 *              @void *pp@ = pointer to octet array
 *              @size_t sz@ = size of octet array
 *
 * Returns:     ---
 *
 * Use:         Stores an MP in an octet array, most significant octet
 *              first.  High-end octets are silently discarded if there
 *              isn't enough space for them.
 */

void mpx_storeb(const mpw *v, const mpw *vl, void *pp, size_t sz)
{
  mpw n, w = 0;
  octet *p = pp, *q = p + sz;
  unsigned bits = 0;

  while (q > p) {
    if (bits < 8) {
      if (v >= vl) {
        *--q = U8(w);
        break;
      }
      n = *v++;
      *--q = U8(w | n << bits);
      w = n >> (8 - bits);
      bits += MPW_BITS - 8;
    } else {
      *--q = U8(w);
      w >>= 8;
      bits -= 8;
    }
  }
  memset(p, 0, q - p);
}

/* --- @mpx_loadb@ --- *
 *
 * Arguments:   @mpw *v, *vl@ = base and limit of destination vector
 *              @const void *pp@ = pointer to octet array
 *              @size_t sz@ = size of octet array
 *
 * Returns:     ---
 *
 * Use:         Loads an MP in an octet array, most significant octet
 *              first.  High-end octets are ignored if there isn't enough
 *              space for them.
 */

void mpx_loadb(mpw *v, mpw *vl, const void *pp, size_t sz)
{
  unsigned n;
  mpw w = 0;
  const octet *p = pp, *q = p + sz;
  unsigned bits = 0;

  if (v >= vl)
    return;
  while (q > p) {
    n = U8(*--q);
    w |= n << bits;
    bits += 8;
    if (bits >= MPW_BITS) {
      *v++ = MPW(w);
      w = n >> (MPW_BITS - bits + 8);
      bits -= MPW_BITS;
      if (v >= vl)
        return;
    }
  }
  *v++ = w;
  MPX_ZERO(v, vl);
}

/*----- Logical shifting --------------------------------------------------*/

/* --- @mpx_lsl@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *avl@ = source vector base and limit
 *              @size_t n@ = number of bit positions to shift by
 *
 * Returns:     ---
 *
 * Use:         Performs a logical shift left operation on an integer.
 */

void mpx_lsl(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl, size_t n)
{
  size_t nw;
  unsigned nb;

  /* --- Trivial special case --- */

  if (n == 0)
    MPX_COPY(dv, dvl, av, avl);

  /* --- Single bit shifting --- */

  else if (n == 1) {
    mpw w = 0;
    while (av < avl) {
      mpw t;
      if (dv >= dvl)
        goto done;
      t = *av++;
      *dv++ = MPW((t << 1) | w);
      w = t >> (MPW_BITS - 1);
    }
    if (dv >= dvl)
      goto done;
    *dv++ = MPW(w);
    MPX_ZERO(dv, dvl);
    goto done;
  }

  /* --- Break out word and bit shifts for more sophisticated work --- */
        
  nw = n / MPW_BITS;
  nb = n % MPW_BITS;

  /* --- Handle a shift by a multiple of the word size --- */

  if (nb == 0) {
    MPX_COPY(dv + nw, dvl, av, avl);
    memset(dv, 0, MPWS(nw));
  }

  /* --- And finally the difficult case --- *
   *
   * This is a little convoluted, because I have to start from the end and
   * work backwards to avoid overwriting the source, if they're both the same
   * block of memory.
   */

  else {
    mpw w;
    size_t nr = MPW_BITS - nb;
    size_t dvn = dvl - dv;
    size_t avn = avl - av;

    if (dvn <= nw) {
      MPX_ZERO(dv, dvl);
      goto done;
    }

    if (dvn > avn + nw) {
      size_t off = avn + nw + 1;
      MPX_ZERO(dv + off, dvl);
      dvl = dv + off;
      w = 0;
    } else {
      avl = av + dvn - nw;
      w = *--avl << nb;
    }

    while (avl > av) {
      mpw t = *--avl;
      *--dvl = (t >> nr) | w;
      w = t << nb;
    }

    *--dvl = w;
    MPX_ZERO(dv, dvl);
  }

done:;
}

/* --- @mpx_lsr@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *avl@ = source vector base and limit
 *              @size_t n@ = number of bit positions to shift by
 *
 * Returns:     ---
 *
 * Use:         Performs a logical shift right operation on an integer.
 */

void mpx_lsr(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl, size_t n)
{
  size_t nw;
  unsigned nb;

  /* --- Trivial special case --- */

  if (n == 0)
    MPX_COPY(dv, dvl, av, avl);

  /* --- Single bit shifting --- */

  else if (n == 1) {
    mpw w = *av++ >> 1;
    while (av < avl) {
      mpw t;
      if (dv >= dvl)
        goto done;
      t = *av++;
      *dv++ = MPW((t << (MPW_BITS - 1)) | w);
      w = t >> 1;
    }
    if (dv >= dvl)
      goto done;
    *dv++ = MPW(w);
    MPX_ZERO(dv, dvl);
    goto done;
  }

  /* --- Break out word and bit shifts for more sophisticated work --- */

  nw = n / MPW_BITS;
  nb = n % MPW_BITS;

  /* --- Handle a shift by a multiple of the word size --- */

  if (nb == 0)
    MPX_COPY(dv, dvl, av + nw, avl);

  /* --- And finally the difficult case --- */

  else {
    mpw w;
    size_t nr = MPW_BITS - nb;

    av += nw;
    w = *av++;
    while (av < avl) {
      mpw t;
      if (dv >= dvl)
        goto done;
      t = *av++;
      *dv++ = MPW((w >> nb) | (t << nr));
      w = t;
    }
    if (dv < dvl) {
      *dv++ = MPW(w >> nb);
      MPX_ZERO(dv, dvl);
    }
  }

done:;
}

/*----- Unsigned arithmetic -----------------------------------------------*/

/* --- @mpx_2c@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector
 *              @const mpw *v, *vl@ = source vector
 *
 * Returns:     ---
 *
 * Use:         Calculates the two's complement of @v@.
 */

void mpx_2c(mpw *dv, mpw *dvl, const mpw *v, const mpw *vl)
{
  mpw c = 0;
  while (dv < dvl && v < vl)
    *dv++ = c = MPW(~*v++);
  if (dv < dvl) {
    if (c > MPW_MAX / 2)
      c = MPW(~0);
    while (dv < dvl)
      *dv++ = c;
  }
  MPX_UADDN(dv, dvl, 1);
}

/* --- @mpx_ucmp@ --- *
 *
 * Arguments:   @const mpw *av, *avl@ = first argument vector base and limit
 *              @const mpw *bv, *bvl@ = second argument vector base and limit
 *
 * Returns:     Less than, equal to, or greater than zero depending on
 *              whether @a@ is less than, equal to or greater than @b@,
 *              respectively.
 *
 * Use:         Performs an unsigned integer comparison.
 */

int mpx_ucmp(const mpw *av, const mpw *avl, const mpw *bv, const mpw *bvl)
{
  MPX_SHRINK(av, avl);
  MPX_SHRINK(bv, bvl);

  if (avl - av > bvl - bv)
    return (+1);
  else if (avl - av < bvl - bv)
    return (-1);
  else while (avl > av) {
    mpw a = *--avl, b = *--bvl;
    if (a > b)
      return (+1);
    else if (a < b)
      return (-1);
  }
  return (0);
}
  
/* --- @mpx_uadd@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *avl@ = first addend vector base and limit
 *              @const mpw *bv, *bvl@ = second addend vector base and limit
 *
 * Returns:     ---
 *
 * Use:         Performs unsigned integer addition.  If the result overflows
 *              the destination vector, high-order bits are discarded.  This
 *              means that two's complement addition happens more or less for
 *              free, although that's more a side-effect than anything else.
 *              The result vector may be equal to either or both source
 *              vectors, but may not otherwise overlap them.
 */

void mpx_uadd(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl,
              const mpw *bv, const mpw *bvl)
{
  mpw c = 0;

  while (av < avl || bv < bvl) {
    mpw a, b;
    mpd x;
    if (dv >= dvl)
      return;
    a = (av < avl) ? *av++ : 0;
    b = (bv < bvl) ? *bv++ : 0;
    x = (mpd)a + (mpd)b + c;
    *dv++ = MPW(x);
    c = x >> MPW_BITS;
  }
  if (dv < dvl) {
    *dv++ = c;
    MPX_ZERO(dv, dvl);
  }
}

/* --- @mpx_uaddn@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = source and destination base and limit
 *              @mpw n@ = other addend
 *
 * Returns:     ---
 *
 * Use:         Adds a small integer to a multiprecision number.
 */

void mpx_uaddn(mpw *dv, mpw *dvl, mpw n) { MPX_UADDN(dv, dvl, n); }

/* --- @mpx_usub@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *avl@ = first argument vector base and limit
 *              @const mpw *bv, *bvl@ = second argument vector base and limit
 *
 * Returns:     ---
 *
 * Use:         Performs unsigned integer subtraction.  If the result
 *              overflows the destination vector, high-order bits are
 *              discarded.  This means that two's complement subtraction
 *              happens more or less for free, althuogh that's more a side-
 *              effect than anything else.  The result vector may be equal to
 *              either or both source vectors, but may not otherwise overlap
 *              them.
 */

void mpx_usub(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl,
              const mpw *bv, const mpw *bvl)
{
  mpw c = 0;

  while (av < avl || bv < bvl) {
    mpw a, b;
    mpd x;
    if (dv >= dvl)
      return;
    a = (av < avl) ? *av++ : 0;
    b = (bv < bvl) ? *bv++ : 0;
    x = (mpd)a - (mpd)b - c;
    *dv++ = MPW(x);
    if (x >> MPW_BITS)
      c = 1;
    else
      c = 0;
  }
  if (c)
    c = MPW_MAX;
  while (dv < dvl)
    *dv++ = c;
}

/* --- @mpx_usubn@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = source and destination base and limit
 *              @n@ = subtrahend
 *
 * Returns:     ---
 *
 * Use:         Subtracts a small integer from a multiprecision number.
 */

void mpx_usubn(mpw *dv, mpw *dvl, mpw n) { MPX_USUBN(dv, dvl, n); }

/* --- @mpx_umul@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *avl@ = multiplicand vector base and limit
 *              @const mpw *bv, *bvl@ = multiplier vector base and limit
 *
 * Returns:     ---
 *
 * Use:         Performs unsigned integer multiplication.  If the result
 *              overflows the desination vector, high-order bits are
 *              discarded.  The result vector may not overlap the argument
 *              vectors in any way.
 */

void mpx_umul(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl,
              const mpw *bv, const mpw *bvl)
{
  /* --- This is probably worthwhile on a multiply --- */

  MPX_SHRINK(av, avl);
  MPX_SHRINK(bv, bvl);

  /* --- Deal with a multiply by zero --- */
  
  if (bv == bvl) {
    MPX_ZERO(dv, dvl);
    return;
  }

  /* --- Do the initial multiply and initialize the accumulator --- */

  MPX_UMULN(dv, dvl, av, avl, *bv++);

  /* --- Do the remaining multiply/accumulates --- */

  while (dv < dvl && bv < bvl) {
    mpw m = *bv++;
    mpw c = 0;
    const mpw *avv = av;
    mpw *dvv = ++dv;

    while (avv < avl) {
      mpd x;
      if (dvv >= dvl)
        goto next;
      x = (mpd)*dvv + (mpd)m * (mpd)*avv++ + c;
      *dvv++ = MPW(x);
      c = x >> MPW_BITS;
    }
    MPX_UADDN(dvv, dvl, c);
  next:;
  }
}

/* --- @mpx_umuln@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *avl@ = multiplicand vector base and limit
 *              @mpw m@ = multiplier
 *
 * Returns:     ---
 *
 * Use:         Multiplies a multiprecision integer by a single-word value.
 *              The destination and source may be equal.  The destination
 *              is completely cleared after use.
 */

void mpx_umuln(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl, mpw m)
{
  MPX_UMULN(dv, dvl, av, avl, m);
}

/* --- @mpx_umlan@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination/accumulator base and limit
 *              @const mpw *av, *avl@ = multiplicand vector base and limit
 *              @mpw m@ = multiplier
 *
 * Returns:     ---
 *
 * Use:         Multiplies a multiprecision integer by a single-word value
 *              and adds the result to an accumulator.
 */

void mpx_umlan(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl, mpw m)
{
  MPX_UMLAN(dv, dvl, av, avl, m);
}

/* --- @mpx_usqr@ --- *
 *
 * Arguments:   @mpw *dv, *dvl@ = destination vector base and limit
 *              @const mpw *av, *av@ = source vector base and limit
 *
 * Returns:     ---
 *
 * Use:         Performs unsigned integer squaring.  The result vector must
 *              not overlap the source vector in any way.
 */

void mpx_usqr(mpw *dv, mpw *dvl, const mpw *av, const mpw *avl)
{
  MPX_ZERO(dv, dvl);

  /* --- Main loop --- */

  while (av < avl) {
    const mpw *avv = av;
    mpw *dvv = dv;
    mpw a = *av;
    mpd c;

    /* --- Stop if I've run out of destination --- */

    if (dvv >= dvl)
      break;

    /* --- Work out the square at this point in the proceedings --- */

    {
      mpd x = (mpd)a * (mpd)a + *dvv;
      *dvv++ = MPW(x);
      c = MPW(x >> MPW_BITS);
    }

    /* --- Now fix up the rest of the vector upwards --- */

    avv++;
    while (dvv < dvl && avv < avl) {
      mpd x = (mpd)a * (mpd)*avv++;
      mpd y = ((x << 1) & MPW_MAX) + c + *dvv;
      c = (x >> (MPW_BITS - 1)) + (y >> MPW_BITS);
      *dvv++ = MPW(y);
    }
    while (dvv < dvl && c) {
      mpd x = c + *dvv;
      *dvv++ = MPW(x);
      c = x >> MPW_BITS;
    }

    /* --- Get ready for the next round --- */

    av++;
    dv += 2;
  }
}

/* --- @mpx_udiv@ --- *
 *
 * Arguments:   @mpw *qv, *qvl@ = quotient vector base and limit
 *              @mpw *rv, *rvl@ = dividend/remainder vector base and limit
 *              @const mpw *dv, *dvl@ = divisor vector base and limit
 *              @mpw *sv, *svl@ = scratch workspace
 *
 * Returns:     ---
 *
 * Use:         Performs unsigned integer division.  If the result overflows
 *              the quotient vector, high-order bits are discarded.  (Clearly
 *              the remainder vector can't overflow.)  The various vectors
 *              may not overlap in any way.  Yes, I know it's a bit odd
 *              requiring the dividend to be in the result position but it
 *              does make some sense really.  The remainder must have
 *              headroom for at least two extra words.  The scratch space
 *              must be at least one word larger than the divisor.
 */

void mpx_udiv(mpw *qv, mpw *qvl, mpw *rv, mpw *rvl,
              const mpw *dv, const mpw *dvl,
              mpw *sv, mpw *svl)
{
  unsigned norm = 0;
  size_t scale;
  mpw d, dd;

  /* --- Initialize the quotient --- */

  MPX_ZERO(qv, qvl);

  /* --- Perform some sanity checks --- */

  MPX_SHRINK(dv, dvl);
  assert(((void)"division by zero in mpx_udiv", dv < dvl));

  /* --- Normalize the divisor --- *
   *
   * The algorithm requires that the divisor be at least two digits long.
   * This is easy to fix.
   */

  {
    unsigned b;

    d = dvl[-1];
    for (b = MPW_BITS / 2; b; b >>= 1) {
      if (d < (MPW_MAX >> b)) {
        d <<= b;
        norm += b;
      }
    }
    if (dv + 1 == dvl)
      norm += MPW_BITS;
  }

  /* --- Normalize the dividend/remainder to match --- */

  if (norm) {
    mpx_lsl(rv, rvl, rv, rvl, norm);
    mpx_lsl(sv, svl, dv, dvl, norm);
    dv = sv;
    dvl = svl;
    MPX_SHRINK(dv, dvl);
  }

  MPX_SHRINK(rv, rvl);
  d = dvl[-1];
  dd = dvl[-2];

  /* --- Work out the relative scales --- */

  {
    size_t rvn = rvl - rv;
    size_t dvn = dvl - dv;

    /* --- If the divisor is clearly larger, notice this --- */

    if (dvn > rvn) {
      mpx_lsr(rv, rvl, rv, rvl, norm);
      return;
    }

    scale = rvn - dvn;
  }

  /* --- Calculate the most significant quotient digit --- *
   *
   * Because the divisor has its top bit set, this can only happen once.  The
   * pointer arithmetic is a little contorted, to make sure that the
   * behaviour is defined.
   */

  if (MPX_UCMP(rv + scale, rvl, >=, dv, dvl)) {
    mpx_usub(rv + scale, rvl, rv + scale, rvl, dv, dvl);
    if (qvl - qv > scale)
      qv[scale] = 1;
  }

  /* --- Now for the main loop --- */

  {
    mpw *rvv = rvl - 2;

    while (scale) {
      mpw q;
      mpd rh;

      /* --- Get an estimate for the next quotient digit --- */

      mpw r = rvv[1];
      mpw rr = rvv[0];
      mpw rrr = *--rvv;

      scale--;
      rh = ((mpd)r << MPW_BITS) | rr;
      if (r == d)
        q = MPW_MAX;
      else
        q = MPW(rh / d);

      /* --- Refine the estimate --- */

      {
        mpd yh = (mpd)d * q;
        mpd yy = (mpd)dd * q;
        mpw yl;

        if (yy > MPW_MAX)
          yh += yy >> MPW_BITS;
        yl = MPW(yy);

        while (yh > rh || (yh == rh && yl > rrr)) {
          q--;
          yh -= d;
          if (yl < dd)
            yh--;
          yl -= dd;
        }
      }

      /* --- Remove a chunk from the dividend --- */

      {
        mpw *svv;
        const mpw *dvv;
        mpw mc = 0, sc = 0;

        /* --- Calculate the size of the chunk --- *
         *
         * This does the whole job of calculating @r >> scale - qd@.
         */

        for (svv = rv + scale, dvv = dv;
             dvv < dvl && svv < rvl;
             svv++, dvv++) {
          mpd x = (mpd)*dvv * (mpd)q + mc;
          mc = x >> MPW_BITS;
          x = (mpd)*svv - MPW(x) - sc;
          *svv = MPW(x);
          if (x >> MPW_BITS)
            sc = 1;
          else
            sc = 0;
        }

        if (svv < rvl) {
          mpd x = (mpd)*svv - mc - sc;
          *svv++ = MPW(x);
          if (x >> MPW_BITS)
            sc = MPW_MAX;
          else
            sc = 0;
          while (svv < rvl)
            *svv++ = sc;
        }

        /* --- Fix if the quotient was too large --- *
         *
         * This doesn't seem to happen very often.
         */

        if (rvl[-1] > MPW_MAX / 2) {
          mpx_uadd(rv + scale, rvl, rv + scale, rvl, dv, dvl);
          q--;
        }
      }

      /* --- Done for another iteration --- */

      if (qvl - qv > scale)
        qv[scale] = q;
      r = rr;
      rr = rrr;
    }
  }

  /* --- Now fiddle with unnormalizing and things --- */

  mpx_lsr(rv, rvl, rv, rvl, norm);
}

/* --- @mpx_udivn@ --- *
 *
 * Arguments:   @mpw *qv, *qvl@ = storage for the quotient (may overlap
 *                      dividend)
 *              @const mpw *rv, *rvl@ = dividend
 *              @mpw d@ = single-precision divisor
 *
 * Returns:     Remainder after divison.
 *
 * Use:         Performs a single-precision division operation.
 */

mpw mpx_udivn(mpw *qv, mpw *qvl, const mpw *rv, const mpw *rvl, mpw d)
{
  size_t i;
  size_t ql = qvl - qv;
  mpd r = 0;

  i = rvl - rv;
  while (i > 0) {
    i--;
    r = (r << MPW_BITS) | rv[i];
    if (i < ql)
      qv[i] = r / d;
    r %= d;
  }
  return (MPW(r));
}

/*----- Test rig ----------------------------------------------------------*/

#ifdef TEST_RIG

#include <mLib/alloc.h>
#include <mLib/dstr.h>
#include <mLib/quis.h>
#include <mLib/testrig.h>

#include "mpscan.h"

#define ALLOC(v, vl, sz) do {                                           \
  size_t _sz = (sz);                                                    \
  mpw *_vv = xmalloc(MPWS(_sz));                                        \
  mpw *_vvl = _vv + _sz;                                                \
  (v) = _vv;                                                            \
  (vl) = _vvl;                                                          \
} while (0)

#define LOAD(v, vl, d) do {                                             \
  const dstr *_d = (d);                                                 \
  mpw *_v, *_vl;                                                        \
  ALLOC(_v, _vl, MPW_RQ(_d->len));                                      \
  mpx_loadb(_v, _vl, _d->buf, _d->len);                                 \
  (v) = _v;                                                             \
  (vl) = _vl;                                                           \
} while (0)

#define MAX(x, y) ((x) > (y) ? (x) : (y))
  
static void dumpbits(const char *msg, const void *pp, size_t sz)
{
  const octet *p = pp;
  fputs(msg, stderr);
  for (; sz; sz--)
    fprintf(stderr, " %02x", *p++);
  fputc('\n', stderr);
}

static void dumpmp(const char *msg, const mpw *v, const mpw *vl)
{
  fputs(msg, stderr);
  MPX_SHRINK(v, vl);
  while (v < vl)
    fprintf(stderr, " %08lx", (unsigned long)*--vl);
  fputc('\n', stderr);
}

static int chkscan(const mpw *v, const mpw *vl,
                   const void *pp, size_t sz, int step)
{
  mpscan mps;
  const octet *p = pp;
  unsigned bit = 0;
  int ok = 1;

  mpscan_initx(&mps, v, vl);
  while (sz) {
    unsigned x = *p;
    int i;
    p += step;
    for (i = 0; i < 8 && MPSCAN_STEP(&mps); i++) {
      if (MPSCAN_BIT(&mps) != (x & 1)) {
        fprintf(stderr,
                "\n*** error, step %i, bit %u, expected %u, found %u\n",
                step, bit, x & 1, MPSCAN_BIT(&mps));
        ok = 0;
      }
      x >>= 1;
      bit++;
    }
    sz--;
  }

  return (ok);
}

static int loadstore(dstr *v)
{
  dstr d = DSTR_INIT;
  size_t sz = MPW_RQ(v->len) * 2, diff;
  mpw *m, *ml;
  int ok = 1;

  dstr_ensure(&d, v->len);
  m = xmalloc(MPWS(sz));

  for (diff = 0; diff < sz; diff += 5) {
    size_t oct;

    ml = m + sz - diff;

    mpx_loadl(m, ml, v->buf, v->len);
    if (!chkscan(m, ml, v->buf, v->len, +1))
      ok = 0;
    MPX_OCTETS(oct, m, ml);
    mpx_storel(m, ml, d.buf, d.sz);
    if (memcmp(d.buf, v->buf, oct) != 0) {
      dumpbits("\n*** storel failed", d.buf, d.sz);
      ok = 0;
    }

    mpx_loadb(m, ml, v->buf, v->len);
    if (!chkscan(m, ml, v->buf + v->len - 1, v->len, -1))
      ok = 0;
    MPX_OCTETS(oct, m, ml);
    mpx_storeb(m, ml, d.buf, d.sz);
    if (memcmp(d.buf + d.sz - oct, v->buf + v->len - oct, oct) != 0) {
      dumpbits("\n*** storeb failed", d.buf, d.sz);
      ok = 0;
    }
  }

  if (!ok)
    dumpbits("input data", v->buf, v->len);

  free(m);
  dstr_destroy(&d);
  return (ok);
}

static int lsl(dstr *v)
{
  mpw *a, *al;
  int n = *(int *)v[1].buf;
  mpw *c, *cl;
  mpw *d, *dl;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(c, cl, &v[2]);
  ALLOC(d, dl, al - a + (n + MPW_BITS - 1) / MPW_BITS);

  mpx_lsl(d, dl, a, al, n);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** lsl(%i) failed\n", n);
    dumpmp("       a", a, al);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(c); free(d);
  return (ok);
}

static int lsr(dstr *v)
{
  mpw *a, *al;
  int n = *(int *)v[1].buf;
  mpw *c, *cl;
  mpw *d, *dl;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(c, cl, &v[2]);
  ALLOC(d, dl, al - a + (n + MPW_BITS - 1) / MPW_BITS + 1);

  mpx_lsr(d, dl, a, al, n);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** lsr(%i) failed\n", n);
    dumpmp("       a", a, al);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(c); free(d);
  return (ok);
}

static int uadd(dstr *v)
{
  mpw *a, *al;
  mpw *b, *bl;
  mpw *c, *cl;
  mpw *d, *dl;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(b, bl, &v[1]);
  LOAD(c, cl, &v[2]);
  ALLOC(d, dl, MAX(al - a, bl - b) + 1);

  mpx_uadd(d, dl, a, al, b, bl);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** uadd failed\n");
    dumpmp("       a", a, al);
    dumpmp("       b", b, bl);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(b); free(c); free(d);
  return (ok);
}

static int usub(dstr *v)
{
  mpw *a, *al;
  mpw *b, *bl;
  mpw *c, *cl;
  mpw *d, *dl;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(b, bl, &v[1]);
  LOAD(c, cl, &v[2]);
  ALLOC(d, dl, al - a);

  mpx_usub(d, dl, a, al, b, bl);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** usub failed\n");
    dumpmp("       a", a, al);
    dumpmp("       b", b, bl);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(b); free(c); free(d);
  return (ok);
}

static int umul(dstr *v)
{
  mpw *a, *al;
  mpw *b, *bl;
  mpw *c, *cl;
  mpw *d, *dl;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(b, bl, &v[1]);
  LOAD(c, cl, &v[2]);
  ALLOC(d, dl, (al - a) + (bl - b));

  mpx_umul(d, dl, a, al, b, bl);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** umul failed\n");
    dumpmp("       a", a, al);
    dumpmp("       b", b, bl);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(b); free(c); free(d);
  return (ok);
}

static int usqr(dstr *v)
{
  mpw *a, *al;
  mpw *c, *cl;
  mpw *d, *dl;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(c, cl, &v[1]);
  ALLOC(d, dl, 2 * (al - a));

  mpx_usqr(d, dl, a, al);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** usqr failed\n");
    dumpmp("       a", a, al);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(c); free(d);
  return (ok);
}

static int udiv(dstr *v)
{
  mpw *a, *al;
  mpw *b, *bl;
  mpw *q, *ql;
  mpw *r, *rl;
  mpw *qq, *qql;
  mpw *s, *sl;
  int ok = 1;

  ALLOC(a, al, MPW_RQ(v[0].len) + 2); mpx_loadb(a, al, v[0].buf, v[0].len);
  LOAD(b, bl, &v[1]);
  LOAD(q, ql, &v[2]);
  LOAD(r, rl, &v[3]);
  ALLOC(qq, qql, al - a);
  ALLOC(s, sl, (bl - b) + 1);

  mpx_udiv(qq, qql, a, al, b, bl, s, sl);
  if (MPX_UCMP(qq, qql, !=, q, ql) ||
      MPX_UCMP(a, al, !=, r, rl)) {
    fprintf(stderr, "\n*** udiv failed\n");
    dumpmp(" divisor", b, bl);
    dumpmp("expect r", r, rl);
    dumpmp("result r", a, al);
    dumpmp("expect q", q, ql);
    dumpmp("result q", qq, qql);
    ok = 0;
  }

  free(a); free(b); free(r); free(q); free(s); free(qq);
  return (ok);
}

static test_chunk defs[] = {
  { "load-store", loadstore, { &type_hex, 0 } },
  { "lsl", lsl, { &type_hex, &type_int, &type_hex, 0 } },
  { "lsr", lsr, { &type_hex, &type_int, &type_hex, 0 } },
  { "uadd", uadd, { &type_hex, &type_hex, &type_hex, 0 } },
  { "usub", usub, { &type_hex, &type_hex, &type_hex, 0 } },
  { "umul", umul, { &type_hex, &type_hex, &type_hex, 0 } },
  { "usqr", usqr, { &type_hex, &type_hex, 0 } },
  { "udiv", udiv, { &type_hex, &type_hex, &type_hex, &type_hex, 0 } },
  { 0, 0, { 0 } }
};

int main(int argc, char *argv[])
{
  test_run(argc, argv, defs, SRCDIR"/tests/mpx");
  return (0);
}


#endif

/*----- That's all, folks -------------------------------------------------*/