/* mpn_mu_divappr_q, mpn_preinv_mu_divappr_q.

   Compute Q = floor(N / D) + e.  N is nn limbs, D is dn limbs and must be

   normalized, and Q must be nn-dn limbs, 0 <= e <= 4.  The requirement that Q

   is nn-dn limbs (and not nn-dn+1 limbs) was put in place in order to allow us

   to let N be unmodified during the operation.

   Contributed to the GNU project by Torbjorn Granlund.

   THE FUNCTIONS IN THIS FILE ARE INTERNAL WITH MUTABLE INTERFACES.  IT IS ONLY

   SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES.  IN FACT, IT IS ALMOST

   GUARANTEED THAT THEY WILL CHANGE OR DISAPPEAR IN A FUTURE GMP RELEASE.

Copyright 2005-2007, 2009, 2010 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

The GNU MP Library is free software; you can redistribute it and/or modify

it under the terms of either:

  * the GNU Lesser General Public License as published by the Free

    Software Foundation; either version 3 of the License, or (at your

    option) any later version.

or

  * the GNU General Public License as published by the Free Software

    Foundation; either version 2 of the License, or (at your option) any

    later version.

or both in parallel, as here.

The GNU MP Library 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 General Public License

for more details.

You should have received copies of the GNU General Public License and the

GNU Lesser General Public License along with the GNU MP Library.  If not,

see https://www.gnu.org/licenses/.  */

/*

   The idea of the algorithm used herein is to compute a smaller inverted value

   than used in the standard Barrett algorithm, and thus save time in the

   Newton iterations, and pay just a small price when using the inverted value

   for developing quotient bits.  This algorithm was presented at ICMS 2006.

*/

/* CAUTION: This code and the code in mu_div_qr.c should be edited in sync.

 Things to work on:

  * The itch/scratch scheme isn't perhaps such a good idea as it once seemed,

    demonstrated by the fact that the mpn_invertappr function's scratch needs

    mean that we need to keep a large allocation long after it is needed.

    Things are worse as mpn_mul_fft does not accept any scratch parameter,

    which means we'll have a large memory hole while in mpn_mul_fft.  In

    general, a peak scratch need in the beginning of a function isn't

    well-handled by the itch/scratch scheme.

*/

#ifdef STAT

#undef STAT

#define STAT(x) x

#else

#define STAT(x)

#endif

#include <stdlib.h>   /* for NULL */

#include "gmp-impl.h"

static mp_limb_t mpn_preinv_mu_divappr_q (mp_ptr, mp_srcptr, mp_size_t,

       mp_srcptr, mp_size_t, mp_srcptr, mp_size_t, mp_ptr);

static mp_size_t mpn_mu_divappr_q_choose_in (mp_size_t, mp_size_t, int);

mp_limb_t

mpn_mu_divappr_q (mp_ptr qp,

      mp_srcptr np,

      mp_size_t nn,

      mp_srcptr dp,

      mp_size_t dn,

      mp_ptr scratch)

{

  mp_size_t qn, in;

  mp_limb_t cy, qh;

  mp_ptr ip, tp;

  ASSERT (dn > 1);

  qn = nn - dn;

  /* If Q is smaller than D, truncate operands. */

  if (qn + 1 < dn)

    {

      np += dn - (qn + 1);

      nn -= dn - (qn + 1);

      dp += dn - (qn + 1);

      dn = qn + 1;

    }

  /* Compute the inverse size.  */

  in = mpn_mu_divappr_q_choose_in (qn, dn, 0);

  ASSERT (in <= dn);

#if 1

  /* This alternative inverse computation method gets slightly more accurate

     results.  FIXMEs: (1) Temp allocation needs not analysed (2) itch function

     not adapted (3) mpn_invertappr scratch needs not met.  */

  ip = scratch;

  tp = scratch + in + 1;

  /* compute an approximate inverse on (in+1) limbs */

  if (dn == in)

    {

      MPN_COPY (tp + 1, dp, in);

      tp[0] = 1;

      mpn_invertappr (ip, tp, in + 1, tp + in + 1);

      MPN_COPY_INCR (ip, ip + 1, in);

    }

  else

    {

      cy = mpn_add_1 (tp, dp + dn - (in + 1), in + 1, 1);

      if (UNLIKELY (cy != 0))

  MPN_ZERO (ip, in);

      else

  {

    mpn_invertappr (ip, tp, in + 1, tp + in + 1);

    MPN_COPY_INCR (ip, ip + 1, in);

  }

    }

#else

  /* This older inverse computation method gets slightly worse results than the

     one above.  */

  ip = scratch;

  tp = scratch + in;

  /* Compute inverse of D to in+1 limbs, then round to 'in' limbs.  Ideally the

     inversion function should do this automatically.  */

  if (dn == in)

    {

      tp[in + 1] = 0;

      MPN_COPY (tp + in + 2, dp, in);

      mpn_invertappr (tp, tp + in + 1, in + 1, NULL);

    }

  else

    {

      mpn_invertappr (tp, dp + dn - (in + 1), in + 1, NULL);

    }

  cy = mpn_sub_1 (tp, tp, in + 1, GMP_NUMB_HIGHBIT);

  if (UNLIKELY (cy != 0))

    MPN_ZERO (tp + 1, in);

  MPN_COPY (ip, tp + 1, in);

#endif

  qh = mpn_preinv_mu_divappr_q (qp, np, nn, dp, dn, ip, in, scratch + in);

  return qh;

}

static mp_limb_t

mpn_preinv_mu_divappr_q (mp_ptr qp,

       mp_srcptr np,

       mp_size_t nn,

       mp_srcptr dp,

       mp_size_t dn,

       mp_srcptr ip,

       mp_size_t in,

       mp_ptr scratch)

{

  mp_size_t qn;

  mp_limb_t cy, cx, qh;

  mp_limb_t r;

  mp_size_t tn, wn;

#define rp           scratch

#define tp           (scratch + dn)

#define scratch_out  (scratch + dn + tn)

  qn = nn - dn;

  np += qn;

  qp += qn;

  qh = mpn_cmp (np, dp, dn) >= 0;

  if (qh != 0)

    mpn_sub_n (rp, np, dp, dn);

  else

    MPN_COPY (rp, np, dn);

  if (UNLIKELY (qn == 0))

    return qh;     /* Degenerate use.  Should we allow this? */

  for (;;) /* The exit condition (qn == 0) is verified in the loop. */

    {

      if (qn < in)

  {

    ip += in - qn;

    in = qn;

  }

      np -= in;

      qp -= in;

      /* Compute the next block of quotient limbs by multiplying the inverse I

   by the upper part of the partial remainder R.  */

      mpn_mul_n (tp, rp + dn - in, ip, in);    /* mulhi  */

      cy = mpn_add_n (qp, tp + in, rp + dn - in, in);  /* I's msb implicit */

      ASSERT_ALWAYS (cy == 0);

      qn -= in;

      if (qn == 0)

  break;

      /* Compute the product of the quotient block and the divisor D, to be

   subtracted from the partial remainder combined with new limbs from the

   dividend N.  We only really need the low dn limbs.  */

      if (BELOW_THRESHOLD (in, MUL_TO_MULMOD_BNM1_FOR_2NXN_THRESHOLD))

  mpn_mul (tp, dp, dn, qp, in);    /* dn+in limbs, high 'in' cancels */

      else

  {

    tn = mpn_mulmod_bnm1_next_size (dn + 1);

    mpn_mulmod_bnm1 (tp, tn, dp, dn, qp, in, scratch_out);

    wn = dn + in - tn;      /* number of wrapped limbs */

    if (wn > 0)

      {

        cy = mpn_sub_n (tp, tp, rp + dn - wn, wn);

        cy = mpn_sub_1 (tp + wn, tp + wn, tn - wn, cy);

        cx = mpn_cmp (rp + dn - in, tp + dn, tn - dn) < 0;

        ASSERT_ALWAYS (cx >= cy);

        mpn_incr_u (tp, cx - cy);

      }

  }

      r = rp[dn - in] - tp[dn];

      /* Subtract the product from the partial remainder combined with new

   limbs from the dividend N, generating a new partial remainder R.  */

      if (dn != in)

  {

    cy = mpn_sub_n (tp, np, tp, in);  /* get next 'in' limbs from N */

    cy = mpn_sub_nc (tp + in, rp, tp + in, dn - in, cy);

    MPN_COPY (rp, tp, dn);    /* FIXME: try to avoid this */

  }

      else

  {

    cy = mpn_sub_n (rp, np, tp, in);  /* get next 'in' limbs from N */

  }

      STAT (int i; int err = 0;

      static int errarr[5]; static int err_rec; static int tot);

      /* Check the remainder R and adjust the quotient as needed.  */

      r -= cy;

      while (r != 0)

  {

    /* We loop 0 times with about 69% probability, 1 time with about 31%

       probability, 2 times with about 0.6% probability, if inverse is

       computed as recommended.  */

    mpn_incr_u (qp, 1);

    cy = mpn_sub_n (rp, rp, dp, dn);

    r -= cy;

    STAT (err++);

  }

      if (mpn_cmp (rp, dp, dn) >= 0)

  {

    /* This is executed with about 76% probability.  */

    mpn_incr_u (qp, 1);

    cy = mpn_sub_n (rp, rp, dp, dn);

    STAT (err++);

  }

      STAT (

      tot++;

      errarr[err]++;

      if (err > err_rec)

        err_rec = err;

      if (tot % 0x10000 == 0)

        {

    for (i = 0; i <= err_rec; i++)

      printf ("  %d(%.1f%%)", errarr[i], 100.0*errarr[i]/tot);

    printf ("\n");

        }

      );

    }

  /* FIXME: We should perhaps be somewhat more elegant in our rounding of the

     quotient.  For now, just make sure the returned quotient is >= the real

     quotient; add 3 with saturating arithmetic.  */

  qn = nn - dn;

  cy += mpn_add_1 (qp, qp, qn, 3);

  if (cy != 0)

    {

      if (qh != 0)

  {

    /* Return a quotient of just 1-bits, with qh set.  */

    mp_size_t i;

    for (i = 0; i < qn; i++)

      qp[i] = GMP_NUMB_MAX;

  }

      else

  {

    /* Propagate carry into qh.  */

    qh = 1;

  }

    }

  return qh;

}

/* In case k=0 (automatic choice), we distinguish 3 cases:

   (a) dn < qn:         in = ceil(qn / ceil(qn/dn))

   (b) dn/3 < qn <= dn: in = ceil(qn / 2)

   (c) qn < dn/3:       in = qn

   In all cases we have in <= dn.

 */

static mp_size_t

mpn_mu_divappr_q_choose_in (mp_size_t qn, mp_size_t dn, int k)

{

  mp_size_t in;

  if (k == 0)

    {

      mp_size_t b;

      if (qn > dn)

  {

    /* Compute an inverse size that is a nice partition of the quotient.  */

    b = (qn - 1) / dn + 1;  /* ceil(qn/dn), number of blocks */

    in = (qn - 1) / b + 1;  /* ceil(qn/b) = ceil(qn / ceil(qn/dn)) */

  }

      else if (3 * qn > dn)

  {

    in = (qn - 1) / 2 + 1;  /* b = 2 */

  }

      else

  {

    in = (qn - 1) / 1 + 1;  /* b = 1 */

  }

    }

  else

    {

      mp_size_t xn;

      xn = MIN (dn, qn);

      in = (xn - 1) / k + 1;

    }

  return in;

}

mp_size_t

mpn_mu_divappr_q_itch (mp_size_t nn, mp_size_t dn, int mua_k)

{

  mp_size_t qn, in, itch_local, itch_out, itch_invapp;

  qn = nn - dn;

  if (qn + 1 < dn)

    {

      dn = qn + 1;

    }

  in = mpn_mu_divappr_q_choose_in (qn, dn, mua_k);

  itch_local = mpn_mulmod_bnm1_next_size (dn + 1);

  itch_out = mpn_mulmod_bnm1_itch (itch_local, dn, in);

  itch_invapp = mpn_invertappr_itch (in + 1) + in + 2; /* 3in + 4 */

  ASSERT (dn + itch_local + itch_out >= itch_invapp);

  return in + MAX (dn + itch_local + itch_out, itch_invapp);

}