path: root/mpn/generic/fib2_ui.c

/* mpn_fib2_ui -- calculate Fibonacci numbers.

Copyright 2001 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 the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.

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 Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library; see the file COPYING.LIB.  If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */

#include <stdio.h>
#include "gmp.h"
#include "gmp-impl.h"
#include "longlong.h"


/* change this to "#define TRACE(x) x" for diagnostics */
#define TRACE(x)


/* The following tables generated by code at the end of this file. */

#if BITS_PER_MP_LIMB == 4
const mp_limb_t __gmp_fib_table[FIB_TABLE_LIMIT+2] = {
 CNST_LIMB (0x1),  /* -1 */
 CNST_LIMB (0x0),  /* 0 */
 CNST_LIMB (0x1),  /* 1 */
 CNST_LIMB (0x1),  /* 2 */
 CNST_LIMB (0x2),  /* 3 */
 CNST_LIMB (0x3),  /* 4 */
 CNST_LIMB (0x5),  /* 5 */
 CNST_LIMB (0x8),  /* 6 */
 CNST_LIMB (0xD),  /* 7 */
};
/* total 9 bytes for BITS_PER_LIMB==4 */
#endif

#if BITS_PER_MP_LIMB == 8
const mp_limb_t __gmp_fib_table[FIB_TABLE_LIMIT+2] = {
 CNST_LIMB (0x1),  /* -1 */
 CNST_LIMB (0x0),  /* 0 */
 CNST_LIMB (0x1),  /* 1 */
 CNST_LIMB (0x1),  /* 2 */
 CNST_LIMB (0x2),  /* 3 */
 CNST_LIMB (0x3),  /* 4 */
 CNST_LIMB (0x5),  /* 5 */
 CNST_LIMB (0x8),  /* 6 */
 CNST_LIMB (0xD),  /* 7 */
 CNST_LIMB (0x15),  /* 8 */
 CNST_LIMB (0x22),  /* 9 */
 CNST_LIMB (0x37),  /* 10 */
 CNST_LIMB (0x59),  /* 11 */
 CNST_LIMB (0x90),  /* 12 */
 CNST_LIMB (0xE9),  /* 13 */
};
/* total 15 bytes for BITS_PER_LIMB==8 */
#endif

#if BITS_PER_MP_LIMB == 16
const mp_limb_t __gmp_fib_table[FIB_TABLE_LIMIT+2] = {
 CNST_LIMB (0x1),  /* -1 */
 CNST_LIMB (0x0),  /* 0 */
 CNST_LIMB (0x1),  /* 1 */
 CNST_LIMB (0x1),  /* 2 */
 CNST_LIMB (0x2),  /* 3 */
 CNST_LIMB (0x3),  /* 4 */
 CNST_LIMB (0x5),  /* 5 */
 CNST_LIMB (0x8),  /* 6 */
 CNST_LIMB (0xD),  /* 7 */
 CNST_LIMB (0x15),  /* 8 */
 CNST_LIMB (0x22),  /* 9 */
 CNST_LIMB (0x37),  /* 10 */
 CNST_LIMB (0x59),  /* 11 */
 CNST_LIMB (0x90),  /* 12 */
 CNST_LIMB (0xE9),  /* 13 */
 CNST_LIMB (0x179),  /* 14 */
 CNST_LIMB (0x262),  /* 15 */
 CNST_LIMB (0x3DB),  /* 16 */
 CNST_LIMB (0x63D),  /* 17 */
 CNST_LIMB (0xA18),  /* 18 */
 CNST_LIMB (0x1055),  /* 19 */
 CNST_LIMB (0x1A6D),  /* 20 */
 CNST_LIMB (0x2AC2),  /* 21 */
 CNST_LIMB (0x452F),  /* 22 */
 CNST_LIMB (0x6FF1),  /* 23 */
 CNST_LIMB (0xB520),  /* 24 */
};
/* total 52 bytes for BITS_PER_LIMB==16 */
#endif

#if BITS_PER_MP_LIMB == 32
const mp_limb_t __gmp_fib_table[FIB_TABLE_LIMIT+2] = {
 CNST_LIMB (0x1),  /* -1 */
 CNST_LIMB (0x0),  /* 0 */
 CNST_LIMB (0x1),  /* 1 */
 CNST_LIMB (0x1),  /* 2 */
 CNST_LIMB (0x2),  /* 3 */
 CNST_LIMB (0x3),  /* 4 */
 CNST_LIMB (0x5),  /* 5 */
 CNST_LIMB (0x8),  /* 6 */
 CNST_LIMB (0xD),  /* 7 */
 CNST_LIMB (0x15),  /* 8 */
 CNST_LIMB (0x22),  /* 9 */
 CNST_LIMB (0x37),  /* 10 */
 CNST_LIMB (0x59),  /* 11 */
 CNST_LIMB (0x90),  /* 12 */
 CNST_LIMB (0xE9),  /* 13 */
 CNST_LIMB (0x179),  /* 14 */
 CNST_LIMB (0x262),  /* 15 */
 CNST_LIMB (0x3DB),  /* 16 */
 CNST_LIMB (0x63D),  /* 17 */
 CNST_LIMB (0xA18),  /* 18 */
 CNST_LIMB (0x1055),  /* 19 */
 CNST_LIMB (0x1A6D),  /* 20 */
 CNST_LIMB (0x2AC2),  /* 21 */
 CNST_LIMB (0x452F),  /* 22 */
 CNST_LIMB (0x6FF1),  /* 23 */
 CNST_LIMB (0xB520),  /* 24 */
 CNST_LIMB (0x12511),  /* 25 */
 CNST_LIMB (0x1DA31),  /* 26 */
 CNST_LIMB (0x2FF42),  /* 27 */
 CNST_LIMB (0x4D973),  /* 28 */
 CNST_LIMB (0x7D8B5),  /* 29 */
 CNST_LIMB (0xCB228),  /* 30 */
 CNST_LIMB (0x148ADD),  /* 31 */
 CNST_LIMB (0x213D05),  /* 32 */
 CNST_LIMB (0x35C7E2),  /* 33 */
 CNST_LIMB (0x5704E7),  /* 34 */
 CNST_LIMB (0x8CCCC9),  /* 35 */
 CNST_LIMB (0xE3D1B0),  /* 36 */
 CNST_LIMB (0x1709E79),  /* 37 */
 CNST_LIMB (0x2547029),  /* 38 */
 CNST_LIMB (0x3C50EA2),  /* 39 */
 CNST_LIMB (0x6197ECB),  /* 40 */
 CNST_LIMB (0x9DE8D6D),  /* 41 */
 CNST_LIMB (0xFF80C38),  /* 42 */
 CNST_LIMB (0x19D699A5),  /* 43 */
 CNST_LIMB (0x29CEA5DD),  /* 44 */
 CNST_LIMB (0x43A53F82),  /* 45 */
 CNST_LIMB (0x6D73E55F),  /* 46 */
 CNST_LIMB (0xB11924E1),  /* 47 */
};
/* total 196 bytes for BITS_PER_LIMB==32 */
#endif

#if BITS_PER_MP_LIMB == 64
const mp_limb_t __gmp_fib_table[FIB_TABLE_LIMIT+2] = {
 CNST_LIMB (0x1),  /* -1 */
 CNST_LIMB (0x0),  /* 0 */
 CNST_LIMB (0x1),  /* 1 */
 CNST_LIMB (0x1),  /* 2 */
 CNST_LIMB (0x2),  /* 3 */
 CNST_LIMB (0x3),  /* 4 */
 CNST_LIMB (0x5),  /* 5 */
 CNST_LIMB (0x8),  /* 6 */
 CNST_LIMB (0xD),  /* 7 */
 CNST_LIMB (0x15),  /* 8 */
 CNST_LIMB (0x22),  /* 9 */
 CNST_LIMB (0x37),  /* 10 */
 CNST_LIMB (0x59),  /* 11 */
 CNST_LIMB (0x90),  /* 12 */
 CNST_LIMB (0xE9),  /* 13 */
 CNST_LIMB (0x179),  /* 14 */
 CNST_LIMB (0x262),  /* 15 */
 CNST_LIMB (0x3DB),  /* 16 */
 CNST_LIMB (0x63D),  /* 17 */
 CNST_LIMB (0xA18),  /* 18 */
 CNST_LIMB (0x1055),  /* 19 */
 CNST_LIMB (0x1A6D),  /* 20 */
 CNST_LIMB (0x2AC2),  /* 21 */
 CNST_LIMB (0x452F),  /* 22 */
 CNST_LIMB (0x6FF1),  /* 23 */
 CNST_LIMB (0xB520),  /* 24 */
 CNST_LIMB (0x12511),  /* 25 */
 CNST_LIMB (0x1DA31),  /* 26 */
 CNST_LIMB (0x2FF42),  /* 27 */
 CNST_LIMB (0x4D973),  /* 28 */
 CNST_LIMB (0x7D8B5),  /* 29 */
 CNST_LIMB (0xCB228),  /* 30 */
 CNST_LIMB (0x148ADD),  /* 31 */
 CNST_LIMB (0x213D05),  /* 32 */
 CNST_LIMB (0x35C7E2),  /* 33 */
 CNST_LIMB (0x5704E7),  /* 34 */
 CNST_LIMB (0x8CCCC9),  /* 35 */
 CNST_LIMB (0xE3D1B0),  /* 36 */
 CNST_LIMB (0x1709E79),  /* 37 */
 CNST_LIMB (0x2547029),  /* 38 */
 CNST_LIMB (0x3C50EA2),  /* 39 */
 CNST_LIMB (0x6197ECB),  /* 40 */
 CNST_LIMB (0x9DE8D6D),  /* 41 */
 CNST_LIMB (0xFF80C38),  /* 42 */
 CNST_LIMB (0x19D699A5),  /* 43 */
 CNST_LIMB (0x29CEA5DD),  /* 44 */
 CNST_LIMB (0x43A53F82),  /* 45 */
 CNST_LIMB (0x6D73E55F),  /* 46 */
 CNST_LIMB (0xB11924E1),  /* 47 */
 CNST_LIMB (0x11E8D0A40),  /* 48 */
 CNST_LIMB (0x1CFA62F21),  /* 49 */
 CNST_LIMB (0x2EE333961),  /* 50 */
 CNST_LIMB (0x4BDD96882),  /* 51 */
 CNST_LIMB (0x7AC0CA1E3),  /* 52 */
 CNST_LIMB (0xC69E60A65),  /* 53 */
 CNST_LIMB (0x1415F2AC48),  /* 54 */
 CNST_LIMB (0x207FD8B6AD),  /* 55 */
 CNST_LIMB (0x3495CB62F5),  /* 56 */
 CNST_LIMB (0x5515A419A2),  /* 57 */
 CNST_LIMB (0x89AB6F7C97),  /* 58 */
 CNST_LIMB (0xDEC1139639),  /* 59 */
 CNST_LIMB (0x1686C8312D0),  /* 60 */
 CNST_LIMB (0x2472D96A909),  /* 61 */
 CNST_LIMB (0x3AF9A19BBD9),  /* 62 */
 CNST_LIMB (0x5F6C7B064E2),  /* 63 */
 CNST_LIMB (0x9A661CA20BB),  /* 64 */
 CNST_LIMB (0xF9D297A859D),  /* 65 */
 CNST_LIMB (0x19438B44A658),  /* 66 */
 CNST_LIMB (0x28E0B4BF2BF5),  /* 67 */
 CNST_LIMB (0x42244003D24D),  /* 68 */
 CNST_LIMB (0x6B04F4C2FE42),  /* 69 */
 CNST_LIMB (0xAD2934C6D08F),  /* 70 */
 CNST_LIMB (0x1182E2989CED1),  /* 71 */
 CNST_LIMB (0x1C5575E509F60),  /* 72 */
 CNST_LIMB (0x2DD8587DA6E31),  /* 73 */
 CNST_LIMB (0x4A2DCE62B0D91),  /* 74 */
 CNST_LIMB (0x780626E057BC2),  /* 75 */
 CNST_LIMB (0xC233F54308953),  /* 76 */
 CNST_LIMB (0x13A3A1C2360515),  /* 77 */
 CNST_LIMB (0x1FC6E116668E68),  /* 78 */
 CNST_LIMB (0x336A82D89C937D),  /* 79 */
 CNST_LIMB (0x533163EF0321E5),  /* 80 */
 CNST_LIMB (0x869BE6C79FB562),  /* 81 */
 CNST_LIMB (0xD9CD4AB6A2D747),  /* 82 */
 CNST_LIMB (0x16069317E428CA9),  /* 83 */
 CNST_LIMB (0x23A367C34E563F0),  /* 84 */
 CNST_LIMB (0x39A9FADB327F099),  /* 85 */
 CNST_LIMB (0x5D4D629E80D5489),  /* 86 */
 CNST_LIMB (0x96F75D79B354522),  /* 87 */
 CNST_LIMB (0xF444C01834299AB),  /* 88 */
 CNST_LIMB (0x18B3C1D91E77DECD),  /* 89 */
 CNST_LIMB (0x27F80DDAA1BA7878),  /* 90 */
 CNST_LIMB (0x40ABCFB3C0325745),  /* 91 */
 CNST_LIMB (0x68A3DD8E61ECCFBD),  /* 92 */
 CNST_LIMB (0xA94FAD42221F2702),  /* 93 */
};
/* total 760 bytes for BITS_PER_LIMB==64 */
#endif


/* Store F[n] at fp and F[n-1] at f1p.  fp and f1p should have room for
   MPN_FIB2_SIZE(n) limbs.

   The return value is the actual number of limbs stored, this will be at
   least 1.  fp[size-1] will be non-zero, except when n==0, in which case
   fp[0] is 0 and f1p[0] is 1.  f1p[size-1] can be zero, since F[n-1]<F[n]
   (for n>0).

   Notes:

   In F[2k+1] with k even, +2 is applied to 4*F[k]^2 just by ORing into the
   low limb.

   In F[2k+1] with k odd, -2 is applied to the low limb of 4*F[k]^2 -
   F[k-1]^2.  This F[2k+1] is an F[4m+3] and such numbers are congruent to
   1, 2 or 5 mod 8, which means no underflow reaching it with a -2 (since
   that would leave 6 or 7 mod 8).

   This property of F[4m+3] can be verified by induction on F[4m+3] =
   7*F[4m-1] - F[4m-5], that formula being a standard lucas sequence
   identity U[i+j] = U[i]*V[j] - U[i-j]*Q^j.

   Enhancements:

   If there was an mpn_addlshift, it'd be possible to eliminate the yp
   temporary, using xp=F[k]^2, fp=F[k-1]^2, f1p=xp+fp, fp+=4*fp, fp-=f1p,
   fp+=2*(-1)^n, etc.  */

mp_size_t
mpn_fib2_ui (mp_ptr fp, mp_ptr f1p, unsigned long int n)
{
  mp_ptr         xp, yp;
  mp_size_t      size;
  unsigned long  nfirst, mask;
  TMP_DECL (marker);

  TRACE (printf ("mpn_fib2_ui n=%lu\n", n));

  ASSERT (! MPN_OVERLAP_P (fp, MPN_FIB2_SIZE(n), f1p, MPN_FIB2_SIZE(n)));

  /* Take a starting pair from the table. */
  mask = 1;
  for (nfirst = n; nfirst > FIB_TABLE_LIMIT; nfirst /= 2)
    mask <<= 1;
  TRACE (printf ("nfirst=%lu mask=0x%lX\n", nfirst, mask));

  f1p[0] = FIB_TABLE ((int) nfirst - 1);
  fp[0]  = FIB_TABLE (nfirst);
  size = 1;

  /* Skip to the end if the table lookup gives the final answer. */
  if (mask != 1)
    {
      mp_size_t  alloc;

      TMP_MARK (marker);
      alloc = MPN_FIB2_SIZE (n);
      TMP_ALLOC_LIMBS_2 (xp,alloc, yp,alloc);

      do
        {
          mp_limb_t  c;

          /* Here fp==F[k] and f1p==F[k-1], with k being the bits of n from
             n&mask upwards.

             The next bit of n is n&(mask>>1) and we'll double to the pair
             fp==F[2k],f1p==F[2k-1] or fp==F[2k+1],f1p==F[2k], according as
             that bit is 0 or 1 respectively.  */

          TRACE (printf ("k=%lu mask=0x%lX size=%ld alloc=%ld\n",
                         n >> refmpn_count_trailing_zeros(mask),
                         mask, size, alloc);
                 mpn_trace ("fp ", fp, size);
                 mpn_trace ("f1p", f1p, size));

          /* fp normalized, f1p at most one high zero */
          ASSERT (fp[size-1] != 0);
          ASSERT (f1p[size-1] != 0 || f1p[size-2] != 0);

          /* f1p[size-1] might be zero, but this occurs rarely, so it's not
             worth bothering checking for it */
          ASSERT (alloc >= 2*size);
          mpn_sqr_n (xp, fp,  size);
          mpn_sqr_n (yp, f1p, size);
          size *= 2;

          /* Shrink if possible.  Since fp was normalized there'll be at
             most one high zero on xp (and if there is then there's one on
             yp too).  */
          ASSERT (xp[size-1] != 0 || yp[size-1] == 0);
          size -= (xp[size-1] == 0);
          ASSERT (xp[size-1] != 0);  /* only one xp high zero */

          /* Calculate F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k.
             n&mask is the low bit of our implied k.  */
          c = mpn_lshift (fp, xp, size, 2);
          fp[0] |= (n & mask ? 0 : 2);   /* possible +2 */
          c -= mpn_sub_n (fp, fp, yp, size);
          ASSERT (n & (mask << 1) ? fp[0] != 0 && fp[0] != 1 : 1);
          fp[0] -= (n & mask ? 2 : 0);   /* possible -2 */
          ASSERT (alloc >= size+1);
          xp[size] = 0;
          yp[size] = 0;
          fp[size] = c;
          size += (c != 0);

          /* Calculate F[2k-1] = F[k]^2 + F[k-1]^2.
             F[2k-1]<F[2k+1] so no carry out of "size" limbs. */
          ASSERT_NOCARRY (mpn_add_n (f1p, xp, yp, size));

          /* now n&mask is the new bit of n being considered */
          mask >>= 1;

          /* Calculate F[2k] = F[2k+1] - F[2k-1], replacing the unwanted one of
             F[2k+1] and F[2k-1].  */
          ASSERT_NOCARRY (mpn_sub_n ((n & mask ? f1p : fp), fp, f1p, size));

          /* Can have a high zero after replacing F[2k+1] with F[2k].
             f1p will have a high zero if fp does. */
          ASSERT (fp[size-1] != 0 || f1p[size-1] == 0);
          size -= (fp[size-1] == 0);
        }
      while (mask != 1);

      TMP_FREE (marker);
    }

  TRACE (printf ("done size=%ld\n", size);
         mpn_trace ("fp ", fp, size);
         mpn_trace ("f1p", f1p, size));

  return size;
}





/* ------------------------------------------------------------------------- */

#if GENERATE_FIB_TABLE
/* Generate the tables of fibonacci data.  This doesn't depend on the limb
   size of the host, and doesn't need mpz_fib_ui working. */


static struct gen_t {
  int    bpml;
  mpz_t  *f;
  int    fnum;
  int    bytes;
  int    lucnum_limit;
} gen[] = {
  { 4 },
  { 8 },
  { 16 },
  { 32 },
  { 64 },
};


void
generate (struct gen_t *p)
{
  mpz_t  fn, fn1, fn2, ln;
  int    falloc, i;

  falloc = p->bpml*2+4;
  p->f = __GMP_ALLOCATE_FUNC_TYPE (falloc, mpz_t);

  /* at n==-1 */
  mpz_init_set_si (fn2,  2L);  /* F[-3] */
  mpz_init_set_si (fn1, -1L);  /* F[-2] */
  mpz_init_set_si (fn,   1L);  /* F[-1] */
  mpz_init (ln);

  for (i = 0; mpz_sizeinbase(fn,2) <= p->bpml; i++)
    {
      /* L[n] = F[n]+2*F[n-1] */
      mpz_add (ln, fn, fn1);
      mpz_add (ln, ln, fn1);
      if (mpz_sizeinbase(ln,2) <= p->bpml)
        p->lucnum_limit = i-1;

      ASSERT_ALWAYS (i < falloc);
      mpz_set (p->f[i], fn);
      p->bytes += MAX (p->bpml, 8) / 8;

      mpz_swap (fn2, fn1);
      mpz_swap (fn1, fn);
      mpz_add (fn, fn1, fn2);  /* F[n+1] = F[n] + F[n-1] */
    }
  p->fnum = i;

  mpz_clear (fn);
  mpz_clear (fn1);
  mpz_clear (fn2);
  mpz_clear (ln);
}

void
print_header (struct gen_t *p)
{
  printf ("#if BITS_PER_MP_LIMB == %d\n", p->bpml);
  printf ("extern const mp_limb_t __gmp_fib_table[];\n");
  printf ("#define FIB_TABLE_LIMIT         %d\n", p->fnum-2);
  printf ("#define FIB_TABLE_LUCNUM_LIMIT  %d\n", p->lucnum_limit);
  printf ("#endif\n");
}

void
print (struct gen_t *p)
{
  int  i;

  printf ("#if BITS_PER_MP_LIMB == %d\n", p->bpml);
  printf ("const mp_limb_t __gmp_fib_table[FIB_TABLE_LIMIT+2] = {\n");
  for (i = 0; i < p->fnum; i++)
    {
      printf (" CNST_LIMB (0x");
      mpz_out_str (stdout, -16, p->f[i]);
      printf("),  /* %d */\n", i-1);
    }
  printf ("};\n");
  printf ("/* total %d bytes for BITS_PER_LIMB==%d */\n", p->bytes, p->bpml);
  printf ("#endif\n");
  printf ("\n");
}

int
main (void)
{
  int  i;

  for (i = 0; i < numberof(gen); i++)
    generate (&gen[i]);

  for (i = 0; i < numberof(gen); i++)
    print_header (&gen[i]);

  printf ("\n\n");

  for (i = 0; i < numberof(gen); i++)
    print (&gen[i]);

  return 0;
}

#endif