source/libs/gmp/gmp-src/mpn/generic/fib2_ui.c

   1 /* mpn_fib2_ui -- calculate Fibonacci numbers.
   2
   3    THE FUNCTIONS IN THIS FILE ARE FOR INTERNAL USE ONLY.  THEY'RE ALMOST
   4    CERTAIN TO BE SUBJECT TO INCOMPATIBLE CHANGES OR DISAPPEAR COMPLETELY IN
   5    FUTURE GNU MP RELEASES.
   6
   7 Copyright 2001, 2002, 2005, 2009 Free Software Foundation, Inc.
   8
   9 This file is part of the GNU MP Library.
  10
  11 The GNU MP Library is free software; you can redistribute it and/or modify
  12 it under the terms of either:
  13
  14   * the GNU Lesser General Public License as published by the Free
  15     Software Foundation; either version 3 of the License, or (at your
  16     option) any later version.
  17
  18 or
  19
  20   * the GNU General Public License as published by the Free Software
  21     Foundation; either version 2 of the License, or (at your option) any
  22     later version.
  23
  24 or both in parallel, as here.
  25
  26 The GNU MP Library is distributed in the hope that it will be useful, but
  27 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
  28 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  29 for more details.
  30
  31 You should have received copies of the GNU General Public License and the
  32 GNU Lesser General Public License along with the GNU MP Library.  If not,
  33 see https://www.gnu.org/licenses/.  */
  34
  35 #include <stdio.h>
  36 #include "gmp.h"
  37 #include "gmp-impl.h"
  38
  39 /* change this to "#define TRACE(x) x" for diagnostics */
  40 #define TRACE(x)
  41
  42
  43 /* Store F[n] at fp and F[n-1] at f1p.  fp and f1p should have room for
  44    MPN_FIB2_SIZE(n) limbs.
  45
  46    The return value is the actual number of limbs stored, this will be at
  47    least 1.  fp[size-1] will be non-zero, except when n==0, in which case
  48    fp[0] is 0 and f1p[0] is 1.  f1p[size-1] can be zero, since F[n-1]<F[n]
  49    (for n>0).
  50
  51    Notes:
  52
  53    In F[2k+1] with k even, +2 is applied to 4*F[k]^2 just by ORing into the
  54    low limb.
  55
  56    In F[2k+1] with k odd, -2 is applied to the low limb of 4*F[k]^2 -
  57    F[k-1]^2.  This F[2k+1] is an F[4m+3] and such numbers are congruent to
  58    1, 2 or 5 mod 8, which means no underflow reaching it with a -2 (since
  59    that would leave 6 or 7 mod 8).
  60
  61    This property of F[4m+3] can be verified by induction on F[4m+3] =
  62    7*F[4m-1] - F[4m-5], that formula being a standard lucas sequence
  63    identity U[i+j] = U[i]*V[j] - U[i-j]*Q^j.
  64 */
  65
  66 mp_size_t
  67 mpn_fib2_ui (mp_ptr fp, mp_ptr f1p, unsigned long int n)
  68 {
  69   mp_size_t      size;
  70   unsigned long  nfirst, mask;
  71
  72   TRACE (printf ("mpn_fib2_ui n=%lu\n", n));
  73
  74   ASSERT (! MPN_OVERLAP_P (fp, MPN_FIB2_SIZE(n), f1p, MPN_FIB2_SIZE(n)));
  75
  76   /* Take a starting pair from the table. */
  77   mask = 1;
  78   for (nfirst = n; nfirst > FIB_TABLE_LIMIT; nfirst /= 2)
  79     mask <<= 1;
  80   TRACE (printf ("nfirst=%lu mask=0x%lX\n", nfirst, mask));
  81
  82   f1p[0] = FIB_TABLE ((int) nfirst - 1);
  83   fp[0]  = FIB_TABLE (nfirst);
  84   size = 1;
  85
  86   /* Skip to the end if the table lookup gives the final answer. */
  87   if (mask != 1)
  88     {
  89       mp_size_t  alloc;
  90       mp_ptr        xp;
  91       TMP_DECL;
  92
  93       TMP_MARK;
  94       alloc = MPN_FIB2_SIZE (n);
  95       xp = TMP_ALLOC_LIMBS (alloc);
  96
  97       do
  98         {
  99           /* Here fp==F[k] and f1p==F[k-1], with k being the bits of n from
 100              n&mask upwards.
 101
 102              The next bit of n is n&(mask>>1) and we'll double to the pair
 103              fp==F[2k],f1p==F[2k-1] or fp==F[2k+1],f1p==F[2k], according as
 104              that bit is 0 or 1 respectively.  */
 105
 106           TRACE (printf ("k=%lu mask=0x%lX size=%ld alloc=%ld\n",
 107                          n >> refmpn_count_trailing_zeros(mask),
 108                          mask, size, alloc);
 109                  mpn_trace ("fp ", fp, size);
 110                  mpn_trace ("f1p", f1p, size));
 111
 112           /* fp normalized, f1p at most one high zero */
 113           ASSERT (fp[size-1] != 0);
 114           ASSERT (f1p[size-1] != 0 || f1p[size-2] != 0);
 115
 116           /* f1p[size-1] might be zero, but this occurs rarely, so it's not
 117              worth bothering checking for it */
 118           ASSERT (alloc >= 2*size);
 119           mpn_sqr (xp, fp,  size);
 120           mpn_sqr (fp, f1p, size);
 121           size *= 2;
 122
 123           /* Shrink if possible.  Since fp was normalized there'll be at
 124              most one high zero on xp (and if there is then there's one on
 125              yp too).  */
 126           ASSERT (xp[size-1] != 0 || fp[size-1] == 0);
 127           size -= (xp[size-1] == 0);
 128           ASSERT (xp[size-1] != 0);  /* only one xp high zero */
 129
 130           /* Calculate F[2k-1] = F[k]^2 + F[k-1]^2. */
 131           f1p[size] = mpn_add_n (f1p, xp, fp, size);
 132
 133           /* Calculate F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k.
 134              n&mask is the low bit of our implied k.  */
 135 #if HAVE_NATIVE_mpn_rsblsh2_n
 136           fp[size] = mpn_rsblsh2_n (fp, fp, xp, size);
 137           if ((n & mask) == 0)
 138             MPN_INCR_U(fp, size + 1, 2);        /* possible +2 */
 139           else
 140           {
 141             ASSERT (fp[0] >= 2);
 142             fp[0] -= 2;                         /* possible -2 */
 143           }
 144 #else
 145           {
 146             mp_limb_t  c;
 147
 148             c = mpn_lshift (xp, xp, size, 2);
 149             xp[0] |= (n & mask ? 0 : 2);        /* possible +2 */
 150             c -= mpn_sub_n (fp, xp, fp, size);
 151             ASSERT (n & mask ? fp[0] != 0 && fp[0] != 1 : 1);
 152             fp[0] -= (n & mask ? 2 : 0);        /* possible -2 */
 153             fp[size] = c;
 154           }
 155 #endif
 156           ASSERT (alloc >= size+1);
 157           size += (fp[size] != 0);
 158
 159           /* now n&mask is the new bit of n being considered */
 160           mask >>= 1;
 161
 162           /* Calculate F[2k] = F[2k+1] - F[2k-1], replacing the unwanted one of
 163              F[2k+1] and F[2k-1].  */
 164           if (n & mask)
 165             ASSERT_NOCARRY (mpn_sub_n (f1p, fp, f1p, size));
 166           else {
 167             ASSERT_NOCARRY (mpn_sub_n ( fp, fp, f1p, size));
 168
 169             /* Can have a high zero after replacing F[2k+1] with F[2k].
 170                f1p will have a high zero if fp does. */
 171             ASSERT (fp[size-1] != 0 || f1p[size-1] == 0);
 172             size -= (fp[size-1] == 0);
 173           }
 174         }
 175       while (mask != 1);
 176
 177       TMP_FREE;
 178     }
 179
 180   TRACE (printf ("done size=%ld\n", size);
 181          mpn_trace ("fp ", fp, size);
 182          mpn_trace ("f1p", f1p, size));
 183
 184   return size;
 185 }