fib_ui.c
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上传日期:2022-08-06
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- /* mpz_fib_ui -- calculate Fibonacci numbers.
- Copyright 2000, 2001, 2002, 2005 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 3 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. If not, see http://www.gnu.org/licenses/. */
- #include <stdio.h>
- #include "gmp.h"
- #include "gmp-impl.h"
- #include "longlong.h"
- /* change to "#define TRACE(x) x" to get some traces */
- #define TRACE(x)
- /* In the F[2k+1] below for k odd, the -2 won't give a borrow from the low
- limb because the result F[2k+1] is an F[4m+3] and such numbers are always
- == 1, 2 or 5 mod 8, whereas an underflow would leave 6 or 7. (This is
- the same as in mpn_fib2_ui.)
- In the F[2k+1] for k even, the +2 won't give a carry out of the low limb
- in normal circumstances. This is an F[4m+1] and we claim that F[3*2^b+1]
- == 1 mod 2^b is the first F[4m+1] congruent to 0 or 1 mod 2^b, and hence
- if n < 2^GMP_NUMB_BITS then F[n] cannot have a low limb of 0 or 1. No
- proof for this claim, but it's been verified up to b==32 and has such a
- nice pattern it must be true :-). Of interest is that F[3*2^b] == 0 mod
- 2^(b+1) seems to hold too.
- When n >= 2^GMP_NUMB_BITS, which can arise in a nails build, then the low
- limb of F[4m+1] can certainly be 1, and an mpn_add_1 must be used. */
- void
- mpz_fib_ui (mpz_ptr fn, unsigned long n)
- {
- mp_ptr fp, xp, yp;
- mp_size_t size, xalloc;
- unsigned long n2;
- mp_limb_t c, c2;
- TMP_DECL;
- if (n <= FIB_TABLE_LIMIT)
- {
- PTR(fn)[0] = FIB_TABLE (n);
- SIZ(fn) = (n != 0); /* F[0]==0, others are !=0 */
- return;
- }
- n2 = n/2;
- xalloc = MPN_FIB2_SIZE (n2) + 1;
- MPZ_REALLOC (fn, 2*xalloc+1);
- fp = PTR (fn);
- TMP_MARK;
- TMP_ALLOC_LIMBS_2 (xp,xalloc, yp,xalloc);
- size = mpn_fib2_ui (xp, yp, n2);
- TRACE (printf ("mpz_fib_ui last step n=%lu size=%ld bit=%lun",
- n >> 1, size, n&1);
- mpn_trace ("xp", xp, size);
- mpn_trace ("yp", yp, size));
- if (n & 1)
- {
- /* F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k */
- mp_size_t xsize, ysize;
- #if HAVE_NATIVE_mpn_add_n_sub_n
- xp[size] = mpn_lshift (xp, xp, size, 1);
- yp[size] = 0;
- ASSERT_NOCARRY (mpn_add_n_sub_n (xp, yp, xp, yp, size+1));
- xsize = size + (xp[size] != 0);
- ysize = size + (yp[size] != 0);
- #else
- c2 = mpn_lshift (fp, xp, size, 1);
- c = c2 + mpn_add_n (xp, fp, yp, size);
- xp[size] = c;
- xsize = size + (c != 0);
- c2 -= mpn_sub_n (yp, fp, yp, size);
- yp[size] = c2;
- ASSERT (c2 <= 1);
- ysize = size + c2;
- #endif
- size = xsize + ysize;
- c = mpn_mul (fp, xp, xsize, yp, ysize);
- #if GMP_NUMB_BITS >= BITS_PER_ULONG
- /* no overflow, see comments above */
- ASSERT (n & 2 ? fp[0] >= 2 : fp[0] <= GMP_NUMB_MAX-2);
- fp[0] += (n & 2 ? -CNST_LIMB(2) : CNST_LIMB(2));
- #else
- if (n & 2)
- {
- ASSERT (fp[0] >= 2);
- fp[0] -= 2;
- }
- else
- {
- ASSERT (c != GMP_NUMB_MAX); /* because it's the high of a mul */
- c += mpn_add_1 (fp, fp, size-1, CNST_LIMB(2));
- fp[size-1] = c;
- }
- #endif
- }
- else
- {
- /* F[2k] = F[k]*(F[k]+2F[k-1]) */
- mp_size_t xsize, ysize;
- c = mpn_lshift (yp, yp, size, 1);
- c += mpn_add_n (yp, yp, xp, size);
- yp[size] = c;
- xsize = size;
- ysize = size + (c != 0);
- size += ysize;
- c = mpn_mul (fp, yp, ysize, xp, xsize);
- }
- /* one or two high zeros */
- size -= (c == 0);
- size -= (fp[size-1] == 0);
- SIZ(fn) = size;
- TRACE (printf ("done special, size=%ldn", size);
- mpn_trace ("fp ", fp, size));
- TMP_FREE;
- }