1 /* mpz_fib_ui -- calculate Fibonacci numbers.
2
3 Copyright 2000-2002, 2005, 2012, 2014, 2015 Free Software Foundation, Inc.
4
5 This file is part of the GNU MP Library.
6
7 The GNU MP Library is free software; you can redistribute it and/or modify
8 it under the terms of either:
9
10 * the GNU Lesser General Public License as published by the Free
11 Software Foundation; either version 3 of the License, or (at your
12 option) any later version.
13
14 or
15
16 * the GNU General Public License as published by the Free Software
17 Foundation; either version 2 of the License, or (at your option) any
18 later version.
19
20 or both in parallel, as here.
21
22 The GNU MP Library is distributed in the hope that it will be useful, but
23 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
24 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
25 for more details.
26
27 You should have received copies of the GNU General Public License and the
28 GNU Lesser General Public License along with the GNU MP Library. If not,
29 see https://www.gnu.org/licenses/. */
30
31 #include <stdio.h>
32 #include "gmp-impl.h"
33 #include "longlong.h"
34
35
36 /* change to "#define TRACE(x) x" to get some traces */
37 #define TRACE(x)
38
39
40 /* In the F[2k+1] below for k odd, the -2 won't give a borrow from the low
41 limb because the result F[2k+1] is an F[4m+3] and such numbers are always
42 == 1, 2 or 5 mod 8, whereas an underflow would leave 6 or 7. (This is
43 the same as in mpn_fib2_ui.)
44
45 In the F[2k+1] for k even, the +2 won't give a carry out of the low limb
46 in normal circumstances. This is an F[4m+1] and we claim that F[3*2^b+1]
47 == 1 mod 2^b is the first F[4m+1] congruent to 0 or 1 mod 2^b, and hence
48 if n < 2^GMP_NUMB_BITS then F[n] cannot have a low limb of 0 or 1. No
49 proof for this claim, but it's been verified up to b==32 and has such a
50 nice pattern it must be true :-). Of interest is that F[3*2^b] == 0 mod
51 2^(b+1) seems to hold too.
52
53 When n >= 2^GMP_NUMB_BITS, which can arise in a nails build, then the low
54 limb of F[4m+1] can certainly be 1, and an mpn_add_1 must be used. */
55
56 void
57 mpz_fib_ui (mpz_ptr fn, unsigned long n)
58 {
59 mp_ptr fp, xp, yp;
60 mp_size_t size, xalloc;
61 unsigned long n2;
62 mp_limb_t c;
63 TMP_DECL;
64
65 if (n <= FIB_TABLE_LIMIT)
66 {
67 MPZ_NEWALLOC (fn, 1)[0] = FIB_TABLE (n);
68 SIZ(fn) = (n != 0); /* F[0]==0, others are !=0 */
69 return;
70 }
71
72 n2 = n/2;
73 xalloc = MPN_FIB2_SIZE (n2) + 1;
74 fp = MPZ_NEWALLOC (fn, 2 * xalloc);
75
76 TMP_MARK;
77 TMP_ALLOC_LIMBS_2 (xp,xalloc, yp,xalloc);
78 size = mpn_fib2_ui (xp, yp, n2);
79
80 TRACE (printf ("mpz_fib_ui last step n=%lu size=%ld bit=%lu\n",
81 n >> 1, size, n&1);
82 mpn_trace ("xp", xp, size);
83 mpn_trace ("yp", yp, size));
84
85 if (n & 1)
86 {
87 /* F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k */
88 mp_size_t xsize, ysize;
89
90 #if HAVE_NATIVE_mpn_add_n_sub_n
91 xp[size] = mpn_lshift (xp, xp, size, 1);
92 yp[size] = 0;
93 ASSERT_NOCARRY (mpn_add_n_sub_n (xp, yp, xp, yp, size+1));
94 xsize = size + (xp[size] != 0);
95 ASSERT (yp[size] <= 1);
96 ysize = size + yp[size];
97 #else
98 mp_limb_t c2;
99
100 c2 = mpn_lshift (fp, xp, size, 1);
101 c = c2 + mpn_add_n (xp, fp, yp, size);
102 xp[size] = c;
103 xsize = size + (c != 0);
104 c2 -= mpn_sub_n (yp, fp, yp, size);
105 yp[size] = c2;
106 ASSERT (c2 <= 1);
107 ysize = size + c2;
108 #endif
109
110 size = xsize + ysize;
111 c = mpn_mul (fp, xp, xsize, yp, ysize);
112
113 #if GMP_NUMB_BITS >= BITS_PER_ULONG
114 /* no overflow, see comments above */
115 ASSERT (n & 2 ? fp[0] >= 2 : fp[0] <= GMP_NUMB_MAX-2);
116 fp[0] += (n & 2 ? -CNST_LIMB(2) : CNST_LIMB(2));
117 #else
118 if (n & 2)
119 {
120 ASSERT (fp[0] >= 2);
121 fp[0] -= 2;
122 }
123 else
124 {
125 ASSERT (c != GMP_NUMB_MAX); /* because it's the high of a mul */
126 c += mpn_add_1 (fp, fp, size-1, CNST_LIMB(2));
127 fp[size-1] = c;
128 }
129 #endif
130 }
131 else
132 {
133 /* F[2k] = F[k]*(F[k]+2F[k-1]) */
134
135 mp_size_t xsize, ysize;
136 #if HAVE_NATIVE_mpn_addlsh1_n
137 c = mpn_addlsh1_n (yp, xp, yp, size);
138 #else
139 c = mpn_lshift (yp, yp, size, 1);
140 c += mpn_add_n (yp, yp, xp, size);
141 #endif
142 yp[size] = c;
143 xsize = size;
144 ysize = size + (c != 0);
145 size += ysize;
146 c = mpn_mul (fp, yp, ysize, xp, xsize);
147 }
148
149 /* one or two high zeros */
150 size -= (c == 0);
151 size -= (fp[size-1] == 0);
152 SIZ(fn) = size;
153
154 TRACE (printf ("done special, size=%ld\n", size);
155 mpn_trace ("fp ", fp, size));
156
157 TMP_FREE;
158 }