1 /* mpz_fib_ui -- calculate Fibonacci numbers. 2 3 Copyright 2000-2002, 2005, 2012, 2014 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.h" 33 #include "gmp-impl.h" 34 #include "longlong.h" 35 36 37 /* change to "#define TRACE(x) x" to get some traces */ 38 #define TRACE(x) 39 40 41 /* In the F[2k+1] below for k odd, the -2 won't give a borrow from the low 42 limb because the result F[2k+1] is an F[4m+3] and such numbers are always 43 == 1, 2 or 5 mod 8, whereas an underflow would leave 6 or 7. (This is 44 the same as in mpn_fib2_ui.) 45 46 In the F[2k+1] for k even, the +2 won't give a carry out of the low limb 47 in normal circumstances. This is an F[4m+1] and we claim that F[3*2^b+1] 48 == 1 mod 2^b is the first F[4m+1] congruent to 0 or 1 mod 2^b, and hence 49 if n < 2^GMP_NUMB_BITS then F[n] cannot have a low limb of 0 or 1. No 50 proof for this claim, but it's been verified up to b==32 and has such a 51 nice pattern it must be true :-). Of interest is that F[3*2^b] == 0 mod 52 2^(b+1) seems to hold too. 53 54 When n >= 2^GMP_NUMB_BITS, which can arise in a nails build, then the low 55 limb of F[4m+1] can certainly be 1, and an mpn_add_1 must be used. */ 56 57 void 58 mpz_fib_ui (mpz_ptr fn, unsigned long n) 59 { 60 mp_ptr fp, xp, yp; 61 mp_size_t size, xalloc; 62 unsigned long n2; 63 mp_limb_t c; 64 TMP_DECL; 65 66 if (n <= FIB_TABLE_LIMIT) 67 { 68 PTR(fn)[0] = FIB_TABLE (n); 69 SIZ(fn) = (n != 0); /* F[0]==0, others are !=0 */ 70 return; 71 } 72 73 n2 = n/2; 74 xalloc = MPN_FIB2_SIZE (n2) + 1; 75 fp = MPZ_NEWALLOC (fn, 2 * xalloc); 76 77 TMP_MARK; 78 TMP_ALLOC_LIMBS_2 (xp,xalloc, yp,xalloc); 79 size = mpn_fib2_ui (xp, yp, n2); 80 81 TRACE (printf ("mpz_fib_ui last step n=%lu size=%ld bit=%lu\n", 82 n >> 1, size, n&1); 83 mpn_trace ("xp", xp, size); 84 mpn_trace ("yp", yp, size)); 85 86 if (n & 1) 87 { 88 /* F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k */ 89 mp_size_t xsize, ysize; 90 91 #if HAVE_NATIVE_mpn_add_n_sub_n 92 xp[size] = mpn_lshift (xp, xp, size, 1); 93 yp[size] = 0; 94 ASSERT_NOCARRY (mpn_add_n_sub_n (xp, yp, xp, yp, size+1)); 95 xsize = size + (xp[size] != 0); 96 ASSERT (yp[size] <= 1); 97 ysize = size + yp[size]; 98 #else 99 mp_limb_t c2; 100 101 c2 = mpn_lshift (fp, xp, size, 1); 102 c = c2 + mpn_add_n (xp, fp, yp, size); 103 xp[size] = c; 104 xsize = size + (c != 0); 105 c2 -= mpn_sub_n (yp, fp, yp, size); 106 yp[size] = c2; 107 ASSERT (c2 <= 1); 108 ysize = size + c2; 109 #endif 110 111 size = xsize + ysize; 112 c = mpn_mul (fp, xp, xsize, yp, ysize); 113 114 #if GMP_NUMB_BITS >= BITS_PER_ULONG 115 /* no overflow, see comments above */ 116 ASSERT (n & 2 ? fp[0] >= 2 : fp[0] <= GMP_NUMB_MAX-2); 117 fp[0] += (n & 2 ? -CNST_LIMB(2) : CNST_LIMB(2)); 118 #else 119 if (n & 2) 120 { 121 ASSERT (fp[0] >= 2); 122 fp[0] -= 2; 123 } 124 else 125 { 126 ASSERT (c != GMP_NUMB_MAX); /* because it's the high of a mul */ 127 c += mpn_add_1 (fp, fp, size-1, CNST_LIMB(2)); 128 fp[size-1] = c; 129 } 130 #endif 131 } 132 else 133 { 134 /* F[2k] = F[k]*(F[k]+2F[k-1]) */ 135 136 mp_size_t xsize, ysize; 137 #if HAVE_NATIVE_mpn_addlsh1_n 138 c = mpn_addlsh1_n (yp, xp, yp, size); 139 #else 140 c = mpn_lshift (yp, yp, size, 1); 141 c += mpn_add_n (yp, yp, xp, size); 142 #endif 143 yp[size] = c; 144 xsize = size; 145 ysize = size + (c != 0); 146 size += ysize; 147 c = mpn_mul (fp, yp, ysize, xp, xsize); 148 } 149 150 /* one or two high zeros */ 151 size -= (c == 0); 152 size -= (fp[size-1] == 0); 153 SIZ(fn) = size; 154 155 TRACE (printf ("done special, size=%ld\n", size); 156 mpn_trace ("fp ", fp, size)); 157 158 TMP_FREE; 159 } 160