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 } 159