1 /* mpz_fib_ui -- calculate Fibonacci numbers. 2 3 Copyright 2000, 2001, 2002, 2005 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 the GNU Lesser General Public License as published by 9 the Free Software Foundation; either version 3 of the License, or (at your 10 option) any later version. 11 12 The GNU MP Library is distributed in the hope that it will be useful, but 13 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 14 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 15 License for more details. 16 17 You should have received a copy of the GNU Lesser General Public License 18 along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */ 19 20 #include <stdio.h> 21 #include "gmp.h" 22 #include "gmp-impl.h" 23 #include "longlong.h" 24 25 26 /* change to "#define TRACE(x) x" to get some traces */ 27 #define TRACE(x) 28 29 30 /* In the F[2k+1] below for k odd, the -2 won't give a borrow from the low 31 limb because the result F[2k+1] is an F[4m+3] and such numbers are always 32 == 1, 2 or 5 mod 8, whereas an underflow would leave 6 or 7. (This is 33 the same as in mpn_fib2_ui.) 34 35 In the F[2k+1] for k even, the +2 won't give a carry out of the low limb 36 in normal circumstances. This is an F[4m+1] and we claim that F[3*2^b+1] 37 == 1 mod 2^b is the first F[4m+1] congruent to 0 or 1 mod 2^b, and hence 38 if n < 2^GMP_NUMB_BITS then F[n] cannot have a low limb of 0 or 1. No 39 proof for this claim, but it's been verified up to b==32 and has such a 40 nice pattern it must be true :-). Of interest is that F[3*2^b] == 0 mod 41 2^(b+1) seems to hold too. 42 43 When n >= 2^GMP_NUMB_BITS, which can arise in a nails build, then the low 44 limb of F[4m+1] can certainly be 1, and an mpn_add_1 must be used. */ 45 46 void 47 mpz_fib_ui (mpz_ptr fn, unsigned long n) 48 { 49 mp_ptr fp, xp, yp; 50 mp_size_t size, xalloc; 51 unsigned long n2; 52 mp_limb_t c, c2; 53 TMP_DECL; 54 55 if (n <= FIB_TABLE_LIMIT) 56 { 57 PTR(fn)[0] = FIB_TABLE (n); 58 SIZ(fn) = (n != 0); /* F[0]==0, others are !=0 */ 59 return; 60 } 61 62 n2 = n/2; 63 xalloc = MPN_FIB2_SIZE (n2) + 1; 64 MPZ_REALLOC (fn, 2*xalloc+1); 65 fp = PTR (fn); 66 67 TMP_MARK; 68 TMP_ALLOC_LIMBS_2 (xp,xalloc, yp,xalloc); 69 size = mpn_fib2_ui (xp, yp, n2); 70 71 TRACE (printf ("mpz_fib_ui last step n=%lu size=%ld bit=%lu\n", 72 n >> 1, size, n&1); 73 mpn_trace ("xp", xp, size); 74 mpn_trace ("yp", yp, size)); 75 76 if (n & 1) 77 { 78 /* F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k */ 79 mp_size_t xsize, ysize; 80 81 #if HAVE_NATIVE_mpn_add_n_sub_n 82 xp[size] = mpn_lshift (xp, xp, size, 1); 83 yp[size] = 0; 84 ASSERT_NOCARRY (mpn_add_n_sub_n (xp, yp, xp, yp, size+1)); 85 xsize = size + (xp[size] != 0); 86 ysize = size + (yp[size] != 0); 87 #else 88 c2 = mpn_lshift (fp, xp, size, 1); 89 c = c2 + mpn_add_n (xp, fp, yp, size); 90 xp[size] = c; 91 xsize = size + (c != 0); 92 c2 -= mpn_sub_n (yp, fp, yp, size); 93 yp[size] = c2; 94 ASSERT (c2 <= 1); 95 ysize = size + c2; 96 #endif 97 98 size = xsize + ysize; 99 c = mpn_mul (fp, xp, xsize, yp, ysize); 100 101 #if GMP_NUMB_BITS >= BITS_PER_ULONG 102 /* no overflow, see comments above */ 103 ASSERT (n & 2 ? fp[0] >= 2 : fp[0] <= GMP_NUMB_MAX-2); 104 fp[0] += (n & 2 ? -CNST_LIMB(2) : CNST_LIMB(2)); 105 #else 106 if (n & 2) 107 { 108 ASSERT (fp[0] >= 2); 109 fp[0] -= 2; 110 } 111 else 112 { 113 ASSERT (c != GMP_NUMB_MAX); /* because it's the high of a mul */ 114 c += mpn_add_1 (fp, fp, size-1, CNST_LIMB(2)); 115 fp[size-1] = c; 116 } 117 #endif 118 } 119 else 120 { 121 /* F[2k] = F[k]*(F[k]+2F[k-1]) */ 122 123 mp_size_t xsize, ysize; 124 c = mpn_lshift (yp, yp, size, 1); 125 c += mpn_add_n (yp, yp, xp, size); 126 yp[size] = c; 127 xsize = size; 128 ysize = size + (c != 0); 129 size += ysize; 130 c = mpn_mul (fp, yp, ysize, xp, xsize); 131 } 132 133 /* one or two high zeros */ 134 size -= (c == 0); 135 size -= (fp[size-1] == 0); 136 SIZ(fn) = size; 137 138 TRACE (printf ("done special, size=%ld\n", size); 139 mpn_trace ("fp ", fp, size)); 140 141 TMP_FREE; 142 } 143