The '''Fibonacci sequence''' is a sequence Fn of natural numbers defined recursively:
<big><big> F<sub>0</sub> = 0 </big></big>
<big><big> F<sub>1</sub> = 1 </big></big>
<big><big> F<sub>n</sub> = F<sub>n-1</sub> + F<sub>n-2</sub>, if n>1 </big></big>
Task
Write a function to generate the nth Fibonacci number.
Solutions can be iterative or recursive (though recursive solutions are generally considered too slow and are mostly used as an exercise in recursion).
The sequence is sometimes extended into negative numbers by using a straightforward inverse of the positive definition:
<big><big> F<sub>n</sub> = F<sub>n+2</sub> - F<sub>n+1</sub>, if n<0 </big></big>
support for negative n in the solution is optional.
Related tasks
- [[Fibonacci n-step number sequences]]
- [[Leonardo numbers]]
References
- [[wp:Fibonacci number|Wikipedia, Fibonacci number]]
- [[wp:Lucas number|Wikipedia, Lucas number]]
- [http://mathworld.wolfram.com/FibonacciNumber.html MathWorld, Fibonacci Number]
- [http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf Some identities for r-Fibonacci numbers]
- [[oeis:A000045|OEIS Fibonacci numbers]]
- [[oeis:A000032|OEIS Lucas numbers]]
0815
%<:0D:>~$<:01:~%>=<:a94fad42221f2702:>~>
}:_s:{x{={~$x+%{=>~>x~-x<:0D:~>~>~^:_s:?
360 Assembly
For maximum compatibility, programs use only the basic instruction set.
using fullword integers
* Fibonacci sequence 05/11/2014
* integer (31 bits) = 10 decimals -> max fibo(46)
FIBONACC CSECT
USING FIBONACC,R12 base register
SAVEAREA B STM-SAVEAREA(R15) skip savearea
DC 17F'0' savearea
DC CL8'FIBONACC' eyecatcher
STM STM R14,R12,12(R13) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R12,R15 set addressability
* ----
LA R1,0 f(n-2)=0
LA R2,1 f(n-1)=1
LA R4,2 n=2
LA R6,1 step
LH R7,NN limit
LOOP EQU * for n=2 to nn
LR R3,R2 f(n)=f(n-1)
AR R3,R1 f(n)=f(n-1)+f(n-2)
CVD R4,PW n convert binary to packed (PL8)
UNPK ZW,PW packed (PL8) to zoned (ZL16)
MVC CW,ZW zoned (ZL16) to char (CL16)
OI CW+L'CW-1,X'F0' zap sign
MVC WTOBUF+5(2),CW+14 output
CVD R3,PW f(n) binary to packed decimal (PL8)
MVC ZN,EM load mask
ED ZN,PW packed dec (PL8) to char (CL20)
MVC WTOBUF+9(14),ZN+6 output
WTO MF=(E,WTOMSG) write buffer
LR R1,R2 f(n-2)=f(n-1)
LR R2,R3 f(n-1)=f(n)
BXLE R4,R6,LOOP endfor n
* ----
LM R14,R12,12(R13) restore previous savearea pointer
XR R15,R15 return code set to 0
BR R14 return to caller
* ---- DATA
NN DC H'46' nn max n
PW DS PL8 15num
ZW DS ZL16
CW DS CL16
ZN DS CL20
* ' b 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0' 15num
EM DC XL20'402020206B2020206B2020206B2020206B202120' mask
WTOMSG DS 0F
DC H'80',XL2'0000'
* fibo(46)=1836311903
WTOBUF DC CL80'fibo(12)=1234567890'
REGEQU
END FIBONACC
...
fibo(41)= 165,580,141
fibo(42)= 267,914,296
fibo(43)= 433,494,437
fibo(44)= 701,408,733
fibo(45)= 1,134,903,170
fibo(46)= 1,836,311,903
using packed decimals
* Fibonacci sequence 31/07/2018
* packed dec (PL8) = 15 decimals => max fibo(73)
FIBOWTOP CSECT
USING FIBOWTOP,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
SAVE (14,12) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
* ----
ZAP FNM2,=P'0' f(0)=0
ZAP FNM1,=P'1' f(1)=1
LA R4,2 n=2
LA R6,1 step
LH R7,NN limit
LOOP EQU * for n=2 to nn
ZAP FN,FNM1 f(n)=f(n-2)
AP FN,FNM2 f(n)=f(n-1)+f(n-2)
CVD R4,PW n
MVC ZN,EM load mask
ED ZN,PW packed dec (PL8) to char (CL16)
MVC WTOBUF+5(2),ZN+L'ZN-2 output
MVC ZN,EM load mask
ED ZN,FN packed dec (PL8) to char (CL16)
MVC WTOBUF+9(L'ZN),ZN output
WTO MF=(E,WTOMSG) write buffer
ZAP FNM2,FNM1 f(n-2)=f(n-1)
ZAP FNM1,FN f(n-1)=f(n)
BXLE R4,R6,LOOP endfor n
* ----
L R13,4(0,R13) restore previous savearea pointer
RETURN (14,12),RC=0 restore registers from calling sav
* ---- DATA
NN DC H'73' nn
FNM2 DS PL8 f(n-2)
FNM1 DS PL8 f(n-1)
FN DS PL8 f(n)
PW DS PL8 15num
ZN DS CL20
* ' b 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0' 15num
EM DC XL20'402020206B2020206B2020206B2020206B202120' mask
WTOMSG DS 0F
DC H'80',XL2'0000'
* fibo(73)=806515533049393
WTOBUF DC CL80'fibo(12)=123456789012345 '
REGEQU
END FIBOWTOP
...
fibo(68)= 72,723,460,248,141
fibo(69)= 117,669,030,460,994
fibo(70)= 190,392,490,709,135
fibo(71)= 308,061,521,170,129
fibo(72)= 498,454,011,879,264
fibo(73)= 806,515,533,049,393
6502 Assembly
This subroutine stores the first n—by default the first ten—Fibonacci numbers in memory, beginning (because, why not?) at address 3867 decimal = F1B hex. Intermediate results are stored in three sequential addresses within the low 256 bytes of memory, which are the most economical to access.
The results are calculated and stored, but are not output to the screen or any other physical device: how to do that would depend on the hardware and the operating system.
LDA #0
STA $F0 ; LOWER NUMBER
LDA #1
STA $F1 ; HIGHER NUMBER
LDX #0
LOOP: LDA $F1
STA $0F1B,X
STA $F2 ; OLD HIGHER NUMBER
ADC $F0
STA $F1 ; NEW HIGHER NUMBER
LDA $F2
STA $F0 ; NEW LOWER NUMBER
INX
CPX #$0A ; STOP AT FIB(10)
BMI LOOP
RTS ; RETURN FROM SUBROUTINE
8080 Assembly
This subroutine expects to be called with the value of in register A, and returns also in A. You may want to take steps to save the previous contents of B, C, and D. The routine only works with fairly small values of .
FIBNCI: MOV C, A ; C will store the counter
DCR C ; decrement, because we know f(1) already
MVI A, 1
MVI B, 0
LOOP: MOV D, A
ADD B ; A := A + B
MOV B, D
DCR C
JNZ LOOP ; jump if not zero
RET ; return from subroutine
8th
An iterative solution:
: fibon \ n -- fib(n)
>r 0 1
( tuck n:+ ) \ fib(n-2) fib(n-1) -- fib(n-1) fib(n)
r> n:1- times ;
: fib \ n -- fib(n)
dup 1 n:= if 1 ;; then
fibon nip ;
ABAP
Iterative
FORM fibonacci_iter USING index TYPE i
CHANGING number_fib TYPE i.
DATA: lv_old type i,
lv_cur type i.
Do index times.
If sy-index = 1 or sy-index = 2.
lv_cur = 1.
lv_old = 0.
endif.
number_fib = lv_cur + lv_old.
lv_old = lv_cur.
lv_cur = number_fib.
enddo.
ENDFORM.
Impure Functional
cl_demo_output=>display( REDUCE #( INIT fibnm = VALUE stringtab( ( |0| ) ( |1| ) )
n TYPE string
x = `0`
y = `1`
FOR i = 1 WHILE i <= 100
NEXT n = ( x + y )
fibnm = VALUE #( BASE fibnm ( n ) )
x = y
y = n ) ).
ACL2
Fast, tail recursive solution:
(defun fast-fib-r (n a b)
(if (or (zp n) (zp (1- n)))
b
(fast-fib-r (1- n) b (+ a b))))
(defun fast-fib (n)
(fast-fib-r n 1 1))
(defun first-fibs-r (n i)
(declare (xargs :measure (nfix (- n i))))
(if (zp (- n i))
nil
(cons (fast-fib i)
(first-fibs-r n (1+ i)))))
(defun first-fibs (n)
(first-fibs-r n 0))
>(first-fibs 20)
(1 1 2 3 5 8 13 21 34 55 89
144 233 377 610 987 1597 2584 4181 6765)
ActionScript
public function fib(n:uint):uint
{
if (n < 2)
return n;
return fib(n - 1) + fib(n - 2);
}
Ada
Recursive
with Ada.Text_IO, Ada.Command_Line;
procedure Fib is
X: Positive := Positive'Value(Ada.Command_Line.Argument(1));
function Fib(P: Positive) return Positive is
begin
if P <= 2 then
return 1;
else
return Fib(P-1) + Fib(P-2);
end if;
end Fib;
begin
Ada.Text_IO.Put("Fibonacci(" & Integer'Image(X) & " ) = ");
Ada.Text_IO.Put_Line(Integer'Image(Fib(X)));
end Fib;
===Iterative, build-in integers===
with Ada.Text_IO; use Ada.Text_IO;
procedure Test_Fibonacci is
function Fibonacci (N : Natural) return Natural is
This : Natural := 0;
That : Natural := 1;
Sum : Natural;
begin
for I in 1..N loop
Sum := This + That;
That := This;
This := Sum;
end loop;
return This;
end Fibonacci;
begin
for N in 0..10 loop
Put_Line (Positive'Image (Fibonacci (N)));
end loop;
end Test_Fibonacci;
0
1
1
2
3
5
8
13
21
34
55
===Iterative, long integers===
Using the big integer implementation from a cryptographic library [https://github.com/cforler/Ada-Crypto-Library/].
with Ada.Text_IO, Ada.Command_Line, Crypto.Types.Big_Numbers;
procedure Fibonacci is
X: Positive := Positive'Value(Ada.Command_Line.Argument(1));
Bit_Length: Positive := 1 + (696 * X) / 1000;
-- that number of bits is sufficient to store the full result.
package LN is new Crypto.Types.Big_Numbers
(Bit_Length + (32 - Bit_Length mod 32));
-- the actual number of bits has to be a multiple of 32
use LN;
function Fib(P: Positive) return Big_Unsigned is
Previous: Big_Unsigned := Big_Unsigned_Zero;
Result: Big_Unsigned := Big_Unsigned_One;
Tmp: Big_Unsigned;
begin
-- Result = 1 = Fibonacci(1)
for I in 1 .. P-1 loop
Tmp := Result;
Result := Previous + Result;
Previous := Tmp;
-- Result = Fibonacci(I+1))
end loop;
return Result;
end Fib;
begin
Ada.Text_IO.Put("Fibonacci(" & Integer'Image(X) & " ) = ");
Ada.Text_IO.Put_Line(LN.Utils.To_String(Fib(X)));
end Fibonacci;
> ./fibonacci 777
Fibonacci( 777 ) = 1081213530912648191985419587942084110095342850438593857649766278346130479286685742885693301250359913460718567974798268702550329302771992851392180275594318434818082
Fast method using fast matrix exponentiation
with ada.text_io;
use ada.text_io;
procedure fast_fibo is
-- We work with biggest natural integers in a 64 bits machine
type Big_Int is mod 2**64;
-- We provide an index type for accessing the fibonacci sequence terms
type Index is new Big_Int;
-- fibo is a generic function that needs a modulus type since it will return
-- the n'th term of the fibonacci sequence modulus this type (use Big_Int to get the
-- expected behaviour in this particular task)
generic
type ring_element is mod <>;
with function "*" (a, b : ring_element) return ring_element is <>;
function fibo (n : Index) return ring_element;
function fibo (n : Index) return ring_element is
type matrix is array (1 .. 2, 1 .. 2) of ring_element;
-- f is the matrix you apply to a column containing (F_n, F_{n+1}) to get
-- the next one containing (F_{n+1},F_{n+2})
-- could be a more general matrix (given as a generic parameter) to deal with
-- other linear sequences of order 2
f : constant matrix := (1 => (0, 1), 2 => (1, 1));
function "*" (a, b : matrix) return matrix is
(1 => (a(1,1)*b(1,1)+a(1,2)*b(2,1), a(1,1)*b(1,2)+a(1,2)*b(2,2)),
2 => (a(2,1)*b(1,1)+a(2,2)*b(2,1), a(2,1)*b(1,2)+a(2,2)*b(2,2)));
function square (m : matrix) return matrix is (m * m);
-- Fast_Pow could be non recursive but it doesn't really matter since
-- the number of calls is bounded up by the size (in bits) of Big_Int (e.g 64)
function fast_pow (m : matrix; n : Index) return matrix is
(if n = 0 then (1 => (1, 0), 2 => (0, 1)) -- = identity matrix
elsif n mod 2 = 0 then square (fast_pow (m, n / 2))
else m * square (fast_pow (m, n / 2)));
begin
return fast_pow (f, n)(2, 1);
end fibo;
function Big_Int_Fibo is new fibo (Big_Int);
begin
-- calculate instantly F_n with n=10^15 (modulus 2^64 )
put_line (Big_Int_Fibo (10**15)'img);
end fast_fibo;
AdvPL
Recursive
#include "totvs.ch"
User Function fibb(a,b,n)
return(if(--n>0,fibb(b,a+b,n),a))
Iterative
#include "totvs.ch"
User Function fibb(n)
local fnow:=0, fnext:=1, tempf
while (--n>0)
tempf:=fnow+fnext
fnow:=fnext
fnext:=tempf
end while
return(fnext)
Aime
integer
fibs(integer n)
{
integer w;
if (n == 0) {
w = 0;
} elif (n == 1) {
w = 1;
} else {
integer a, b, i;
i = 1;
a = 0;
b = 1;
while (i < n) {
w = a + b;
a = b;
b = w;
i += 1;
}
}
return w;
}
ALGOL 68
Analytic
PROC analytic fibonacci = (LONG INT n)LONG INT:(
LONG REAL sqrt 5 = long sqrt(5);
LONG REAL p = (1 + sqrt 5) / 2;
LONG REAL q = 1/p;
ROUND( (p**n + q**n) / sqrt 5 )
);
FOR i FROM 1 TO 30 WHILE
print(whole(analytic fibonacci(i),0));
# WHILE # i /= 30 DO
print(", ")
OD;
print(new line)
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040
Iterative
PROC iterative fibonacci = (INT n)INT:
CASE n+1 IN
0, 1, 1, 2, 3, 5
OUT
INT even:=3, odd:=5;
FOR i FROM odd+1 TO n DO
(ODD i|odd|even) := odd + even
OD;
(ODD n|odd|even)
ESAC;
FOR i FROM 0 TO 30 WHILE
print(whole(iterative fibonacci(i),0));
# WHILE # i /= 30 DO
print(", ")
OD;
print(new line)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040
Recursive
PROC recursive fibonacci = (INT n)INT:
( n < 2 | n | fib(n-1) + fib(n-2));
Generative
MODE YIELDINT = PROC(INT)VOID;
PROC gen fibonacci = (INT n, YIELDINT yield)VOID: (
INT even:=0, odd:=1;
yield(even);
yield(odd);
FOR i FROM odd+1 TO n DO
yield( (ODD i|odd|even) := odd + even )
OD
);
main:(
# FOR INT n IN # gen fibonacci(30, # ) DO ( #
## (INT n)VOID:(
print((" ",whole(n,0)))
# OD # ));
print(new line)
)
1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
===Array (Table) Lookup=== This uses a pre-generated list, requiring much less run-time processor usage, but assumes that INT is only 31 bits wide.
[]INT const fibonacci = []INT( -1836311903, 1134903170,
-701408733, 433494437, -267914296, 165580141, -102334155,
63245986, -39088169, 24157817, -14930352, 9227465, -5702887,
3524578, -2178309, 1346269, -832040, 514229, -317811, 196418,
-121393, 75025, -46368, 28657, -17711, 10946, -6765, 4181,
-2584, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13,
-8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,
28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040,
1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817,
39088169, 63245986, 102334155, 165580141, 267914296, 433494437,
701408733, 1134903170, 1836311903
)[@-46];
PROC VOID value error := stop;
PROC lookup fibonacci = (INT i)INT: (
IF LWB const fibonacci <= i AND i<= UPB const fibonacci THEN
const fibonacci[i]
ELSE
value error; SKIP
FI
);
FOR i FROM 0 TO 30 WHILE
print(whole(lookup fibonacci(i),0));
# WHILE # i /= 30 DO
print(", ")
OD;
print(new line)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040
=={{header|ALGOL-M}}== Note that the 21st Fibonacci number (= 10946) is the largest that can be calculated without overflowing ALGOL-M's integer data type.
=Iterative=
INTEGER FUNCTION FIBONACCI( X ); INTEGER X;
BEGIN
INTEGER M, N, A, I;
M := 0;
N := 1;
FOR I := 2 STEP 1 UNTIL X DO
BEGIN
A := N;
N := M + N;
M := A;
END;
FIBONACCI := N;
END;
=Naively recursive=
INTEGER FUNCTION FIBONACCI( X ); INTEGER X;
BEGIN
IF X < 3 THEN
FIBONACCI := 1
ELSE
FIBONACCI := FIBONACCI( X - 2 ) + FIBONACCI( X - 1 );
END;
ALGOL W
begin
% return the nth Fibonacci number %
integer procedure Fibonacci( integer value n ) ;
begin
integer fn, fn1, fn2;
fn2 := 1;
fn1 := 0;
fn := 0;
for i := 1 until n do begin
fn := fn1 + fn2;
fn2 := fn1;
fn1 := fn
end ;
fn
end Fibonacci ;
for i := 0 until 10 do writeon( i_w := 3, s_w := 0, Fibonacci( i ) )
end.
0 1 1 2 3 5 8 13 21 34 55
Alore
def fib(n as Int) as Int
if n < 2
return 1
end
return fib(n-1) + fib(n-2)
end
AntLang
/Sequence
fib:{<0;1> {x,<x[-1]+x[-2]>}/ range[x]}
/nth
fibn:{fib[x][x]}
Apex
/*
author: snugsfbay
date: March 3, 2016
description: Create a list of x numbers in the Fibonacci sequence.
- user may specify the length of the list
- enforces a minimum of 2 numbers in the sequence because any fewer is not a sequence
- enforces a maximum of 47 because further values are too large for integer data type
- Fibonacci sequence always starts with 0 and 1 by definition
*/
public class FibNumbers{
final static Integer MIN = 2; //minimum length of sequence
final static Integer MAX = 47; //maximum length of sequence
/*
description: method to create a list of numbers in the Fibonacci sequence
param: user specified integer representing length of sequence should be 2-47, inclusive.
- Sequence starts with 0 and 1 by definition so the minimum length could be as low as 2.
- For 48th number in sequence or greater, code would require a Long data type rather than an Integer.
return: list of integers in sequence.
*/
public static List<Integer> makeSeq(Integer len){
List<Integer> fib = new List<Integer>{0,1}; // initialize list with first two values
Integer i;
if(len<MIN || len==null || len>MAX) {
if (len>MAX){
len=MAX; //set length to maximum if user entered too high a value
}else{
len=MIN; //set length to minimum if user entered too low a value or none
}
} //This could be refactored using teneray operator, but we want code coverage to be reflected for each condition
//start with initial list size to find previous two values in the sequence, continue incrementing until list reaches user defined length
for(i=fib.size(); i<len; i++){
fib.add(fib[i-1]+fib[i-2]); //create new number based on previous numbers and add that to the list
}
return fib;
}
}
APL
=
Dyalog APL
=
=Naive Recursive=
fib←{⍵≤1:⍵ ⋄ (∇ ⍵-1)+∇ ⍵-2}
Read this as: In the variable "fib", store the function that says, if the argument is less than or equal to 1, return the argument. Else, calculate the value you get when you recursively call the current function with the argument of the current argument minus one and add that to the value you get when you recursively call the current function with the argument of the current function minus two.
This naive solution requires Dyalog APL because GNU APL does not support this syntax for conditional guards.
=
GNU APL/Dyalog APL
=
=Array=
Since APL is an array language we'll use the following identity: : In APL:
↑+.×/N/⊂2 2⍴1 1 1 0
Plugging in 4 for N gives the following result: : Here's what happens: We replicate the 2-by-2 matrix N times and then apply inner product-replication. The ''First'' removes the shell from the ''Enclose''. At this point we're basically done, but we need to pick out only in order to complete the task. Here's one way:
↑0 1↓↑+.×/N/⊂2 2⍴1 1 1 0
=Analytic=
An alternative approach, using Binet's formula (which was apparently known long before Binet):
⌊.5+(((1+PHI)÷2)*⍳N)÷PHI←5*.5
AppleScript
Imperative
set fibs to {}
set x to (text returned of (display dialog "What fibbonaci number do you want?" default answer "3"))
set x to x as integer
repeat with y from 1 to x
if (y = 1 or y = 2) then
copy 1 to the end of fibs
else
copy ((item (y - 1) of fibs) + (item (y - 2) of fibs)) to the end of fibs
end if
end repeat
return item x of fibs
Functional
The simple recursive version is famously slow:
on fib(n)
if n < 1 then
0
else if n < 3 then
1
else
fib(n - 2) + fib(n - 1)
end if
end fib
but we can combine '''enumFromTo(m, n)''' with the accumulator of a higher-order '''fold/reduce''' function to memoize the series:
{{Trans|JavaScript}} (ES6 memoized fold example) {{Trans|Haskell}} (Memoized fold example)
-- fib :: Int -> Int
on fib(n)
-- lastTwo : (Int, Int) -> (Int, Int)
script lastTwo
on |λ|([a, b])
[b, a + b]
end |λ|
end script
item 1 of foldl(lastTwo, {0, 1}, enumFromTo(1, n))
end fib
-- TEST -----------------------------------------------------------------------
on run
fib(32)
--> 2178309
end run
-- GENERIC FUNCTIONS ----------------------------------------------------------
-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m, n)
if n < m then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo
-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
2178309
Arendelle
( fibonacci , 1; 1 )
[ 98 , // 100 numbers of fibonacci
( fibonacci[ @fibonacci? ] ,
@fibonacci[ @fibonacci - 1 ] + @fibonacci[ @fibonacci - 2 ]
)
"Index: | @fibonacci? | => | @fibonacci[ @fibonacci? - 1 ] |"
]
ARM Assembly
Expects to be called with in R0, and will return in the same register.
fibonacci:
push {r1-r3}
mov r1, #0
mov r2, #1
fibloop:
mov r3, r2
add r2, r1, r2
mov r1, r3
sub r0, r0, #1
cmp r0, #1
bne fibloop
mov r0, r2
pop {r1-r3}
mov pc, lr
ArnoldC
IT'S SHOWTIME
HEY CHRISTMAS TREE f1
YOU SET US UP @I LIED
TALK TO THE HAND f1
HEY CHRISTMAS TREE f2
YOU SET US UP @NO PROBLEMO
HEY CHRISTMAS TREE f3
YOU SET US UP @I LIED
STICK AROUND @NO PROBLEMO
GET TO THE CHOPPER f3
HERE IS MY INVITATION f1
GET UP f2
ENOUGH TALK
TALK TO THE HAND f3
GET TO THE CHOPPER f1
HERE IS MY INVITATION f2
ENOUGH TALK
GET TO THE CHOPPER f2
HERE IS MY INVITATION f3
ENOUGH TALK
CHILL
YOU HAVE BEEN TERMINATED
Arturo
Recursive
Fib [x]{
if x<2 { 1 }{
$(Fib x-1) + $(Fib x-2)
}
}
Recursive with Memoization
Fib $(memoize [x]{
if x<2 { 1 }{
$(Fib x-1) + $(Fib x-2)
}
})
AsciiDots
/--#$--\
| |
>-*>{+}/
| \+-/
1 |
# 1
| #
| |
. .
ATS
Recursive
fun fib_rec(n: int): int =
if n >= 2 then fib_rec(n-1) + fib_rec(n-2) else n
Iterative
(*
** This one is also referred to as being tail-recursive
*)
fun
fib_trec(n: int): int =
if
n > 0
then (fix loop (i:int, r0:int, r1:int): int => if i > 1 then loop (i-1, r1, r0+r1) else r1)(n, 0, 1)
else 0
Iterative and Verified
(*
** This implementation is verified!
*)
dataprop FIB (int, int) =
| FIB0 (0, 0) | FIB1 (1, 1)
| {n:nat} {r0,r1:int} FIB2 (n+2, r0+r1) of (FIB (n, r0), FIB (n+1, r1))
// end of [FIB] // end of [dataprop]
fun
fibats{n:nat}
(n: int (n))
: [r:int] (FIB (n, r) | int r) = let
fun loop
{i:nat | i <= n}{r0,r1:int}
(
pf0: FIB (i, r0), pf1: FIB (i+1, r1)
| ni: int (n-i), r0: int r0, r1: int r1
) : [r:int] (FIB (n, r) | int r) =
if (ni > 0)
then loop{i+1}(pf1, FIB2 (pf0, pf1) | ni - 1, r1, r0 + r1)
else (pf0 | r0)
// end of [if]
// end of [loop]
in
loop {0} (FIB0 (), FIB1 () | n, 0, 1)
end // end of [fibats]
===Matrix-based===
(* ****** ****** *)
//
// How to compile:
// patscc -o fib fib.dats
//
(* ****** ****** *)
//
#include
"share/atspre_staload.hats"
//
(* ****** ****** *)
//
abst@ype
int3_t0ype =
(int, int, int)
//
typedef int3 = int3_t0ype
//
(* ****** ****** *)
extern
fun int3 : (int, int, int) -<> int3
extern
fun int3_1 : int3 -<> int
extern
fun mul_int3_int3: (int3, int3) -<> int3
(* ****** ****** *)
local
assume
int3_t0ype = (int, int, int)
in (* in-of-local *)
//
implement
int3 (x, y, z) = @(x, y, z)
//
implement int3_1 (xyz) = xyz.1
//
implement
mul_int3_int3
(
@(a,b,c), @(d,e,f)
) =
(a*d + b*e, a*e + b*f, b*e + c*f)
//
end // end of [local]
(* ****** ****** *)
//
implement
gnumber_int<int3> (n) = int3(n, 0, n)
//
implement gmul_val<int3> = mul_int3_int3
//
(* ****** ****** *)
//
fun
fib (n: intGte(0)): int =
int3_1(gpow_int_val<int3> (n, int3(1, 1, 0)))
//
(* ****** ****** *)
implement
main0 () =
{
//
val N = 10
val () = println! ("fib(", N, ") = ", fib(N))
val N = 20
val () = println! ("fib(", N, ") = ", fib(N))
val N = 30
val () = println! ("fib(", N, ") = ", fib(N))
val N = 40
val () = println! ("fib(", N, ") = ", fib(N))
//
} (* end of [main0] *)
AutoHotkey
Iterative
Loop, 5
MsgBox % fib(A_Index)
Return
fib(n)
{
If (n < 2)
Return n
i := last := this := 1
While (i <= n)
{
new := last + this
last := this
this := new
i++
}
Return this
}
Recursive and iterative
Source: [http://www.autohotkey.com/forum/topic44657.html AutoHotkey forum] by Laszlo
/*
Important note: the recursive version would be very slow
without a global or static array. The iterative version
handles also negative arguments properly.
*/
FibR(n) { ; n-th Fibonacci number (n>=0, recursive with static array Fibo)
Static
Return n<2 ? n : Fibo%n% ? Fibo%n% : Fibo%n% := FibR(n-1)+FibR(n-2)
}
Fib(n) { ; n-th Fibonacci number (n < 0 OK, iterative)
a := 0, b := 1
Loop % abs(n)-1
c := b, b += a, a := c
Return n=0 ? 0 : n>0 || n&1 ? b : -b
}
AutoIt
Iterative
#AutoIt Version: 3.2.10.0
$n0 = 0
$n1 = 1
$n = 10
MsgBox (0,"Iterative Fibonacci ", it_febo($n0,$n1,$n))
Func it_febo($n_0,$n_1,$N)
$first = $n_0
$second = $n_1
$next = $first + $second
$febo = 0
For $i = 1 To $N-3
$first = $second
$second = $next
$next = $first + $second
Next
if $n==0 Then
$febo = 0
ElseIf $n==1 Then
$febo = $n_0
ElseIf $n==2 Then
$febo = $n_1
Else
$febo = $next
EndIf
Return $febo
EndFunc
Recursive
#AutoIt Version: 3.2.10.0
$n0 = 0
$n1 = 1
$n = 10
MsgBox (0,"Recursive Fibonacci ", rec_febo($n0,$n1,$n))
Func rec_febo($r_0,$r_1,$R)
if $R<3 Then
if $R==2 Then
Return $r_1
ElseIf $R==1 Then
Return $r_0
ElseIf $R==0 Then
Return 0
EndIf
Return $R
Else
Return rec_febo($r_0,$r_1,$R-1) + rec_febo($r_0,$r_1,$R-2)
EndIf
EndFunc
AWK
As in many examples, this one-liner contains the function as well as testing with input from stdin, output to stdout.
$ awk 'func fib(n){return(n<2?n:fib(n-1)+fib(n-2))}{print "fib("$1")="fib($1)}'
10
fib(10)=55
Axe
A recursive solution is not practical in Axe because there is no concept of variable scope in Axe.
Iterative solution:
Lbl FIB
r₁→N
0→I
1→J
For(K,1,N)
I+J→T
J→I
T→J
End
J
Return
bash
Iterative
$ fib=1;j=1;while((fib<100));do echo $fib;((k=fib+j,fib=j,j=k));done
1
1
2
3
5
8
13
21
34
55
89
Recursive
fib()
{
if [ $1 -le 0 ]
then
echo 0
return 0
fi
if [ $1 -le 2 ]
then
echo 1
else
a=$(fib $[$1-1])
b=$(fib $[$1-2])
echo $(($a+$b))
fi
}
Babel
In Babel, we can define fib using a stack-based approach that is not recursive:
fib { <- 0 1 { dup <- + -> swap } -> times zap } <
foo x < puts x in foo. In this case, x is the code list between the curly-braces. This is how you define callable code in Babel. The definition works by initializing the stack with 0, 1. On each iteration of the times loop, the function duplicates the top element. In the first iteration, this gives 0, 1, 1. Then it moves down the stack with the <- operator, giving 0, 1 again. It adds, giving 1. then it moves back up the stack, giving 1, 1. Then it swaps. On the next iteration this gives:
1, 1, 1 (dup)
1, 1, (<-)
2 (+)
2, 1 (->)
1, 2 (swap)
And so on. To test fib:
{19 iter - fib !} 20 times collect ! lsnum !
( 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 )
BASIC
=
Applesoft BASIC
= Same code as [[#Commodore_BASIC|Commodore BASIC]]
Entering a value of N > 183, produces an error message:
?OVERFLOW ERROR IN 40
=
BASIC256
=
# Basic-256 ver 1.1.4
# iterative Fibonacci sequence
# Matches sequence A000045 in the OEIS, https://oeis.org/A000045/list
# Return the Nth Fibonacci number
input "N = ",f
limit = 500 # set upper limit - can be changed, removed
f = int(f)
if f > limit then f = limit
a = 0 : b = 1 : c = 0 : n = 0 # initial values
while n < f
print n + chr(9) + c # chr(9) = tab
a = b
b = c
c = a + b
n += 1
end while
print " "
print n + chr(9) + c
=
Commodore BASIC
=
10 INPUT "ENTER VALUE OF N"; N
20 N1 = 0 : N2 = 1
30 FOR K=1 TO N
40 SUM = N1+N2
50 N1 = N2
60 N2 = SUM
70 NEXT K
80 PRINT N1
=
Integer BASIC
= Only works with quite small values of .
10 INPUT N
20 A=0
30 B=1
40 FOR I=2 TO N
50 C=B
60 B=A+B
70 A=C
80 NEXT I
90 PRINT B
100 END
==={{header|IS-BASIC}}===
=
## QBasic
=
### =Iterative=
```qbasic
FUNCTION itFib (n)
n1 = 0
n2 = 1
FOR k = 1 TO ABS(n)
sum = n1 + n2
n1 = n2
n2 = sum
NEXT k
IF n < 0 THEN
itFib = n1 * ((-1) ^ ((-n) + 1))
ELSE
itFib = n1
END IF
END FUNCTION
Next version calculates each value once, as needed, and stores the results in an array for later retreival (due to the use of REDIM PRESERVE, it requires [[QuickBASIC]] 4.5 or newer):
DECLARE FUNCTION fibonacci& (n AS INTEGER)
REDIM SHARED fibNum(1) AS LONG
fibNum(1) = 1
'*****sample inputs*****
PRINT fibonacci(0) 'no calculation needed
PRINT fibonacci(13) 'figure F(2)..F(13)
PRINT fibonacci(-42) 'figure F(14)..F(42)
PRINT fibonacci(47) 'error: too big
'*****sample inputs*****
FUNCTION fibonacci& (n AS INTEGER)
DIM a AS INTEGER
a = ABS(n)
SELECT CASE a
CASE 0 TO 46
SHARED fibNum() AS LONG
DIM u AS INTEGER, L0 AS INTEGER
u = UBOUND(fibNum)
IF a > u THEN
REDIM PRESERVE fibNum(a) AS LONG
FOR L0 = u + 1 TO a
fibNum(L0) = fibNum(L0 - 1) + fibNum(L0 - 2)
NEXT
END IF
IF n < 0 THEN
fibonacci = fibNum(a) * ((-1) ^ (a + 1))
ELSE
fibonacci = fibNum(n)
END IF
CASE ELSE
'limited to signed 32-bit int (LONG)
'F(47)=&hB11924E1
ERROR 6 'overflow
END SELECT
END FUNCTION
{{out}} (unhandled error in final input prevents output):
0
233
-267914296
=Recursive=
This example can't handle n < 0.
FUNCTION recFib (n)
IF (n < 2) THEN
recFib = n
ELSE
recFib = recFib(n - 1) + recFib(n - 2)
END IF
END FUNCTION
====Array (Table) Lookup====
This uses a pre-generated list, requiring much less run-time processor usage. (Since the sequence never changes, this is probably the best way to do this in "the real world". The same applies to other sequences like prime numbers, and numbers like pi and e.)
DATA -1836311903,1134903170,-701408733,433494437,-267914296,165580141,-102334155
DATA 63245986,-39088169,24157817,-14930352,9227465,-5702887,3524578,-2178309
DATA 1346269,-832040,514229,-317811,196418,-121393,75025,-46368,28657,-17711
DATA 10946,-6765,4181,-2584,1597,-987,610,-377,233,-144,89,-55,34,-21,13,-8,5,-3
DATA 2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765
DATA 10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,1346269
DATA 2178309,3524578,5702887,9227465,14930352,24157817,39088169,63245986
DATA 102334155,165580141,267914296,433494437,701408733,1134903170,1836311903
DIM fibNum(-46 TO 46) AS LONG
FOR n = -46 TO 46
READ fibNum(n)
NEXT
'*****sample inputs*****
FOR n = -46 TO 46
PRINT fibNum(n),
NEXT
PRINT
'*****sample inputs*****
=
Sinclair ZX81 BASIC
=
=Analytic=
10 INPUT N
20 PRINT INT (0.5+(((SQR 5+1)/2)**N)/SQR 5)
=Iterative=
10 INPUT N
20 LET A=0
30 LET B=1
40 FOR I=2 TO N
50 LET C=B
60 LET B=A+B
70 LET A=C
80 NEXT I
90 PRINT B
=Tail recursive=
10 INPUT N
20 LET A=0
30 LET B=1
40 GOSUB 70
50 PRINT B
60 STOP
70 IF N=1 THEN RETURN
80 LET C=B
90 LET B=A+B
100 LET A=C
110 LET N=N-1
120 GOSUB 70
130 RETURN
Batch File
Recursive version
::fibo.cmd
@echo off
if "%1" equ "" goto :eof
call :fib %1
echo %errorlevel%
goto :eof
:fib
setlocal enabledelayedexpansion
if %1 geq 2 goto :ge2
exit /b %1
:ge2
set /a r1 = %1 - 1
set /a r2 = %1 - 2
call :fib !r1!
set r1=%errorlevel%
call :fib !r2!
set r2=%errorlevel%
set /a r0 = r1 + r2
exit /b !r0!
>for /L %i in (1,5,20) do fibo.cmd %i
>fibo.cmd 1
1
>fibo.cmd 6
8
>fibo.cmd 11
89
>fibo.cmd 16
987
Battlestar
// Fibonacci sequence, recursive version
fun fibb
loop
a = funparam[0]
break (a < 2)
a--
// Save "a" while calling fibb
a -> stack
// Set the parameter and call fibb
funparam[0] = a
call fibb
// Handle the return value and restore "a"
b = funparam[0]
stack -> a
// Save "b" while calling fibb again
b -> stack
a--
// Set the parameter and call fibb
funparam[0] = a
call fibb
// Handle the return value and restore "b"
c = funparam[0]
stack -> b
// Sum the results
b += c
a = b
funparam[0] = a
break
end
end
// vim: set syntax=c ts=4 sw=4 et:
BBC BASIC
PRINT FNfibonacci_r(1), FNfibonacci_i(1)
PRINT FNfibonacci_r(13), FNfibonacci_i(13)
PRINT FNfibonacci_r(26), FNfibonacci_i(26)
END
DEF FNfibonacci_r(N)
IF N < 2 THEN = N
= FNfibonacci_r(N-1) + FNfibonacci_r(N-2)
DEF FNfibonacci_i(N)
LOCAL F, I, P, T
IF N < 2 THEN = N
P = 1
FOR I = 1 TO N
T = F
F += P
P = T
NEXT
= F
1 1
233 233
121393 121393
bc
iterative
#! /usr/bin/bc -q
define fib(x) {
if (x <= 0) return 0;
if (x == 1) return 1;
a = 0;
b = 1;
for (i = 1; i < x; i++) {
c = a+b; a = b; b = c;
}
return c;
}
fib(1000)
quit
beeswax
'#{;
_`Enter n: `TN`Fib(`{`)=`X~P~K#{;
#>~P~L#MM@>+@'q@{;
b~@M<
Example output:
Notice the UInt64 wrap-around at Fib(94)!
beeswax("n-th Fibonacci number.bswx")
Enter n: i0
Fib(0)=0
Program finished!
julia> beeswax("n-th Fibonacci number.bswx")
Enter n: i10
Fib(10)=55
Program finished!
julia> beeswax("n-th Fibonacci number.bswx")
Enter n: i92
Fib(92)=7540113804746346429
Program finished!
julia> beeswax("n-th Fibonacci number.bswx")
Enter n: i93
Fib(93)=12200160415121876738
Program finished!
julia> beeswax("n-th Fibonacci number.bswx")
Enter n: i94
Fib(94)=1293530146158671551
Program finished!
Befunge
:"@"8**++\1+:67+`#@_v
^ .:\/*8"@"\%*8"@":\ <
=={{header|Brainfuck}}== The first cell contains ''n'' (10), the second cell will contain ''fib(n)'' (55), and the third cell will contain ''fib(n-1)'' (34).
++++++++++
>>+<<[->[->+>+<<]>[-<+>]>[-<+>]<<<]
The following generates n fibonacci numbers and prints them, though not in ascii. It does have a limit due to the cells usually being 1 byte in size.
+++++ +++++ #0 set to n
>> + Init #2 to 1
<<
[
- #Decrement counter in #0
>>. Notice: This doesn't print it in ascii
To look at results you can pipe into a file and look with a hex editor
Copying sequence to save #2 in #4 using #5 as restore space
>>[-] Move to #4 and clear
>[-] Clear #5
<<< #2
[ Move loop
- >> + > + <<< Subtract #2 and add #4 and #5
]
>>>
[ Restore loop
- <<< + >>> Subtract from #5 and add to #2
]
<<<< Back to #1
Non destructive add sequence using #3 as restore value
[ Loop to add
- > + > + << Subtract #1 and add to value #2 and restore space #3
]
>>
[ Loop to restore #1 from #3
- << + >> Subtract from restore space #3 and add in #1
]
<< [-] Clear #1
>>>
[ Loop to move #4 to #1
- <<< + >>> Subtract from #4 and add to #1
]
<<<< Back to #0
]
Bracmat
Recursive
fib=.!arg:<2|fib$(!arg+-2)+fib$(!arg+-1)
fib$30 832040
Iterative
(fib=
last i this new
. !arg:<2
| 0:?last:?i
& 1:?this
& whl
' ( !i+1:<!arg:?i
& !last+!this:?new
& !this:?last
& !new:?this
)
& !this
)
fib$777 1081213530912648191985419587942084110095342850438593857649766278346130479286685742885693301250359913460718567974798268702550329302771992851392180275594318434818082
Brat
Recursive
fibonacci = { x |
true? x < 2, x, { fibonacci(x - 1) + fibonacci(x - 2) }
}
Tail Recursive
fib_aux = { x, next, result |
true? x == 0,
result,
{ fib_aux x - 1, next + result, next }
}
fibonacci = { x |
fib_aux x, 1, 0
}
Memoization
cache = hash.new
fibonacci = { x |
true? cache.key?(x)
{ cache[x] }
{true? x < 2, x, { cache[x] = fibonacci(x - 1) + fibonacci(x - 2) }}
}
Burlesque
{0 1}{^^++[+[-^^-]\/}30.*\[e!vv
0 1{{.+}c!}{1000.<}w!
C
Recursive
long long fibb(long long a, long long b, int n) {
return (--n>0)?(fibb(b, a+b, n)):(a);
}
Iterative
long long int fibb(int n) {
int fnow = 0, fnext = 1, tempf;
while(--n>0){
tempf = fnow + fnext;
fnow = fnext;
fnext = tempf;
}
return fnext;
}
Analytic
#include <tgmath.h>
#define PHI ((1 + sqrt(5))/2)
long long unsigned fib(unsigned n) {
return floor( (pow(PHI, n) - pow(1 - PHI, n))/sqrt(5) );
}
Generative
#include <stdio.h>
typedef enum{false=0, true=!0} bool;
typedef void iterator;
#include <setjmp.h>
/* declare label otherwise it is not visible in sub-scope */
#define LABEL(label) jmp_buf label; if(setjmp(label))goto label;
#define GOTO(label) longjmp(label, true)
/* the following line is the only time I have ever required "auto" */
#define FOR(i, iterator) { auto bool lambda(i); yield_init = (void *)λ iterator; bool lambda(i)
#define DO {
#define YIELD(x) if(!yield(x))return
#define BREAK return false
#define CONTINUE return true
#define OD CONTINUE; } }
static volatile void *yield_init; /* not thread safe */
#define YIELDS(type) bool (*yield)(type) = yield_init
iterator fibonacci(int stop){
YIELDS(int);
int f[] = {0, 1};
int i;
for(i=0; i<stop; i++){
YIELD(f[i%2]);
f[i%2]=f[0]+f[1];
}
}
main(){
printf("fibonacci: ");
FOR(int i, fibonacci(16)) DO
printf("%d, ",i);
OD;
printf("...\n");
}
fibonacci: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ...
Fast method for a single large value
#include <iostream>
#include <stdio.h>
#include <gmp.h>
typedef struct node node;
struct node {
int n;
mpz_t v;
node *next;
};
#define CSIZE 37
node *cache[CSIZE];
// very primitive linked hash table
node * find_cache(int n)
{
int idx = n % CSIZE;
node *p;
for (p = cache[idx]; p && p->n != n; p = p->next);
if (p) return p;
p = malloc(sizeof(node));
p->next = cache[idx];
cache[idx] = p;
if (n < 2) {
p->n = n;
mpz_init_set_ui(p->v, 1);
} else {
p->n = -1; // -1: value not computed yet
mpz_init(p->v);
}
return p;
}
mpz_t tmp1, tmp2;
mpz_t *fib(int n)
{
int x;
node *p = find_cache(n);
if (p->n < 0) {
p->n = n;
x = n / 2;
mpz_mul(tmp1, *fib(x-1), *fib(n - x - 1));
mpz_mul(tmp2, *fib(x), *fib(n - x));
mpz_add(p->v, tmp1, tmp2);
}
return &p->v;
}
int main(int argc, char **argv)
{
int i, n;
if (argc < 2) return 1;
mpz_init(tmp1);
mpz_init(tmp2);
for (i = 1; i < argc; i++) {
n = atoi(argv[i]);
if (n < 0) {
printf("bad input: %s\n", argv[i]);
continue;
}
// about 75% of time is spent in printing
gmp_printf("%Zd\n", *fib(n));
}
return 0;
}
% ./a.out 0 1 2 3 4 5
1
1
2
3
5
8
% ./a.out 10000000 | wc -c # count length of output, including the newline
1919488
C++
Using unsigned int, this version only works up to 48 before fib overflows.
#include <iostream>
int main()
{
unsigned int a = 1, b = 1;
unsigned int target = 48;
for(unsigned int n = 3; n <= target; ++n)
{
unsigned int fib = a + b;
std::cout << "F("<< n << ") = " << fib << std::endl;
a = b;
b = fib;
}
return 0;
}
This version does not have an upper bound.
#include <iostream>
#include <gmpxx.h>
int main()
{
mpz_class a = mpz_class(1), b = mpz_class(1);
mpz_class target = mpz_class(100);
for(mpz_class n = mpz_class(3); n <= target; ++n)
{
mpz_class fib = b + a;
if ( fib < b )
{
std::cout << "Overflow at " << n << std::endl;
break;
}
std::cout << "F("<< n << ") = " << fib << std::endl;
a = b;
b = fib;
}
return 0;
}
Version using transform:
#include <algorithm>
#include <vector>
#include <functional>
#include <iostream>
unsigned int fibonacci(unsigned int n) {
if (n == 0) return 0;
std::vector<int> v(n+1);
v[1] = 1;
transform(v.begin(), v.end()-2, v.begin()+1, v.begin()+2, std::plus<int>());
// "v" now contains the Fibonacci sequence from 0 up
return v[n];
}
Far-fetched version using adjacent_difference:
#include <numeric>
#include <vector>
#include <functional>
#include <iostream>
unsigned int fibonacci(unsigned int n) {
if (n == 0) return 0;
std::vector<int> v(n, 1);
adjacent_difference(v.begin(), v.end()-1, v.begin()+1, std::plus<int>());
// "array" now contains the Fibonacci sequence from 1 up
return v[n-1];
}
Version which computes at compile time with metaprogramming:
#include <iostream>
template <int n> struct fibo
{
enum {value=fibo<n-1>::value+fibo<n-2>::value};
};
template <> struct fibo<0>
{
enum {value=0};
};
template <> struct fibo<1>
{
enum {value=1};
};
int main(int argc, char const *argv[])
{
std::cout<<fibo<12>::value<<std::endl;
std::cout<<fibo<46>::value<<std::endl;
return 0;
}
The following version is based on fast exponentiation:
#include <iostream>
inline void fibmul(int* f, int* g)
{
int tmp = f[0]*g[0] + f[1]*g[1];
f[1] = f[0]*g[1] + f[1]*(g[0] + g[1]);
f[0] = tmp;
}
int fibonacci(int n)
{
int f[] = { 1, 0 };
int g[] = { 0, 1 };
while (n > 0)
{
if (n & 1) // n odd
{
fibmul(f, g);
--n;
}
else
{
fibmul(g, g);
n >>= 1;
}
}
return f[1];
}
int main()
{
for (int i = 0; i < 20; ++i)
std::cout << fibonacci(i) << " ";
std::cout << std::endl;
}
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
Using Zeckendorf Numbers
The nth fibonacci is represented as Zeckendorf 1 followed by n-1 zeroes. [[Zeckendorf number representation#Using a C++11 User Defined Literal|Here]] I define a class N which defines the operations increment ++() and comparison <=(other N) for Zeckendorf Numbers.
// Use Zeckendorf numbers to display Fibonacci sequence.
// Nigel Galloway October 23rd., 2012
int main(void) {
char NG[22] = {'1',0};
int x = -1;
N G;
for (int fibs = 1; fibs <= 20; fibs++) {
for (;G <= N(NG); ++G) x++;
NG[fibs] = '0';
NG[fibs+1] = 0;
std::cout << x << " ";
}
std::cout << std::endl;
return 0;
}
1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946
Using Standard Template Library
Possibly less "Far-fetched version".
// Use Standard Template Library to display Fibonacci sequence.
// Nigel Galloway March 30th., 2013
#include <algorithm>
#include <iostream>
#include <iterator>
int main()
{
int x = 1, y = 1;
generate_n(std::ostream_iterator<int>(std::cout, " "), 21, [&]{int n=x; x=y; y+=n; return n;});
return 0;
}
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946
C#
Recursive
public static ulong Fib(uint n) {
return (n < 2)? n : Fib(n - 1) + Fib(n - 2);
}
=== Tail-Recursive ===
public static ulong Fib(uint n) {
return Fib(0, 1, n);
}
private static ulong Fib(ulong a, ulong b, uint n) {
return (n < 1)? a :(n == 1)? b : Fib(b, a + b, n - 1);
}
Iterative
public static ulong Fib(uint x) {
if (x == 0) return 0;
ulong prev = 0;
ulong next = 1;
for (int i = 1; i < x; i++)
{
ulong sum = prev + next;
prev = next;
next = sum;
}
return next;
}
=== Eager-Generative ===
public static IEnumerable<long> Fibs(uint x) {
IList<ulong> fibs = new List<ulong>();
ulong prev = -1;
ulong next = 1;
for (int i = 0; i < x; i++)
{
long sum = prev + next;
prev = next;
next = sum;
fibs.Add(sum);
}
return fibs;
}
=== Lazy-Generative ===
public static IEnumerable<ulong> Fibs(uint x) {
ulong prev = -1;
ulong next = 1;
for (uint i = 0; i < x; i++) {
ulong sum = prev + next;
prev = next;
next = sum;
yield return sum;
}
}
Analytic
Only works to the 92th fibonacci number.
private static double Phi = ((1d + Math.Sqrt(5d))/2d);
private static double D = 1d/Math.Sqrt(5d);
ulong Fib(uint n) {
if(n > 92) throw new ArgumentOutOfRangeException("n", n, "Needs to be smaller than 93.");
return (ulong)((Phi^n) - (1d - Phi)^n))*D);
}
Matrix
Algorithm is based on :.
Needs System.Windows.Media.Matrix or similar Matrix class.
Calculates in .
public static ulong Fib(uint n) {
var M = new Matrix(1,0,0,1);
var N = new Matrix(1,1,1,0);
for (uint i = 1; i < n; i++) M *= N;
return (ulong)M[0][0];
}
Needs System.Windows.Media.Matrix or similar Matrix class.
Calculates in .
private static Matrix M;
private static readonly Matrix N = new Matrix(1,1,1,0);
public static ulong Fib(uint n) {
M = new Matrix(1,0,0,1);
MatrixPow(n-1);
return (ulong)M[0][0];
}
private static void MatrixPow(double n){
if (n > 1) {
MatrixPow(n/2);
M *= M;
}
if (n % 2 == 0) M *= N;
}
=== Array (Table) Lookup ===
private static int[] fibs = new int[]{ -1836311903, 1134903170,
-701408733, 433494437, -267914296, 165580141, -102334155,
63245986, -39088169, 24157817, -14930352, 9227465, -5702887,
3524578, -2178309, 1346269, -832040, 514229, -317811, 196418,
-121393, 75025, -46368, 28657, -17711, 10946, -6765, 4181,
-2584, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13,
-8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,
28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040,
1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817,
39088169, 63245986, 102334155, 165580141, 267914296, 433494437,
701408733, 1134903170, 1836311903};
public static int Fib(int n) {
if(n < -46 || n > 46) throw new ArgumentOutOfRangeException("n", n, "Has to be between -46 and 47.")
return fibs[n+46];
}
Arbitrary Precision
This large step recurrence routine can calculate the two millionth Fibonacci number in under 1 / 5 second at tio.run. This routine can generate the fifty millionth Fibonacci number in under 30 seconds at tio.run. The unused conventional iterative method times out at two million on tio.run, you can only go to around 1,290,000 or so to keep the calculation time (plus string conversion time) under the 60 second timeout limit there. When using this large step recurrence method, it takes around 5 seconds to convert the two millionth Fibonacci number (417975 digits) into a string (so that one may count those digits).
using System;
using System.Collections.Generic;
using System.Numerics;
static class QuikFib
{
// A sparse array of values calculated along the way
private static SortedList<int, BigInteger> sl = new SortedList<int, BigInteger>();
// Square a BigInteger
public static BigInteger sqr(BigInteger n)
{
return n * n;
}
// Helper routine for Fsl(). It adds an entry to the sorted list when necessary
public static void IfNec(int n)
{
if (!sl.ContainsKey(n)) sl.Add(n, Fsl(n));
}
// This routine is semi-recursive, but doesn't need to evaluate every number up to n.
// Algorithm from here: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html#section3
public static BigInteger Fsl(int n)
{
if (n < 2) return n;
int n2 = n >> 1, pm = n2 + ((n & 1) << 1) - 1; IfNec(n2); IfNec(pm);
return n2 > pm ? (2 * sl[pm] + sl[n2]) * sl[n2] : sqr(sl[n2]) + sqr(sl[pm]);
}
// Conventional iteration method (not used here)
public static BigInteger Fm(BigInteger n)
{
if (n < 2) return n; BigInteger cur = 0, pre = 1;
for (int i = 0; i <= n - 1; i++) { BigInteger sum = cur + pre; pre = cur; cur = sum; }
return cur;
}
public static void Main()
{
int num = 2_000_000;
DateTime st = DateTime.Now;
BigInteger v = Fsl(num);
Console.WriteLine("{0:n3} ms to calculate the {1:n0}th Fibonacci number,",
(DateTime.Now - st).TotalMilliseconds, num);
st = DateTime.Now;
string vs = v.ToString();
Console.WriteLine("{0:n3} seconds to convert to a string.", (DateTime.Now - st).TotalSeconds);
Console.WriteLine("number of digits is {0}", vs.Length);
if (vs.Length < 10000)
{
st = DateTime.Now;
Console.WriteLine(vs);
Console.WriteLine("{0:n3} ms to write it to the console.", (DateTime.Now - st).TotalMilliseconds);
}
else
Console.WriteLine("partial: {0}...{1}", vs.Substring(1, 35), vs.Substring(vs.Length - 35));
}
}
179.978 ms to calculate the 2,000,000th Fibonacci number,
4.728 seconds to convert to a string.
number of digits is 417975
partial: 53129491750764154305166065450382516...91799493108960825129188777803453125
Cat
define fib {
dup 1 <=
[]
[dup 1 - fib swap 2 - fib +]
if
}
Chapel
iter fib() {
var a = 0, b = 1;
while true {
yield a;
(a, b) = (b, b + a);
}
}
Chef
Stir-Fried Fibonacci Sequence.
An unobfuscated iterative implementation.
It prints the first N + 1 Fibonacci numbers,
where N is taken from standard input.
Ingredients.
0 g last
1 g this
0 g new
0 g input
Method.
Take input from refrigerator.
Put this into 4th mixing bowl.
Loop the input.
Clean the 3rd mixing bowl.
Put last into 3rd mixing bowl.
Add this into 3rd mixing bowl.
Fold new into 3rd mixing bowl.
Clean the 1st mixing bowl.
Put this into 1st mixing bowl.
Fold last into 1st mixing bowl.
Clean the 2nd mixing bowl.
Put new into 2nd mixing bowl.
Fold this into 2nd mixing bowl.
Put new into 4th mixing bowl.
Endloop input until looped.
Pour contents of the 4th mixing bowl into baking dish.
Serves 1.
CMake
Iteration uses a while() loop. Memoization uses global properties.
set_property(GLOBAL PROPERTY fibonacci_0 0)
set_property(GLOBAL PROPERTY fibonacci_1 1)
set_property(GLOBAL PROPERTY fibonacci_next 2)
# var = nth number in Fibonacci sequence.
function(fibonacci var n)
# If the sequence is too short, compute more Fibonacci numbers.
get_property(next GLOBAL PROPERTY fibonacci_next)
if(NOT next GREATER ${n})
# a, b = last 2 Fibonacci numbers
math(EXPR i "${next} - 2")
get_property(a GLOBAL PROPERTY fibonacci_${i})
math(EXPR i "${next} - 1")
get_property(b GLOBAL PROPERTY fibonacci_${i})
while(NOT next GREATER ${n})
math(EXPR i "${a} + ${b}") # i = next Fibonacci number
set_property(GLOBAL PROPERTY fibonacci_${next} ${i})
set(a ${b})
set(b ${i})
math(EXPR next "${next} + 1")
endwhile()
set_property(GLOBAL PROPERTY fibonacci_next ${next})
endif()
get_property(answer GLOBAL PROPERTY fibonacci_${n})
set(${var} ${answer} PARENT_SCOPE)
endfunction(fibonacci)
# Test program: print 0th to 9th and 25th to 30th Fibonacci numbers.
set(s "")
foreach(i RANGE 0 9)
fibonacci(f ${i})
set(s "${s} ${f}")
endforeach(i)
set(s "${s} ... ")
foreach(i RANGE 25 30)
fibonacci(f ${i})
set(s "${s} ${f}")
endforeach(i)
message(${s})
0 1 1 2 3 5 8 13 21 34 ... 75025 121393 196418 317811 514229 832040
Clio
Clio is pure and functions are lazy and memoized by default
fn fib n:
if n < 2: n
else: (n - 1 -> fib) + (n - 2 -> fib)
[0:100] -> * fib -> * print
Clojure
Lazy Sequence
This is implemented idiomatically as an infinitely long, lazy sequence of all Fibonacci numbers:
(defn fibs []
(map first (iterate (fn [[a b]] [b (+ a b)]) [0 1])))
Thus to get the nth one:
(nth (fibs) 5)
So long as one does not hold onto the head of the sequence, this is unconstrained by length.
The one-line implementation may look confusing at first, but on pulling it apart it actually solves the problem more "directly" than a more explicit looping construct.
(defn fibs []
(map first ;; throw away the "metadata" (see below) to view just the fib numbers
(iterate ;; create an infinite sequence of [prev, curr] pairs
(fn [[a b]] ;; to produce the next pair, call this function on the current pair
[b (+ a b)]) ;; new prev is old curr, new curr is sum of both previous numbers
[0 1]))) ;; recursive base case: prev 0, curr 1
A more elegant solution is inspired by the Haskell implementation of an infinite list of Fibonacci numbers:
(def fib (lazy-cat [0 1] (map + fib (rest fib))))
Then, to see the first ten,
(take 10 fib)
(0 1 1 2 3 5 8 13 21 34)
Iterative
Here's a simple interative process (using a recursive function) that carries state along with it (as args) until it reaches a solution:
;; max is which fib number you'd like computed (0th, 1st, 2nd, etc.)
;; n is which fib number you're on for this call (0th, 1st, 2nd, etc.)
;; j is the nth fib number (ex. when n = 5, j = 5)
;; i is the nth - 1 fib number
(defn- fib-iter
[max n i j]
(if (= n max)
j
(recur max
(inc n)
j
(+ i j))))
(defn fib
[max]
(if (< max 2)
max
(fib-iter max 1 0N 1N)))
"defn-" means that the function is private (for use only inside this library). The "N" suffixes on integers tell Clojure to use arbitrary precision ints for those.
===Doubling Algorithm (Fast)=== Based upon the doubling algorithm which computes in O(log (n)) time as described here https://www.nayuki.io/page/fast-fibonacci-algorithms Implementation credit: https://stackoverflow.com/questions/27466311/how-to-implement-this-fast-doubling-fibonacci-algorithm-in-clojure/27466408#27466408
(defn fib [n]
(letfn [(fib* [n]
(if (zero? n)
[0 1]
(let [[a b] (fib* (quot n 2))
c (*' a (-' (*' 2 b) a))
d (+' (*' b b) (*' a a))]
(if (even? n)
[c d]
[d (+' c d)]))))]
(first (fib* n))))
Recursive
A naive slow recursive solution:
(defn fib [n]
(case n
0 0
1 1
(+ (fib (- n 1))
(fib (- n 2)))))
This can be improved to an O(n) solution, like the iterative solution, by memoizing the function so that numbers that have been computed are cached. Like a lazy sequence, this also has the advantage that subsequent calls to the function use previously cached results rather than recalculating.
(def fib
(memoize
(fn [n]
(case n
0 0
1 1
(+ (fib (- n 1))
(fib (- n 2)))))))
Using core.async
(ns fib.core)
(require '[clojure.core.async
:refer [<! >! >!! <!! timeout chan alt! go]])
(defn fib [c]
(loop [a 0 b 1]
(>!! c a)
(recur b (+ a b))))
(defn -main []
(let [c (chan)]
(go (fib c))
(dorun
(for [i (range 10)]
(println (<!! c))))))
COBOL
Iterative
Program-ID. Fibonacci-Sequence.
Data Division.
Working-Storage Section.
01 FIBONACCI-PROCESSING.
05 FIBONACCI-NUMBER PIC 9(36) VALUE 0.
05 FIB-ONE PIC 9(36) VALUE 0.
05 FIB-TWO PIC 9(36) VALUE 1.
01 DESIRED-COUNT PIC 9(4).
01 FORMATTING.
05 INTERM-RESULT PIC Z(35)9.
05 FORMATTED-RESULT PIC X(36).
05 FORMATTED-SPACE PIC x(35).
Procedure Division.
000-START-PROGRAM.
Display "What place of the Fibonacci Sequence would you like (<173)? " with no advancing.
Accept DESIRED-COUNT.
If DESIRED-COUNT is less than 1
Stop run.
If DESIRED-COUNT is less than 2
Move FIBONACCI-NUMBER to INTERM-RESULT
Move INTERM-RESULT to FORMATTED-RESULT
Unstring FORMATTED-RESULT delimited by all spaces into FORMATTED-SPACE,FORMATTED-RESULT
Display FORMATTED-RESULT
Stop run.
Subtract 1 from DESIRED-COUNT.
Move FIBONACCI-NUMBER to INTERM-RESULT.
Move INTERM-RESULT to FORMATTED-RESULT.
Unstring FORMATTED-RESULT delimited by all spaces into FORMATTED-SPACE,FORMATTED-RESULT.
Display FORMATTED-RESULT.
Perform 100-COMPUTE-FIBONACCI until DESIRED-COUNT = zero.
Stop run.
100-COMPUTE-FIBONACCI.
Compute FIBONACCI-NUMBER = FIB-ONE + FIB-TWO.
Move FIB-TWO to FIB-ONE.
Move FIBONACCI-NUMBER to FIB-TWO.
Subtract 1 from DESIRED-COUNT.
Move FIBONACCI-NUMBER to INTERM-RESULT.
Move INTERM-RESULT to FORMATTED-RESULT.
Unstring FORMATTED-RESULT delimited by all spaces into FORMATTED-SPACE,FORMATTED-RESULT.
Display FORMATTED-RESULT.
Recursive
SOURCE FREE
IDENTIFICATION DIVISION.
PROGRAM-ID. fibonacci-main.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 num PIC 9(6) COMP.
01 fib-num PIC 9(6) COMP.
PROCEDURE DIVISION.
ACCEPT num
CALL "fibonacci" USING CONTENT num RETURNING fib-num
DISPLAY fib-num
.
END PROGRAM fibonacci-main.
IDENTIFICATION DIVISION.
PROGRAM-ID. fibonacci RECURSIVE.
DATA DIVISION.
LOCAL-STORAGE SECTION.
01 1-before PIC 9(6) COMP.
01 2-before PIC 9(6) COMP.
LINKAGE SECTION.
01 num PIC 9(6) COMP.
01 fib-num PIC 9(6) COMP BASED.
PROCEDURE DIVISION USING num RETURNING fib-num.
ALLOCATE fib-num
EVALUATE num
WHEN 0
MOVE 0 TO fib-num
WHEN 1
MOVE 1 TO fib-num
WHEN OTHER
SUBTRACT 1 FROM num
CALL "fibonacci" USING CONTENT num RETURNING 1-before
SUBTRACT 1 FROM num
CALL "fibonacci" USING CONTENT num RETURNING 2-before
ADD 1-before TO 2-before GIVING fib-num
END-EVALUATE
.
END PROGRAM fibonacci.
CoffeeScript
Analytic
fib_ana = (n) ->
sqrt = Math.sqrt
phi = ((1 + sqrt(5))/2)
Math.round((Math.pow(phi, n)/sqrt(5)))
Iterative
fib_iter = (n) ->
return n if n < 2
[prev, curr] = [0, 1]
[prev, curr] = [curr, curr + prev] for i in [1..n]
curr
Recursive
fib_rec = (n) ->
if n < 2 then n else fib_rec(n-1) + fib_rec(n-2)
Comefrom0x10
Recursion is is not possible in Comefrom0x10.
Iterative
stop = 6
a = 1
i = 1 # start
a # print result
fib
comefrom if i is 1 # start
b = 1
comefrom fib # start of loop
i = i + 1
next_b = a + b
a = b
b = next_b
comefrom fib if i > stop
Common Lisp
Note that Common Lisp uses bignums, so this will never overflow.
Iterative
(defun fibonacci-iterative (n &aux (f0 0) (f1 1))
(case n
(0 f0)
(1 f1)
(t (loop for n from 2 to n
for a = f0 then b and b = f1 then result
for result = (+ a b)
finally (return result)))))
Simpler one:
(defun fibonacci (n)
(let ((a 0) (b 1) (c n))
(loop for i from 2 to n do
(setq c (+ a b)
a b
b c))
c))
Not a function, just printing out the entire (for some definition of "entire") sequence with a for var = loop:
(loop for x = 0 then y and y = 1 then (+ x y) do (print x))
Recursive
(defun fibonacci-recursive (n)
(if (< n 2)
n
(+ (fibonacci-recursive (- n 2)) (fibonacci-recursive (- n 1)))))
(defun fibonacci-tail-recursive ( n &optional (a 0) (b 1))
(if (= n 0)
a
(fibonacci-tail-recursive (- n 1) b (+ a b))))
Tail recursive and squaring:
(defun fib (n &optional (a 1) (b 0) (p 0) (q 1))
(if (= n 1) (+ (* b p) (* a q))
(fib (ash n -1)
(if (evenp n) a (+ (* b q) (* a (+ p q))))
(if (evenp n) b (+ (* b p) (* a q)))
(+ (* p p) (* q q))
(+ (* q q) (* 2 p q))))) ;p is Fib(2^n-1), q is Fib(2^n).
(print (fib 100000))
Alternate solution
I use [https://franz.com/downloads/clp/survey Allegro CL 10.1]
;; Project : Fibonacci sequence
(defun fibonacci (nr)
(cond ((= nr 0) 1)
((= nr 1) 1)
(t (+ (fibonacci (- nr 1))
(fibonacci (- nr 2))))))
(format t "~a" "First 10 Fibonacci numbers")
(dotimes (n 10)
(if (< n 1) (terpri))
(if (< n 9) (format t "~a" " "))
(write(+ n 1)) (format t "~a" ": ")
(write (fibonacci n)) (terpri))
Output:
First 10 Fibonacci numbers
1: 1
2: 1
3: 2
4: 3
5: 5
6: 8
7: 13
8: 21
9: 34
10: 55
Solution with methods and eql specializers
(defmethod fib (n)
(declare ((integer 0 *) n))
(+ (fib (- n 1))
(fib (- n 2))))
(defmethod fib ((n (eql 0))) 0)
(defmethod fib ((n (eql 1))) 1)
Computer/zero Assembly
To find the th Fibonacci number, set the initial value of count equal to –2 and run the program. The machine will halt with the answer stored in the accumulator. Since Computer/zero's word length is only eight bits, the program will not work with values of greater than 13.
loop: LDA y ; higher No.
STA temp
ADD x ; lower No.
STA y
LDA temp
STA x
LDA count
SUB one
BRZ done
STA count
JMP loop
done: LDA y
STP
one: 1
count: 8 ; n = 10
x: 1
y: 1
temp: 0
D
Here are four versions of Fibonacci Number calculating functions. ''FibD'' has an argument limit of magnitude 84 due to floating point precision, the others have a limit of 92 due to overflow (long).The traditional recursive version is inefficient. It is optimized by supplying a static storage to store intermediate results. A Fibonacci Number generating function is added. All functions have support for negative arguments.
import std.stdio, std.conv, std.algorithm, std.math;
long sgn(alias unsignedFib)(int n) { // break sign manipulation apart
immutable uint m = (n >= 0) ? n : -n;
if (n < 0 && (n % 2 == 0))
return -unsignedFib(m);
else
return unsignedFib(m);
}
long fibD(uint m) { // Direct Calculation, correct for abs(m) <= 84
enum sqrt5r = 1.0L / sqrt(5.0L); // 1 / sqrt(5)
enum golden = (1.0L + sqrt(5.0L)) / 2.0L; // (1 + sqrt(5)) / 2
return roundTo!long(pow(golden, m) * sqrt5r);
}
long fibI(in uint m) pure nothrow { // Iterative
long thisFib = 0;
long nextFib = 1;
foreach (i; 0 .. m) {
long tmp = nextFib;
nextFib += thisFib;
thisFib = tmp;
}
return thisFib;
}
long fibR(uint m) { // Recursive
return (m < 2) ? m : fibR(m - 1) + fibR(m - 2);
}
long fibM(uint m) { // memoized Recursive
static long[] fib = [0, 1];
while (m >= fib.length )
fib ~= fibM(m - 2) + fibM(m - 1);
return fib[m];
}
alias sgn!fibD sfibD;
alias sgn!fibI sfibI;
alias sgn!fibR sfibR;
alias sgn!fibM sfibM;
auto fibG(in int m) { // generator(?)
immutable int sign = (m < 0) ? -1 : 1;
long yield;
return new class {
final int opApply(int delegate(ref int, ref long) dg) {
int idx = -sign; // prepare for pre-increment
foreach (f; this)
if (dg(idx += sign, f))
break;
return 0;
}
final int opApply(int delegate(ref long) dg) {
long f0, f1 = 1;
foreach (p; 0 .. m * sign + 1) {
if (sign == -1 && (p % 2 == 0))
yield = -f0;
else
yield = f0;
if (dg(yield)) break;
auto temp = f1;
f1 = f0 + f1;
f0 = temp;
}
return 0;
}
};
}
void main(in string[] args) {
int k = args.length > 1 ? to!int(args[1]) : 10;
writefln("Fib(%3d) = ", k);
writefln("D : %20d <- %20d + %20d",
sfibD(k), sfibD(k - 1), sfibD(k - 2));
writefln("I : %20d <- %20d + %20d",
sfibI(k), sfibI(k - 1), sfibI(k - 2));
if (abs(k) < 36 || args.length > 2)
// set a limit for recursive version
writefln("R : %20d <- %20d + %20d",
sfibR(k), sfibM(k - 1), sfibM(k - 2));
writefln("O : %20d <- %20d + %20d",
sfibM(k), sfibM(k - 1), sfibM(k - 2));
foreach (i, f; fibG(-9))
writef("%d:%d | ", i, f);
}
{{out}} for n = 85:
Fib( 85) =
D : 259695496911122586 <- 160500643816367088 + 99194853094755497
I : 259695496911122585 <- 160500643816367088 + 99194853094755497
O : 259695496911122585 <- 160500643816367088 + 99194853094755497
0:0 | -1:1 | -2:-1 | -3:2 | -4:-3 | -5:5 | -6:-8 | -7:13 | -8:-21 | -9:34 |
Matrix Exponentiation Version
import std.bigint;
T fibonacciMatrix(T=BigInt)(size_t n) {
int[size_t.sizeof * 8] binDigits;
size_t nBinDigits;
while (n > 0) {
binDigits[nBinDigits] = n % 2;
n /= 2;
nBinDigits++;
}
T x=1, y, z=1;
foreach_reverse (b; binDigits[0 .. nBinDigits]) {
if (b) {
x = (x + z) * y;
y = y ^^ 2 + z ^^ 2;
} else {
auto x_old = x;
x = x ^^ 2 + y ^^ 2;
y = (x_old + z) * y;
}
z = x + y;
}
return y;
}
void main() {
10_000_000.fibonacciMatrix;
}
Faster Version
For N = 10_000_000 this is about twice faster (run-time about 2.20 seconds) than the matrix exponentiation version.
import std.bigint, std.math;
// Algorithm from: Takahashi, Daisuke,
// "A fast algorithm for computing large Fibonacci numbers".
// Information Processing Letters 75.6 (30 November 2000): 243-246.
// Implementation from:
// pythonista.wordpress.com/2008/07/03/pure-python-fibonacci-numbers
BigInt fibonacci(in ulong n)
in {
assert(n > 0, "fibonacci(n): n must be > 0.");
} body {
if (n <= 2)
return 1.BigInt;
BigInt F = 1;
BigInt L = 1;
int sign = -1;
immutable uint n2 = cast(uint)n.log2.floor;
auto mask = 2.BigInt ^^ (n2 - 1);
foreach (immutable i; 1 .. n2) {
auto temp = F ^^ 2;
F = (F + L) / 2;
F = 2 * F ^^ 2 - 3 * temp - 2 * sign;
L = 5 * temp + 2 * sign;
sign = 1;
if (n & mask) {
temp = F;
F = (F + L) / 2;
L = F + 2 * temp;
sign = -1;
}
mask /= 2;
}
if ((n & mask) == 0) {
F *= L;
} else {
F = (F + L) / 2;
F = F * L - sign;
}
return F;
}
void main() {
10_000_000.fibonacci;
}
Dart
int fib(int n) {
if (n==0 || n==1) {
return n;
}
var prev=1;
var current=1;
for (var i=2; i<n; i++) {
var next = prev + current;
prev = current;
current = next;
}
return current;
}
int fibRec(int n) => n==0 || n==1 ? n : fibRec(n-1) + fibRec(n-2);
main() {
print(fib(11));
print(fibRec(11));
}
Dc
This needs a modern Dc with r (swap) and # (comment).
It easily can be adapted to an older Dc, but it will impact readability a lot.
[ # todo: n(<2) -- 1 and break 2 levels
d - # 0
1 + # 1
q
] s1
[ # todo: n(>-1) -- F(n)
d 0=1 # n(!=0)
d 1=1 # n(!in {0,1})
2 - d 1 + # (n-2) (n-1)
lF x # (n-2) F(n-1)
r # F(n-1) (n-2)
lF x # F(n-1)+F(n-2)
+
] sF
33 lF x f
5702887
Delphi
Iterative
function FibonacciI(N: Word): UInt64;
var
Last, New: UInt64;
I: Word;
begin
if N < 2 then
Result := N
else begin
Last := 0;
Result := 1;
for I := 2 to N do
begin
New := Last + Result;
Last := Result;
Result := New;
end;
end;
end;
Recursive
function Fibonacci(N: Word): UInt64;
begin
if N < 2 then
Result := N
else
Result := Fibonacci(N - 1) + Fibonacci(N - 2);
end;
Matrix
Algorithm is based on :.
function fib(n: Int64): Int64;
type TFibMat = array[0..1] of array[0..1] of Int64;
function FibMatMul(a,b: TFibMat): TFibMat;
var i,j,k: integer;
tmp: TFibMat;
begin
for i := 0 to 1 do
for j := 0 to 1 do
begin
tmp[i,j] := 0;
for k := 0 to 1 do tmp[i,j] := tmp[i,j] + a[i,k] * b[k,j];
end;
FibMatMul := tmp;
end;
function FibMatExp(a: TFibMat; n: Int64): TFibmat;
begin
if n <= 1 then fibmatexp := a
else if (n mod 2 = 0) then FibMatExp := FibMatExp(FibMatMul(a,a), n div 2)
else if (n mod 2 = 1) then FibMatExp := FibMatMul(a, FibMatExp(FibMatMul(a,a), n div 2));
end;
var
matrix: TFibMat;
begin
matrix[0,0] := 1;
matrix[0,1] := 1;
matrix[1,0] := 1;
matrix[1,1] := 0;
if n > 1 then
matrix := fibmatexp(matrix,n-1);
fib := matrix[0,0];
end;
DWScript
function fib(N : Integer) : Integer;
begin
if N < 2 then Result := 1
else Result := fib(N-2) + fib(N-1);
End;
Dyalect
func fib(n) {
if n < 2 {
return n
} else {
return fib(n - 1) + fib(n - 2)
}
}
print(fib(30))
E
def fib(n) {
var s := [0, 1]
for _ in 0..!n {
def [a, b] := s
s := [b, a+b]
}
return s[0]
}
(This version defines fib(0) = 0 because [http://www.research.att.com/~njas/sequences/A000045 OEIS A000045] does.)
EasyLang
## EchoLisp
Use '''memoization''' with the recursive version.
```scheme
(define (fib n)
(if (< n 2) n
(+ (fib (- n 2)) (fib (1- n)))))
(remember 'fib #(0 1))
(for ((i 12)) (write (fib i)))
0 1 1 2 3 5 8 13 21 34 55 89
ECL
Analytic
//Calculates Fibonacci sequence up to n steps using Binet's closed form solution
FibFunction(UNSIGNED2 n) := FUNCTION
REAL Sqrt5 := Sqrt(5);
REAL Phi := (1+Sqrt(5))/2;
REAL Phi_Inv := 1/Phi;
UNSIGNED FibValue := ROUND( ( POWER(Phi,n)-POWER(Phi_Inv,n) ) /Sqrt5);
RETURN FibValue;
END;
FibSeries(UNSIGNED2 n) := FUNCTION
Fib_Layout := RECORD
UNSIGNED5 FibNum;
UNSIGNED5 FibValue;
END;
FibSeq := DATASET(n+1,
TRANSFORM
( Fib_Layout
, SELF.FibNum := COUNTER-1
, SELF.FibValue := IF(SELF.FibNum<2,SELF.FibNum, FibFunction(SELF.FibNum) )
)
);
RETURN FibSeq;
END; }
EDSAC order code
This program calculates the nth—by default the tenth—number in the Fibonacci sequence and displays it (in binary) in the first word of storage tank 3.
[ Fibonacci sequence
### ============
A program for the EDSAC
Calculates the nth Fibonacci
number and displays it at the
top of storage tank 3
The default value of n is 10
To calculate other Fibonacci
numbers, set the starting value
of the count to n-2
Works with Initial Orders 2 ]
T56K [ set load point ]
GK [ set theta ]
[ Orders ]
[ 0 ] T20@ [ a = 0 ]
A17@ [ a += y ]
U18@ [ temp = a ]
A16@ [ a += x ]
T17@ [ y = a; a = 0 ]
A18@ [ a += temp ]
T16@ [ x = a; a = 0 ]
A19@ [ a = count ]
S15@ [ a -= 1 ]
U19@ [ count = a ]
E@ [ if a>=0 go to θ ]
T20@ [ a = 0 ]
A17@ [ a += y ]
T96F [ C(96) = a; a = 0]
ZF [ halt ]
[ Data ]
[ 15 ] P0D [ const: 1 ]
[ 16 ] P0F [ var: x = 0 ]
[ 17 ] P0D [ var: y = 1 ]
[ 18 ] P0F [ var: temp = 0 ]
[ 19 ] P4F [ var: count = 8 ]
[ 20 ] P0F [ used to clear a ]
EZPF [ begin execution ]
00000000000110111
Eiffel
class
APPLICATION
create
make
feature
fibonacci (n: INTEGER): INTEGER
require
non_negative: n >= 0
local
i, n2, n1, tmp: INTEGER
do
n2 := 0
n1 := 1
from
i := 1
until
i >= n
loop
tmp := n1
n1 := n2 + n1
n2 := tmp
i := i + 1
end
Result := n1
if n = 0 then
Result := 0
end
end
feature {NONE} -- Initialization
make
-- Run application.
do
print (fibonacci (0))
print (" ")
print (fibonacci (1))
print (" ")
print (fibonacci (2))
print (" ")
print (fibonacci (3))
print (" ")
print (fibonacci (4))
print ("%N")
end
end
Ela
Tail-recursive function:
fib = fib' 0 1
where fib' a b 0 = a
fib' a b n = fib' b (a + b) (n - 1)
Infinite (lazy) list:
fib = fib' 1 1
where fib' x y = & x :: fib' y (x + y)
Elena
ELENA 4.1 :
import extensions;
fibu(n)
{
int[] ac := new int[]::( 0,1 );
if (n < 2)
{
^ ac[n]
}
else
{
for(int i := 2, i <= n, i+=1)
{
int t := ac[1];
ac[1] := ac[0] + ac[1];
ac[0] := t
};
^ ac[1]
}
}
public program()
{
for(int i := 0, i <= 10, i+=1)
{
console.printLine(fibu(i))
}
}
0
1
1
2
3
5
8
13
21
34
55
Elixir
defmodule Fibonacci do
def fib(0), do: 0
def fib(1), do: 1
def fib(n), do: fib(0, 1, n-2)
def fib(_, prv, -1), do: prv
def fib(prvprv, prv, n) do
next = prv + prvprv
fib(prv, next, n-1)
end
end
IO.inspect Enum.map(0..10, fn i-> Fibonacci.fib(i) end)
Using Stream:
Stream.unfold({0,1}, fn {a,b} -> {a,{b,a+b}} end) |> Enum.take(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
Elm
Naïve recursive implementation.
fibonacci : Int -> Int
fibonacci n = if n < 2 then
n
else
fibonacci(n - 2) + fibonacci(n - 1)
Emacs Lisp
version 1
(defun fib (n a b c)
(if (< c n) (fib n b (+ a b) (+ 1 c) )
(if (= c n) b a) ))
(defun fibonacci (n) (if (< n 2) n (fib n 0 1 1) ))
version 2
(defun fibonacci (n)
(let ( (vec) (i) (j) (k) )
(if (< n 2) n
(progn
(setq vec (make-vector (+ n 1) 0) i 0 j 1 k 2)
(setf (aref vec 1) 1)
(while (<= k n)
(setf (aref vec k) (+ (elt vec i) (elt vec j) ))
(setq i (+ 1 i) j (+ 1 j) k (+ 1 k) ))
(elt vec n) ))))
Eval:
(insert
(mapconcat '(lambda (n) (format "%d" (fibonacci n) ))
(number-sequence 0 15) " ") )
Output:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
Erlang
Recursive
-module(fib).
-export([fib/1]).
fib(0) -> 0;
fib(1) -> 1;
fib(N) -> fib(N-1) + fib(N-2).
Iterative
-module(fiblin).
-export([fib/1])
fib(0) -> 0;
fib(1) -> 1;
fib(2) -> 1;
fib(3) -> 2;
fib(4) -> 3;
fib(5) -> 5;
fib(N) when is_integer(N) -> fib(N - 6, 5, 8).
fib(N, A, B) -> if N < 1 -> B; true -> fib(N-1, B, A+B) end.
Evaluate:
io:write([fiblin:fib(X) || X <- lists:seq(1,10) ]).
Output:
[1,1,2,3,5,8,13,21,34,55]ok
Iterative 2
fib(N) -> fib(N, 0, 1).
fib(0, Result, _Next) -> Result;
fib(Iter, Result, Next) -> fib(Iter-1, Next, Result+Next).
ERRE
!-------------------------------------------
! derived from my book "PROGRAMMARE IN ERRE"
! iterative solution
!-------------------------------------------
PROGRAM FIBONACCI
!$DOUBLE
!VAR F1#,F2#,TEMP#,COUNT%,N%
BEGIN !main
INPUT("Number",N%)
F1=0
F2=1
REPEAT
TEMP=F2
F2=F1+F2
F1=TEMP
COUNT%=COUNT%+1
UNTIL COUNT%=N%
PRINT("FIB(";N%;")=";F2)
! Obviously a FOR loop or a WHILE loop can
! be used to solve this problem
END PROGRAM
Number? 20
FIB( 20 )= 6765
Euphoria
==='Recursive' version===
function fibor(integer n)
if n<2 then return n end if
return fibor(n-1)+fibor(n-2)
end function
==='Iterative' version===
function fiboi(integer n)
integer f0=0, f1=1, f
if n<2 then return n end if
for i=2 to n do
f=f0+f1
f0=f1
f1=f
end for
return f
end function
==='Tail recursive' version===
function fibot(integer n, integer u = 1, integer s = 0)
if n < 1 then
return s
else
return fibot(n-1,u+s,u)
end if
end function
-- example:
? fibot(10) -- says 55
==='Paper tape' version===
include std/mathcons.e -- for PINF constant
enum ADD, MOVE, GOTO, OUT, TEST, TRUETO
global sequence tape = { 0,
1,
{ ADD, 2, 1 },
{ TEST, 1, PINF },
{ TRUETO, 0 },
{ OUT, 1, "%.0f\n" },
{ MOVE, 2, 1 },
{ MOVE, 0, 2 },
{ GOTO, 3 } }
global integer ip
global integer test
global atom accum
procedure eval( sequence cmd )
atom i = 1
while i <= length( cmd ) do
switch cmd[ i ] do
case ADD then
accum = tape[ cmd[ i + 1 ] ] + tape[ cmd[ i + 2 ] ]
i += 2
case OUT then
printf( 1, cmd[ i + 2], tape[ cmd[ i + 1 ] ] )
i += 2
case MOVE then
if cmd[ i + 1 ] = 0 then
tape[ cmd[ i + 2 ] ] = accum
else
tape[ cmd[ i + 2 ] ] = tape[ cmd[ i + 1 ] ]
end if
i += 2
case GOTO then
ip = cmd[ i + 1 ] - 1 -- due to ip += 1 in main loop
i += 1
case TEST then
if tape[ cmd[ i + 1 ] ] = cmd[ i + 2 ] then
test = 1
else
test = 0
end if
i += 2
case TRUETO then
if test then
if cmd[ i + 1 ] = 0 then
abort(0)
else
ip = cmd[ i + 1 ] - 1
end if
end if
end switch
i += 1
end while
end procedure
test = 0
accum = 0
ip = 1
while 1 do
-- embedded sequences (assumed to be code) are evaluated
-- atoms (assumed to be data) are ignored
if sequence( tape[ ip ] ) then
eval( tape[ ip ] )
end if
ip += 1
end while
FALSE
[[$0=~][1-@@\$@@+\$44,.@]#]f:
20n: {First 20 numbers}
0 1 n;f;!%%44,. {Output: "0,1,1,2,3,5..."}
Factor
Iterative
: fib ( n -- m )
dup 2 < [
[ 0 1 ] dip [ swap [ + ] keep ] times
drop
] unless ;
Recursive
: fib ( n -- m )
dup 2 < [
[ 1 - fib ] [ 2 - fib ] bi +
] unless ;
===Tail-Recursive===
: fib2 ( x y n -- a )
dup 1 <
[ 2drop ]
[ [ swap [ + ] keep ] dip 1 - fib2 ]
if ;
: fib ( n -- m ) [ 0 1 ] dip fib2 ;
Matrix
USE: math.matrices
: fib ( n -- m )
dup 2 < [
[ { { 0 1 } { 1 1 } } ] dip 1 - m^n
second second
] unless ;
Fancy
class Fixnum {
def fib {
match self -> {
case 0 -> 0
case 1 -> 1
case _ -> self - 1 fib + (self - 2 fib)
}
}
}
15 times: |x| {
x fib println
}
Falcon
Iterative
function fib_i(n)
if n < 2: return n
fibPrev = 1
fib = 1
for i in [2:n]
tmp = fib
fib += fibPrev
fibPrev = tmp
end
return fib
end
Recursive
function fib_r(n)
if n < 2 : return n
return fib_r(n-1) + fib_r(n-2)
end
Tail Recursive
function fib_tr(n)
return fib_aux(n,0,1)
end
function fib_aux(n,a,b)
switch n
case 0 : return a
default: return fib_aux(n-1,a+b,a)
end
end
Fantom
Ints have a limit of 64-bits, so overflow errors occur after computing Fib(92) = 7540113804746346429.
class Main
{
static Int fib (Int n)
{
if (n < 2) return n
fibNums := [1, 0]
while (fibNums.size <= n)
{
fibNums.insert (0, fibNums[0] + fibNums[1])
}
return fibNums.first
}
public static Void main ()
{
20.times |n|
{
echo ("Fib($n) is ${fib(n)}")
}
}
}
Fexl
# (fib n) = the nth Fibonacci number
\fib=
(
\loop==
(\x\y\n
le n 0 x;
\z=(+ x y)
\n=(- n 1)
loop y z n
)
loop 0 1
)
# Now test it:
for 0 20 (\n say (fib n))
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
FOCAL
01.10 TYPE "FIBONACCI NUMBERS" !
01.20 ASK "N =", N
01.30 SET A=0
01.40 SET B=1
01.50 FOR I=2,N; DO 2.0
01.60 TYPE "F(N) ", %8, B, !
01.70 QUIT
02.10 SET T=B
02.20 SET B=A+B
02.30 SET A=T
FIBONACCI NUMBERS
N =:20
F(N) = 6765
Forth
: fib ( n -- fib )
0 1 rot 0 ?do over + swap loop drop ;
Since there are only a fixed and small amount of Fibonacci numbers that fit in a machine word, this FORTH version creates a table of Fibonacci numbers at compile time. It stops compiling numbers when there is arithmetic overflow (the number turns negative, indicating overflow.)
: F-start, here 1 0 dup , ;
: F-next, over + swap
dup 0> IF dup , true ELSE false THEN ;
: computed-table ( compile: 'start 'next / run: i -- x )
create
>r execute
BEGIN r@ execute not UNTIL rdrop
does>
swap cells + @ ;
' F-start, ' F-next, computed-table fibonacci 2drop
here swap - cell/ Constant #F/64 \ # of fibonacci numbers generated
16 fibonacci . 987 ok
#F/64 . 93 ok
92 fibonacci . 7540113804746346429 ok \ largest number generated.
Fortran
FORTRAN IV
C FIBONACCI SEQUENCE - FORTRAN IV
NN=46
DO 1 I=0,NN
1 WRITE(*,300) I,IFIBO(I)
300 FORMAT(1X,I2,1X,I10)
END
C
FUNCTION IFIBO(N)
IF(N) 9,1,2
1 IFN=0
GOTO 9
2 IF(N-1) 9,3,4
3 IFN=1
GOTO 9
4 IFNM1=0
IFN=1
DO 5 I=2,N
IFNM2=IFNM1
IFNM1=IFN
5 IFN=IFNM1+IFNM2
9 IFIBO=IFN
END
0 0
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
...
45 1134903170
46 1836311903
FORTRAN 77
FUNCTION IFIB(N)
IF (N.EQ.0) THEN
ITEMP0=0
ELSE IF (N.EQ.1) THEN
ITEMP0=1
ELSE IF (N.GT.1) THEN
ITEMP1=0
ITEMP0=1
DO 1 I=2,N
ITEMP2=ITEMP1
ITEMP1=ITEMP0
ITEMP0=ITEMP1+ITEMP2
1 CONTINUE
ELSE
ITEMP1=1
ITEMP0=0
DO 2 I=-1,N,-1
ITEMP2=ITEMP1
ITEMP1=ITEMP0
ITEMP0=ITEMP2-ITEMP1
2 CONTINUE
END IF
IFIB=ITEMP0
END
Test program
EXTERNAL IFIB
CHARACTER*10 LINE
PARAMETER ( LINE = '----------' )
WRITE(*,900) 'N', 'F[N]', 'F[-N]'
WRITE(*,900) LINE, LINE, LINE
DO 1 N = 0, 10
WRITE(*,901) N, IFIB(N), IFIB(-N)
1 CONTINUE
900 FORMAT(3(X,A10))
901 FORMAT(3(X,I10))
END
N F[N] F[-N]
---------- ---------- ----------
0 0 0
1 1 1
2 1 -1
3 2 2
4 3 -3
5 5 5
6 8 -8
7 13 13
8 21 -21
9 34 34
10 55 -55
Recursive
In ISO Fortran 90 or later, use a RECURSIVE function:
module fibonacci
contains
recursive function fibR(n) result(fib)
integer, intent(in) :: n
integer :: fib
select case (n)
case (:0); fib = 0
case (1); fib = 1
case default; fib = fibR(n-1) + fibR(n-2)
end select
end function fibR
Iterative
In ISO Fortran 90 or later:
function fibI(n)
integer, intent(in) :: n
integer, parameter :: fib0 = 0, fib1 = 1
integer :: fibI, back1, back2, i
select case (n)
case (:0); fibI = fib0
case (1); fibI = fib1
case default
fibI = fib1
back1 = fib0
do i = 2, n
back2 = back1
back1 = fibI
fibI = back1 + back2
end do
end select
end function fibI
end module fibonacci
Test program
program fibTest
use fibonacci
do i = 0, 10
print *, fibr(i), fibi(i)
end do
end program fibTest
0 0
1 1
1 1
2 2
3 3
5 5
8 8
13 13
21 21
34 34
55 55
FreeBASIC
Extended sequence coded big integer.
'Fibonacci extended
'Freebasic version 24 Windows
Dim Shared ADDQmod(0 To 19) As Ubyte
Dim Shared ADDbool(0 To 19) As Ubyte
For z As Integer=0 To 19
ADDQmod(z)=(z Mod 10+48)
ADDbool(z)=(-(10<=z))
Next z
Function plusINT(NUM1 As String,NUM2 As String) As String
Dim As Byte flag
#macro finish()
three=Ltrim(three,"0")
If three="" Then Return "0"
If flag=1 Then Swap NUM2,NUM1
Return three
Exit Function
#endmacro
var lenf=Len(NUM1)
var lens=Len(NUM2)
If lens>lenf Then
Swap NUM2,NUM1
Swap lens,lenf
flag=1
End If
var diff=lenf-lens-Sgn(lenf-lens)
var three="0"+NUM1
var two=String(lenf-lens,"0")+NUM2
Dim As Integer n2
Dim As Ubyte addup,addcarry
addcarry=0
For n2=lenf-1 To diff Step -1
addup=two[n2]+NUM1[n2]-96
three[n2+1]=addQmod(addup+addcarry)
addcarry=addbool(addup+addcarry)
Next n2
If addcarry=0 Then
finish()
End If
If n2=-1 Then
three[0]=addcarry+48
finish()
End If
For n2=n2 To 0 Step -1
addup=two[n2]+NUM1[n2]-96
three[n2+1]=addQmod(addup+addcarry)
addcarry=addbool(addup+addcarry)
Next n2
three[0]=addcarry+48
finish()
End Function
Function fibonacci(n As Integer) As String
Dim As String sl,l,term
sl="0": l="1"
If n=1 Then Return "0"
If n=2 Then Return "1"
n=n-2
For x As Integer= 1 To n
term=plusINT(l,sl)
sl=l
l=term
Next x
Function =term
End Function
'
### =========== EXAMPLE ============
print "THE SEQUENCE TO 10:"
print
For n As Integer=1 To 10
Print "term";n;": "; fibonacci(n)
Next n
print
print "Selected Fibonacci number"
print "Fibonacci 500"
print
print fibonacci(500)
Sleep
THE SEQUENCE TO 10:
term 1: 0
term 2: 1
term 3: 1
term 4: 2
term 5: 3
term 6: 5
term 7: 8
term 8: 13
term 9: 21
term 10: 34
Selected Fibonacci number
Fibonacci 500
86168291600238450732788312165664788095941068326060883324529903470149056115823592
713458328176574447204501
Free Pascal
''See also: [[#Pascal|Pascal]]''
type
/// domain for Fibonacci function
/// where result is within nativeUInt
// You can not name it fibonacciDomain,
// since the Fibonacci function itself
// is defined for all whole numbers
// but the result beyond F(n) exceeds high(nativeUInt).
fibonacciLeftInverseRange =
{$ifdef CPU64} 0..93 {$else} 0..47 {$endif};
{**
implements Fibonacci sequence iteratively
\param n the index of the Fibonacci number to calculate
\returns the Fibonacci value at n
}
function fibonacci(const n: fibonacciLeftInverseRange): nativeUInt;
type
/// more meaningful identifiers than simple integers
relativePosition = (previous, current, next);
var
/// temporary iterator variable
i: longword;
/// holds preceding fibonacci values
f: array[relativePosition] of nativeUInt;
begin
f[previous] := 0;
f[current] := 1;
// note, in Pascal for-loop-limits are inclusive
for i := 1 to n do
begin
f[next] := f[previous] + f[current];
f[previous] := f[current];
f[current] := f[next];
end;
// assign to previous, bc f[current] = f[next] for next iteration
fibonacci := f[previous];
end;
Frink
All of Frink's integers can be arbitrarily large.
fibonacciN[n] :=
{
a = 0
b = 1
count = 0
while count < n
{
[a,b] = [b, a + b]
count = count + 1
}
return a
}
FRISC Assembly
To find the nth Fibonacci number, call this subroutine with n in register R0: the answer will be returned in R0 too. Contents of other registers are preserved.
FIBONACCI PUSH R1
PUSH R2
PUSH R3
MOVE 0, R1
MOVE 1, R2
FIB_LOOP SUB R0, 1, R0
JP_Z FIB_DONE
MOVE R2, R3
ADD R1, R2, R2
MOVE R3, R1
JP FIB_LOOP
FIB_DONE MOVE R2, R0
POP R3
POP R2
POP R1
RET
=={{header|F_Sharp|F#}}== This is a fast [tail-recursive] approach using the F# big integer support:
let fibonacci n : bigint =
let rec f a b n =
match n with
| 0 -> a
| 1 -> b
| n -> (f b (a + b) (n - 1))
f (bigint 0) (bigint 1) n
> fibonacci 100;;
val it : bigint = 354224848179261915075I
Lazy evaluated using sequence workflow:
let rec fib = seq { yield! [0;1];
for (a,b) in Seq.zip fib (Seq.skip 1 fib) -> a+b}
The above is extremely slow due to the nested recursions on sequences, which aren't very efficient at the best of times. The above takes seconds just to compute the 30th Fibonacci number!
Lazy evaluation using the sequence unfold anamorphism is much much better as to efficiency:
let fibonacci = Seq.unfold (fun (x, y) -> Some(x, (y, x + y))) (0I,1I)
fibonacci |> Seq.nth 10000
Approach similar to the Matrix algorithm in C#, with some shortcuts involved. Since it uses exponentiation by squaring, calculations of fib(n) where n is a power of 2 are particularly quick. Eg. fib(2^20) was calculated in a little over 4 seconds on this poster's laptop.
open System
open System.Diagnostics
open System.Numerics
/// Finds the highest power of two which is less than or equal to a given input.
let inline prevPowTwo (x : int) =
let mutable n = x
n <- n - 1
n <- n ||| (n >>> 1)
n <- n ||| (n >>> 2)
n <- n ||| (n >>> 4)
n <- n ||| (n >>> 8)
n <- n ||| (n >>> 16)
n <- n + 1
match x with
| x when x = n -> x
| _ -> n/2
/// Evaluates the nth Fibonacci number using matrix arithmetic and
/// exponentiation by squaring.
let crazyFib (n : int) =
let powTwo = prevPowTwo n
/// Applies 2n rule repeatedly until another application of the rule would
/// go over the target value (or the target value has been reached).
let rec iter1 i q r s =
match i with
| i when i < powTwo ->
iter1 (i*2) (q*q + r*r) (r * (q+s)) (r*r + s*s)
| _ -> i, q, r, s
/// Applies n+1 rule until the target value is reached.
let rec iter2 (i, q, r, s) =
match i with
| i when i < n ->
iter2 ((i+1), (q+r), q, r)
| _ -> q
match n with
| 0 -> 1I
| _ ->
iter1 1 1I 1I 0I
|> iter2
FunL
Recursive
def
fib( 0 ) = 0
fib( 1 ) = 1
fib( n ) = fib( n - 1 ) + fib( n - 2 )
Tail Recursive
def fib( n ) =
def
_fib( 0, prev, _ ) = prev
_fib( 1, _, next ) = next
_fib( n, prev, next ) = _fib( n - 1, next, next + prev )
_fib( n, 0, 1 )
Lazy List
val fib =
def _fib( a, b ) = a # _fib( b, a + b )
_fib( 0, 1 )
println( fib(10000) )
33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875
Iterative
def fib( n ) =
a, b = 0, 1
for i <- 1..n
a, b = b, a+b
a
=== Binet's Formula ===
import math.sqrt
def fib( n ) =
phi = (1 + sqrt( 5 ))/2
int( (phi^n - (-phi)^-n)/sqrt(5) + .5 )
Matrix Exponentiation
def mul( a, b ) =
res = array( a.length(), b(0).length() )
for i <- 0:a.length(), j <- 0:b(0).length()
res( i, j ) = sum( a(i, k)*b(k, j) | k <- 0:b.length() )
vector( res )
def
pow( _, 0 ) = ((1, 0), (0, 1))
pow( x, 1 ) = x
pow( x, n )
| 2|n = pow( mul(x, x), n\2 )
| otherwise = mul(x, pow( mul(x, x), (n - 1)\2 ) )
def fib( n ) = pow( ((0, 1), (1, 1)), n )(0, 1)
for i <- 0..10
println( fib(i) )
0
1
1
2
3
5
8
13
21
34
55
Futhark
Iterative
fun main(n: int): int =
loop((a,b) = (0,1)) = for _i < n do
(b, a + b)
in a
FutureBasic
Iterative
include "Tlbx Timer.incl"
include "ConsoleWindow"
local fn Fibonacci( n as long ) as Long
begin globals
dim as long s1, s2// static
end globals
dim as long temp
if ( n < 2 )
s1 = n
exit fn
else
temp = s1 + s2
s2 = s1
s1 = temp
exit fn
end if
end fn = s1
dim as long i
dim as UnsignedWide start, finish
Microseconds( @start )
for i = 0 to 40
print i; ". "; fn Fibonacci(i)
next i
Microseconds( @finish )
print "Compute time:"; (finish.lo - start.lo ) / 1000; " ms"
Output:
0. 0
1. 1
2. 1
3. 2
4. 3
5. 5
6. 8
7. 13
8. 21
9. 34
10. 55
11. 89
12. 144
13. 233
14. 377
15. 610
16. 987
17. 1597
18. 2584
19. 4181
20. 6765
21. 10946
22. 17711
23. 28657
24. 46368
25. 75025
26. 121393
27. 196418
28. 317811
29. 514229
30. 832040
31. 1346269
32. 2178309
33. 3524578
34. 5702887
35. 9227465
36. 14930352
37. 24157817
38. 39088169
39. 63245986
40. 102334155
41. 165580141
42. 267914296
43. 433494437
44. 701408733
45. 1134903170
46. 1836311903
47. 2971215073
48. 4.80752698e+9
49. 7.77874205e+9
50. 1.2586269e+10
51. 2.03650111e+10
52. 3.29512801e+10
53. 5.33162912e+10
54. 8.62675713e+10
55. 1.39583862e+11
56. 2.25851434e+11
57. 3.65435296e+11
58. 5.9128673e+11
59. 9.56722026e+11
60. 1.54800876e+12
61. 2.50473078e+12
62. 4.05273954e+12
63. 6.55747032e+12
64. 1.06102099e+13
65. 1.71676802e+13
66. 2.777789e+13
67. 4.49455702e+13
68. 7.27234602e+13
69. 1.1766903e+14
70. 1.90392491e+14
71. 3.08061521e+14
72. 4.98454012e+14
73. 8.06515533e+14
74. 1.30496954e+15
75. 2.11148508e+15
76. 3.41645462e+15
77. 5.5279397e+15
78. 8.94439432e+15
79. 1.4472334e+16
80. 2.34167283e+16
81. 3.78890624e+16
82. 6.13057907e+16
83. 9.91948531e+16
84. 1.60500644e+17
85. 2.59695497e+17
86. 4.20196141e+17
87. 6.79891638e+17
88. 1.10008778e+18
89. 1.77997942e+18
90. 2.88006719e+18
91. 4.66004661e+18
92. 7.5401138e+18
93. 1.22001604e+19
94. 1.97402742e+19
95. 3.19404346e+19
96. 5.16807089e+19
97. 8.36211435e+19
98. 1.35301852e+20
99. 2.18922996e+20
100. 3.54224848e+20
Compute time: 15 ms
GAP
fib := function(n)
local a;
a := [[0, 1], [1, 1]]^n;
return a[1][2];
end;
GAP has also a buit-in function for that.
Fibonacci(n);
Gecho
Prints the first several fibonacci numbers...
## GFA Basic
<lang>
'
' Compute nth Fibonacci number
'
' open a window for display
OPENW 1
CLEARW 1
' Display some fibonacci numbers
' Fib(46) is the largest number GFA Basic can reach
' (long integers are 4 bytes)
FOR i%=0 TO 46
PRINT "fib(";i%;")=";@fib(i%)
NEXT i%
' wait for a key press and tidy up
~INP(2)
CLOSEW 1
'
' Function to compute nth fibonacci number
' n must be in range 0 to 46, inclusive
'
FUNCTION fib(n%)
LOCAL n0%,n1%,nn%,i%
n0%=0
n1%=1
SELECT n%
CASE 0
RETURN n0%
CASE 1
RETURN n1%
DEFAULT
FOR i%=2 TO n%
nn%=n0%+n1%
n0%=n1%
n1%=nn%
NEXT i%
RETURN nn%
ENDSELECT
ENDFUNC
GML
///fibonacci(n)
//Returns the nth fibonacci number
var n, numb;
n = argument0;
if (n == 0)
{
numb = 0;
}
else
{
var fm2, fm1;
fm2 = 0;
fm1 = 1;
numb = 1;
repeat(n-1)
{
numb = fm2+fm1;
fm2 = fm1;
fm1 = numb;
}
}
return numb;
Go
Recursive
func fib(a int) int {
if a < 2 {
return a
}
return fib(a - 1) + fib(a - 2)
}
Iterative
import (
"math/big"
)
func fib(n uint64) *big.Int {
if n < 2 {
return big.NewInt(int64(n))
}
a, b := big.NewInt(0), big.NewInt(1)
for n--; n > 0; n-- {
a.Add(a, b)
a, b = b, a
}
return b
}
Iterative using a closure
func fibNumber() func() int {
fib1, fib2 := 0, 1
return func() int {
fib1, fib2 = fib2, fib1 + fib2
return fib1
}
}
func fibSequence(n int) int {
f := fibNumber()
fib := 0
for i := 0; i < n; i++ {
fib = f()
}
return fib
}
Using a goroutine and channel
func fib(c chan int) {
a, b := 0, 1
for {
c <- a
a, b = b, a+b
}
}
func main() {
c := make(chan int)
go fib(c)
for i := 0; i < 10; i++ {
fmt.Println(<-c)
}
}
Groovy
Full "extra credit" solutions.
Recursive
A recursive closure must be ''pre-declared''.
def rFib
rFib = {
it == 0 ? 0
: it == 1 ? 1
: it > 1 ? rFib(it-1) + rFib(it-2)
/*it < 0*/: rFib(it+2) - rFib(it+1)
}
Iterative
def iFib = {
it == 0 ? 0
: it == 1 ? 1
: it > 1 ? (2..it).inject([0,1]){i, j -> [i[1], i[0]+i[1]]}[1]
/*it < 0*/: (-1..it).inject([0,1]){i, j -> [i[1]-i[0], i[0]]}[0]
}
Analytic
final φ = (1 + 5**(1/2))/2
def aFib = { (φ**it - (-φ)**(-it))/(5**(1/2)) as BigInteger }
Test program:
def time = { Closure c ->
def start = System.currentTimeMillis()
def result = c()
def elapsedMS = (System.currentTimeMillis() - start)/1000
printf '(%6.4fs elapsed)', elapsedMS
result
}
print " F(n) elapsed time "; (-10..10).each { printf ' %3d', it }; println()
print "--------- -----------------"; (-10..10).each { print ' ---' }; println()
[recursive:rFib, iterative:iFib, analytic:aFib].each { name, fib ->
printf "%9s ", name
def fibList = time { (-10..10).collect {fib(it)} }
fibList.each { printf ' %3d', it }
println()
}
F(n) elapsed time -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
--------- ----------------- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
recursive (0.0080s elapsed) -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55
iterative (0.0040s elapsed) -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55
analytic (0.0030s elapsed) -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55
Harbour
Recursive
#include "harbour.ch"
Function fibb(a,b,n)
return(if(--n>0,fibb(b,a+b,n),a))
Iterative
#include "harbour.ch"
Function fibb(n)
local fnow:=0, fnext:=1, tempf
while (--n>0)
tempf:=fnow+fnext
fnow:=fnext
fnext:=tempf
end while
return(fnext)
Haskell
Analytic
main :: IO ()
main =
print
[ floor (0.01 + (1 / p ** n + p ** n) / sqrt 5)
| let p = (1 + sqrt 5) / 2
, n <- [0 .. 42] ]
Recursive
Simple definition, very inefficient.
fib x =
if x < 1
then 0
else if x < 2
then 1
else fib (x - 1) + fib (x - 2)
Recursive with Memoization
Very fast.
fib x =
if x < 1
then 0
else if x == 1
then 1
else fibs !! (x - 1) + fibs !! (x - 2)
where
fibs = map fib [0 ..]
Recursive with Memoization using memoized library
Even faster and simpler is to use a defined memoizer (e.g. from MemoTrie package):
import Data.MemoTrie
fib :: Integer -> Integer
fib = memo f where
f 0 = 0
f 1 = 1
f n = fib (n-1) + fib (n-2)
You can rewrite this without introducing f explicitly
import Data.MemoTrie
fib :: Integer -> Integer
fib = memo $ \x -> case x of
0 -> 0
1 -> 1
n -> fib (n-1) + fib (n-2)
Or using LambdaCase extension you can write it even shorter:
{-# Language LambdaCase #-}
import Data.MemoTrie
fib :: Integer -> Integer
fib = memo $ \case
0 -> 0
1 -> 1
n -> fib (n-1) + fib (n-2)
The version that supports negative numbers:
{-# Language LambdaCase #-}
import Data.MemoTrie
fib :: Integer -> Integer
fib = memo $ \case
0 -> 0
1 -> 1
n | n>0 -> fib (n-1) + fib (n-2)
| otherwise -> fib (n+2) - fib (n+1)
Iterative
fib n = go n 0 1
where
go n a b
| n == 0 = a
| otherwise = go (n - 1) b (a + b)
= With lazy lists =
This is a standard example how to use lazy lists. Here's the (infinite) list of all Fibonacci numbers:
fib = 0 : 1 : zipWith (+) fib (tail fib)
Or alternatively:
fib = 0 : 1 : (zipWith (+) <*> tail) fib
The ''n''th Fibonacci number is then just fib !! n. The above is equivalent to
fib = 0 : 1 : next fib where next (a: t@(b:_)) = (a+b) : next t
Also
fib = 0 : scanl (+) 1 fib
= As a fold =
Accumulator holds last two members of the series:
import Data.List (foldl') --'
fib :: Integer -> Integer
fib n =
fst $
foldl' --'
(\(a, b) _ -> (b, a + b))
(0, 1)
[1 .. n]
With matrix exponentiation
Adapting the (rather slow) code from [[Matrix exponentiation operator]], we can simply write:
import Data.List (transpose)
fib
:: (Integral b, Num a)
=> b -> a
fib 0 = 0 -- this line is necessary because "something ^ 0" returns "fromInteger 1", which unfortunately
-- in our case is not our multiplicative identity (the identity matrix) but just a 1x1 matrix of 1
fib n = (last . head . unMat) (Mat [[1, 1], [1, 0]] ^ n)
-- Code adapted from Matrix exponentiation operator task ---------------------
(<+>)
:: Num c
=> [c] -> [c] -> [c]
(<+>) = zipWith (+)
(<*>)
:: Num a
=> [a] -> [a] -> a
(<*>) = (sum .) . zipWith (*)
newtype Mat a = Mat
{ unMat :: [[a]]
} deriving (Eq)
instance Show a =>
Show (Mat a) where
show xm = "Mat " ++ show (unMat xm)
instance Num a =>
Num (Mat a) where
negate xm = Mat $ map (map negate) $ unMat xm
xm + ym = Mat $ zipWith (<+>) (unMat xm) (unMat ym)
xm * ym =
Mat
[ [ xs Main.<*> ys -- to distinguish from standard applicative operator
| ys <- transpose $ unMat ym ]
| xs <- unMat xm ]
fromInteger n = Mat [[fromInteger n]]
abs = undefined
signum = undefined
-- TEST ----------------------------------------------------------------------
main :: IO ()
main = (print . take 10 . show . fib) (10 ^ 5)
So, for example, the hundred-thousandth Fibonacci number starts with the digits:
"2597406934"
With recurrence relations
Using Fib[m=3n+r] [http://en.wikipedia.org/wiki/Fibonacci_number#Other_identities recurrence identities]:
import Control.Arrow ((&&&))
fibstep :: (Integer, Integer) -> (Integer, Integer)
fibstep (a, b) = (b, a + b)
fibnums :: [Integer]
fibnums = map fst $ iterate fibstep (0, 1)
fibN2 :: Integer -> (Integer, Integer)
fibN2 m
| m < 10 = iterate fibstep (0, 1) !! fromIntegral m
fibN2 m = fibN2_next (n, r) (fibN2 n)
where
(n, r) = quotRem m 3
fibN2_next (n, r) (f, g)
| r == 0 = (a, b) -- 3n ,3n+1
| r == 1 = (b, c) -- 3n+1,3n+2
| r == 2 = (c, d) -- 3n+2,3n+3 (*)
where
a =
5 * f ^ 3 +
if even n
then 3 * f
else (-3 * f) -- 3n
b = g ^ 3 + 3 * g * f ^ 2 - f ^ 3 -- 3n+1
c = g ^ 3 + 3 * g ^ 2 * f + f ^ 3 -- 3n+2
d =
5 * g ^ 3 +
if even n
then (-3 * g)
else 3 * g -- 3(n+1) (*)
main :: IO ()
main = print $ (length &&& take 20) . show . fst $ fibN2 (10 ^ 2)
(21,"35422484817926191507")
(fibN2 n) directly calculates a pair (f,g) of two consecutive Fibonacci numbers, (Fib[n], Fib[n+1]), from recursively calculated such pair at about n/3:
*Main> (length &&& take 20) . show . fst $ fibN2 (10^6)
(208988,"19532821287077577316")
The above should take less than 0.1s to calculate on a modern box.
Other identities that could also be used are [http://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form here]. In particular, for (n-1,n) ---> (2n-1,2n) transition which is equivalent to the matrix exponentiation scheme, we have
f (n,(a,b)) = (2*n,(a*a+b*b,2*a*b+b*b)) -- iterate f (1,(0,1)) ; b is nth
and for (n,n+1) ---> (2n,2n+1) (derived from d'Ocagne's identity, for example),
g (n,(a,b)) = (2*n,(2*a*b-a*a,a*a+b*b)) -- iterate g (1,(1,1)) ; a is nth
Haxe
Iterative
static function fib(steps:Int, handler:Int->Void)
{
var current = 0;
var next = 1;
for (i in 1...steps)
{
handler(current);
var temp = current + next;
current = next;
next = temp;
}
handler(current);
}
As Iterator
class FibIter
{
private var current = 0;
private var nextItem = 1;
private var limit:Int;
public function new(limit) this.limit = limit;
public function hasNext() return limit > 0;
public function next() {
limit--;
var ret = current;
var temp = current + nextItem;
current = nextItem;
nextItem = temp;
return ret;
}
}
Used like:
for (i in new FibIter(10))
Sys.println(i);
Hope
Recursive
dec f : num -> num;
--- f 0 <= 0;
--- f 1 <= 1;
--- f(n+2) <= f n + f(n+1);
===Tail-recursive===
dec fib : num -> num;
--- fib n <= l (1, 0, n)
whererec l == \(a,b,succ c) => if c<1 then a else l((a+b),a,c)
|(a,b,0) => 0;
With lazy lists
This language, being one of Haskell's ancestors, also has lazy lists. Here's the (infinite) list of all Fibonacci numbers:
dec fibs : list num;
--- fibs <= fs whererec fs == 0::1::map (+) (tail fs||fs);
The ''n''th Fibonacci number is then just fibs @ n.
HicEst
REAL :: Fibonacci(10)
Fibonacci = ($==2) + Fibonacci($-1) + Fibonacci($-2)
WRITE(ClipBoard) Fibonacci ! 0 1 1 2 3 5 8 13 21 34
Hy
Recursive implementation.
(defn fib [n]
(if (< n 2)
n
(+ (fib (- n 2)) (fib (- n 1)))))
=={{header|Icon}} and {{header|Unicon}}== Icon has built-in support for big numbers. First, a simple recursive solution augmented by caching for non-negative input. This examples computes fib(1000) if there is no integer argument.
procedure main(args)
write(fib(integer(!args) | 1000)
end
procedure fib(n)
static fCache
initial {
fCache := table()
fCache[0] := 0
fCache[1] := 1
}
/fCache[n] := fib(n-1) + fib(n-2)
return fCache[n]
end
The above solution is similar to the one provided [http://www.cs.arizona.edu/icon/library/src/procs/memrfncs.icn fib in memrfncs]
Now, an O(logN) solution. For large N, it takes far longer to convert the result to a string for output than to do the actual computation. This example computes fib(1000000) if there is no integer argument.
procedure main(args)
write(fib(integer(!args) | 1000000))
end
procedure fib(n)
return fibMat(n)[1]
end
procedure fibMat(n)
if n <= 0 then return [0,0]
if n = 1 then return [1,0]
fp := fibMat(n/2)
c := fp[1]*fp[1] + fp[2]*fp[2]
d := fp[1]*(fp[1]+2*fp[2])
if n%2 = 1 then return [c+d, d]
else return [d, c]
end
IDL
Recursive
function fib,n
if n lt 3 then return,1L else return, fib(n-1)+fib(n-2)
end
Execution time O(2^n) until memory is exhausted and your machine starts swapping. Around fib(35) on a 2GB Core2Duo.
Iterative
function fib,n
psum = (csum = 1uL)
if n lt 3 then return,csum
for i = 3,n do begin
nsum = psum + csum
psum = csum
csum = nsum
endfor
return,nsum
end
Execution time O(n). Limited by size of uLong to fib(49)
Analytic
function fib,n
q=1/( p=(1+sqrt(5))/2 )
return,round((p^n+q^n)/sqrt(5))
end
Execution time O(1), only limited by the range of LongInts to fib(48).
Idris
Analytic
fibAnalytic : Nat -> Double
fibAnalytic n =
floor $ ((pow goldenRatio n) - (pow (-1.0/goldenRatio) n)) / sqrt(5)
where goldenRatio : Double
goldenRatio = (1.0 + sqrt(5)) / 2.0
Recursive
fibRecursive : Nat -> Nat
fibRecursive Z = Z
fibRecursive (S Z) = (S Z)
fibRecursive (S (S n)) = fibRecursive (S n) + fibRecursive n
Iterative
fibIterative : Nat -> Nat
fibIterative n = fibIterative' n Z (S Z)
where fibIterative' : Nat -> Nat -> Nat -> Nat
fibIterative' Z a _ = a
fibIterative' (S n) a b = fibIterative' n b (a + b)
Lazy
fibLazy : Lazy (List Nat)
fibLazy = 0 :: 1 :: zipWith (+) fibLazy (
case fibLazy of
(x::xs) => xs
[] => [])
J
The [[j:Essays/Fibonacci_Sequence|Fibonacci Sequence essay]] on the J Wiki presents a number of different ways of obtaining the nth Fibonacci number. Here is one:
fibN=: (-&2 +&$: -&1)^:(1&<) M."0
'''Examples:'''
fibN 12
144
fibN i.31
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
(This implementation is doubly recursive except that results are cached across function calls.)
Java
Iterative
public static long itFibN(int n)
{
if (n < 2)
return n;
long ans = 0;
long n1 = 0;
long n2 = 1;
for(n--; n > 0; n--)
{
ans = n1 + n2;
n1 = n2;
n2 = ans;
}
return ans;
}
/**
* O(log(n))
*/
public static long fib(long n) {
if (n <= 0)
return 0;
long i = (int) (n - 1);
long a = 1, b = 0, c = 0, d = 1, tmp1,tmp2;
while (i > 0) {
if (i % 2 != 0) {
tmp1 = d * b + c * a;
tmp2 = d * (b + a) + c * b;
a = tmp1;
b = tmp2;
}
tmp1 = (long) (Math.pow(c, 2) + Math.pow(d, 2));
tmp2 = d * (2 * c + d);
c = tmp1;
d = tmp2;
i = i / 2;
}
return a + b;
}
Recursive
public static long recFibN(final int n)
{
return (n < 2) ? n : recFibN(n - 1) + recFibN(n - 2);
}
Analytic
This method works up to the 92nd Fibonacci number. After that, it goes out of range.
public static long anFibN(final long n)
{
double p = (1 + Math.sqrt(5)) / 2;
double q = 1 / p;
return (long) ((Math.pow(p, n) + Math.pow(q, n)) / Math.sqrt(5));
}
===Tail-recursive===
public static long fibTailRec(final int n)
{
return fibInner(0, 1, n);
}
private static long fibInner(final long a, final long b, final int n)
{
return n < 1 ? a : n == 1 ? b : fibInner(b, a + b, n - 1);
}
Streams
import java.util.function.LongUnaryOperator;
import java.util.stream.LongStream;
public class FibUtil {
public static LongStream fibStream() {
return LongStream.iterate( 1l, new LongUnaryOperator() {
private long lastFib = 0;
@Override public long applyAsLong( long operand ) {
long ret = operand + lastFib;
lastFib = operand;
return ret;
}
});
}
public static long fib(long n) {
return fibStream().limit( n ).reduce((prev, last) -> last).getAsLong();
}
}
JavaScript
ES5
=Recursive=
Basic recursive function:
function fib(n) {
return n<2?n:fib(n-1)+fib(n-2);
}
Can be rewritten as:
function fib(n) {
if (n<2) { return n; } else { return fib(n-1)+fib(n-2); }
}
One possibility familiar to Scheme programmers is to define an internal function for iteration through anonymous tail recursion:
function fib(n) {
return function(n,a,b) {
return n>0 ? arguments.callee(n-1,b,a+b) : a;
}(n,0,1);
}
Iterative
function fib(n) {
var a = 0, b = 1, t;
while (n-- > 0) {
t = a;
a = b;
b += t;
console.log(a);
}
return a;
}
=Memoization=
With the keys of a dictionary,
var fib = (function(cache){
return cache = cache || {}, function(n){
if (cache[n]) return cache[n];
else return cache[n] = n == 0 ? 0 : n < 0 ? -fib(-n)
: n <= 2 ? 1 : fib(n-2) + fib(n-1);
};
})();
with the indices of an array,
(function () {
'use strict';
function fib(n) {
return Array.apply(null, Array(n + 1))
.map(function (_, i, lst) {
return lst[i] = (
i ? i < 2 ? 1 :
lst[i - 2] + lst[i - 1] :
0
);
})[n];
}
return fib(32);
})();
2178309
====Y-Combinator====
function Y(dn) {
return (function(fn) {
return fn(fn);
}(function(fn) {
return dn(function() {
return fn(fn).apply(null, arguments);
});
}));
}
var fib = Y(function(fn) {
return function(n) {
if (n === 0 || n === 1) {
return n;
}
return fn(n - 1) + fn(n - 2);
};
});
=Generators=
function* fibonacciGenerator() {
var prev = 0;
var curr = 1;
while (true) {
yield curr;
curr = curr + prev;
prev = curr - prev;
}
}
var fib = fibonacciGenerator();
ES6
=Memoized=
If we want access to the whole preceding series, as well as a memoized route to a particular member, we can use an accumulating fold.
(() => {
'use strict';
// Nth member of fibonacci series
// fib :: Int -> Int
function fib(n) {
return mapAccumL(([a, b]) => [
[b, a + b], b
], [0, 1], range(1, n))[0][0];
};
// GENERIC FUNCTIONS
// mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y])
let mapAccumL = (f, acc, xs) => {
return xs.reduce((a, x) => {
let pair = f(a[0], x);
return [pair[0], a[1].concat(pair[1])];
}, [acc, []]);
}
// range :: Int -> Int -> Maybe Int -> [Int]
let range = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);
// TEST
return fib(32);
// --> 2178309
})();
Otherwise, a simple fold will suffice.
{{Trans|Haskell}} (Memoized fold example)
(() => {
'use strict';
// fib :: Int -> Int
let fib = n => range(1, n)
.reduce(([a, b]) => [b, a + b], [0, 1])[0];
// GENERIC [m..n]
// range :: Int -> Int -> [Int]
let range = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);
// TEST
return fib(32);
// --> 2178309
})();
2178309
Joy
Recursive
DEFINE fib == [small] [] [pred dup pred] [+] binrec.
Iterative
DEFINE fib == [1 0] dip [swap [+] unary] times popd.
jq
jq does not (yet) have infinite-precision integer arithmetic, and currently the following algorithms only give exact answers up to fib(78). At a certain point, integers are converted to floats, but floating point precision for fib(n) fails after n = 1476: in jq, fib(1476) evaluates to 1.3069892237633987e+308
Recursive
def nth_fib_naive(n):
if (n < 2) then n
else nth_fib_naive(n - 1) + nth_fib_naive(n - 2)
end;
Tail Recursive
Recent versions of jq (after July 1, 2014) include basic optimizations for tail recursion, and nth_fib is defined here to take advantage of TCO. For example, nth_fib(10000000) completes with only 380KB (that's K) of memory. However nth_fib can also be used with earlier versions of jq.
def nth_fib(n):
# input: [f(i-2), f(i-1), countdown]
def fib: (.[0] + .[1]) as $sum
| .[2] as $n
| if ($n <= 0) then $sum
else [ .[1], $sum, $n - 1 ]
| fib end;
[-1, 1, n] | fib;
Example:
(range(0;5), 50) | [., nth_fib(.)]
yields:
[0,0]
[1,1]
[2,1]
[3,2]
[4,3]
[50,12586269025]
===Binet's Formula===
def fib_binet(n):
(5|sqrt) as $rt
| ((1 + $rt)/2) as $phi
| (($phi | log) * n | exp) as $phin
| (if 0 == (n % 2) then 1 else -1 end) as $sign
| ( ($phin - ($sign / $phin) ) / $rt ) + .5
| floor;
Generator
The following is a jq generator which produces the first n terms of the Fibonacci sequence efficiently, one by one. Notice that it is simply a variant of the above tail-recursive function. The function is in effect turned into a generator by changing "( _ | fib )" to "$sum, (_ | fib)".
# Generator
def fibonacci(n):
# input: [f(i-2), f(i-1), countdown]
def fib: (.[0] + .[1]) as $sum
| if .[2] == 0 then $sum
else $sum, ([ .[1], $sum, .[2] - 1 ] | fib)
end;
[-1, 1, n] | fib;
Julia
Recursive
fib(n) = n < 2 ? n : fib(n-1) + fib(n-2)
Iterative
function fib(n)
x,y = (0,1)
for i = 1:n x,y = (y, x+y) end
x
end
Matrix form
fib(n) = ([1 1 ; 1 0]^n)[1,2]
K
Recursive
{:[x<3;1;_f[x-1]+_f[x-2]]}
Recursive with memoization
Using a (global) dictionary c.
{c::.();{v:c[a:`$$x];:[x<3;1;:[_n~v;c[a]:_f[x-1]+_f[x-2];v]]}x}
Analytic
phi:(1+_sqrt(5))%2
{_((phi^x)-((1-phi)^x))%_sqrt[5]}
Sequence to n
{(x(|+\)\1 1)[;1]}
{x{x,+/-2#x}/!2}
Kabap
Sequence to n
// Calculate the $n'th Fibonacci number
// Set this to how many in the sequence to generate
$n = 10;
// These are what hold the current calculation
$a = 0;
$b = 1;
// This holds the complete sequence that is generated
$sequence = "";
// Prepare a loop
$i = 0;
:calcnextnumber;
$i = $i++;
// Do the calculation for this loop iteration
$b = $a + $b;
$a = $b - $a;
// Add the result to the sequence
$sequence = $sequence << $a;
// Make the loop run a fixed number of times
if $i < $n; {
$sequence = $sequence << ", ";
goto calcnextnumber;
}
// Use the loop counter as the placeholder
$i--;
// Return the sequence
return = "Fibonacci number " << $i << " is " << $a << " (" << $sequence << ")";
Kotlin
enum class Fibonacci {
ITERATIVE {
override fun invoke(n: Long) = if (n < 2) {
n
} else {
var n1 = 0L
var n2 = 1L
var i = n
do {
val sum = n1 + n2
n1 = n2
n2 = sum
} while (i-- > 1)
n1
}
},
RECURSIVE {
override fun invoke(n: Long): Long = if (n < 2) n else this(n - 1) + this(n - 2)
};
abstract operator fun invoke(n: Long): Long
}
fun main(a: Array<String>) {
val r = 0..30L
Fibonacci.values().forEach {
print("${it.name}: ")
r.forEach { i -> print(" " + it(i)) }
println()
}
}
ITERATIVE: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
RECURSIVE: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
L++
(defn int fib (int n) (return (? (< n 2) n (+ (fib (- n 1)) (fib (- n 2))))))
(main (prn (fib 30)))
LabVIEW
Lang5
[] '__A set : dip swap __A swap 2 compress collapse '__A set execute
__A -1 extract nip ; : nip swap drop ; : tuck swap over ;
: -rot rot rot ; : 0= 0 == ; : 1+ 1 + ; : 1- 1 - ; : sum '+ reduce ;
: bi 'keep dip execute ; : keep over 'execute dip ;
: fib dup 1 > if dup 1- fib swap 2 - fib + then ;
: fib dup 1 > if "1- fib" "2 - fib" bi + then ;
Langur
val .fibonacci = f if(.x < 2: .x ; self(.x - 1) + self(.x - 2))
writeln map .fibonacci, series 2..20
[1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]
lambdatalk
1) basic version
{def fib1
{lambda {:n}
{if {< :n 3}
then 1
else {+ {fib1 {- :n 1}} {fib1 {- :n 2}}} }}}
{fib1 16} -> 987 (CPU ~ 16ms)
{fib1 30} = 832040 (CPU > 12000ms)
2) tail-recursive version
{def fib2
{def fib2.r
{lambda {:a :b :i}
{if {< :i 1}
then :a
else {fib2.r :b {+ :a :b} {- :i 1}} }}}
{lambda {:n} {fib2.r 0 1 :n}}}
{fib2 16} -> 987 (CPU ~ 1ms)
{fib2 30} -> 832040 (CPU ~2ms)
{fib2 1000} -> 4.346655768693743e+208 (CPU ~ 22ms)
3) Dijkstra Algorithm
{def fib3
{def fib3.r
{lambda {:a :b :p :q :count}
{if {= :count 0}
then :b
else {if {= {% :count 2} 0}
then {fib3.r :a :b
{+ {* :p :p} {* :q :q}}
{+ {* :q :q} {* 2 :p :q}}
{/ :count 2}}
else {fib3.r {+ {* :b :q} {* :a :q} {* :a :p}}
{+ {* :b :p} {* :a :q}}
:p :q
{- :count 1}} }}}}
{lambda {:n}
{fib3.r 1 0 0 1 :n} }}
{fib3 16} -> 987 (CPU ~ 2ms)
{fib3 30} -> 832040 (CPU ~ 2ms)
{fib3 1000} -> 4.346655768693743e+208 (CPU ~ 3ms)
4) memoization
{def fib4
{def fib4.m {array.new}} // init an empty array
{def fib4.r {lambda {:n}
{if {< :n 2}
then {array.get {array.set! {fib4.m} :n 1} :n} // init with 1,1
else {if {equal? {array.get {fib4.m} :n} undefined} // if not exists
then {array.get {array.set! {fib4.m} :n
{+ {fib4.r {- :n 1}}
{fib4.r {- :n 2}}}} :n} // compute it
else {array.get {fib4.m} :n} }}}} // else get it
{lambda {:n}
{fib4.r :n}
{fib4.m} }} // display the number AND all its predecessors
-> fib4
{fib4 90}
-> 4660046610375530000
[1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,
317811,514229,832040,1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169,63245986,102334155,
165580141,267914296,433494437,701408733,1134903170,1836311903,2971215073,4807526976,7778742049,12586269025,
20365011074,32951280099,53316291173,86267571272,139583862445,225851433717,365435296162,591286729879,956722026041,
1548008755920,2504730781961,4052739537881,6557470319842,10610209857723,17167680177565,27777890035288,44945570212853,
72723460248141,117669030460994,190392490709135,308061521170129,498454011879264,806515533049393,1304969544928657,
2111485077978050,3416454622906707,5527939700884757,8944394323791464,14472334024676220,23416728348467684,
37889062373143900,61305790721611580,99194853094755490,160500643816367070,259695496911122560,420196140727489660,
679891637638612200,1100087778366101900,1779979416004714000,2880067194370816000,4660046610375530000]
5) Binet's formula (non recursive)
{def fib5
{lambda {:n}
{let { {:n :n} {:sqrt5 {sqrt 5}} }
{round {/ {- {pow {/ {+ 1 :sqrt5} 2} :n}
{pow {/ {- 1 :sqrt5} 2} :n}} :sqrt5}}} }}
{fib5 16} -> 987 (CPU ~ 1ms)
{fib5 30} -> 832040 (CPU ~ 1ms)
{fib5 1000} -> 4.346655768693743e+208 (CPU ~ 1ms)
Lasso
define fibonacci(n::integer) => {
#n < 1 ? return false
local(
swap = 0,
n1 = 0,
n2 = 1
)
loop(#n) => {
#swap = #n1 + #n2;
#n2 = #n1;
#n1 = #swap;
}
return #n1
}
fibonacci(0) //->output false
fibonacci(1) //->output 1
fibonacci(2) //->output 1
fibonacci(3) //->output 2
Latitude
Recursive
fibo := {
takes '[n].
if { n <= 1. } then {
n.
} else {
fibo (n - 1) + fibo (n - 2).
}.
}.
Memoization
fibo := {
takes '[n].
cache := here cache.
{ cache slot? (n ordinal). } ifFalse {
cache slot (n ordinal) =
if { n <= 1. } then {
n.
} else {
fibo (n - 1) + fibo (n - 2).
}.
}.
cache slot (n ordinal).
} tap {
;; Attach the cache to the method object itself.
#'self cache := Object clone.
}.
LFE
Recursive
(defun fib
((0) 0)
((1) 1)
((n)
(+ (fib (- n 1))
(fib (- n 2)))))
Iterative
(defun fib
((n) (when (>= n 0))
(fib n 0 1)))
(defun fib
((0 result _)
result)
((n result next)
(fib (- n 1) next (+ result next))))
Liberty BASIC
Iterative/Recursive
for i = 0 to 15
print fiboR(i),fiboI(i)
next i
function fiboR(n)
if n <= 1 then
fiboR = n
else
fiboR = fiboR(n-1) + fiboR(n-2)
end if
end function
function fiboI(n)
a = 0
b = 1
for i = 1 to n
temp = a + b
a = b
b = temp
next i
fiboI = a
end function
0 0
1 1
1 1
2 2
3 3
5 5
8 8
13 13
21 21
34 34
55 55
89 89
144 144
233 233
377 377
610 610
Iterative/Negative
print "Rosetta Code - Fibonacci sequence": print
print " n Fn"
for x=-12 to 12 '68 max
print using("### ", x); using("##############", FibonacciTerm(x))
next x
print
[start]
input "Enter a term#: "; n$
n$=lower$(trim$(n$))
if n$="" then print "Program complete.": end
print FibonacciTerm(val(n$))
goto [start]
function FibonacciTerm(n)
n=int(n)
FTa=0: FTb=1: FTc=-1
select case
case n=0 : FibonacciTerm=0 : exit function
case n=1 : FibonacciTerm=1 : exit function
case n=-1 : FibonacciTerm=-1 : exit function
case n>1
for x=2 to n
FibonacciTerm=FTa+FTb
FTa=FTb: FTb=FibonacciTerm
next x
exit function
case n<-1
for x=-2 to n step -1
FibonacciTerm=FTa+FTc
FTa=FTc: FTc=FibonacciTerm
next x
exit function
end select
end function
Rosetta Code - Fibonacci sequence
n Fn
-12 -144
-11 -89
-10 -55
-9 -34
-8 -21
-7 -13
-6 -8
-5 -5
-4 -3
-3 -2
-2 -1
-1 -1
0 0
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
Enter a term#: 12
144
Enter a term#:
Program complete.
Lingo
Recursive
on fib (n)
if n<2 then return n
return fib(n-1)+fib(n-2)
end
Iterative
on fib (n)
if n<2 then return n
fibPrev = 0
fib = 1
repeat with i = 2 to n
tmp = fib
fib = fib + fibPrev
fibPrev = tmp
end repeat
return fib
end
Analytic
on fib (n)
sqrt5 = sqrt(5.0)
p = (1+sqrt5)/2
q = 1 - p
return integer((power(p,n)-power(q,n))/sqrt5)
end
Lisaac
- fib(n : UINTEGER_32) : UINTEGER_64 <- (
+ result : UINTEGER_64;
(n < 2).if {
result := n;
} else {
result := fib(n - 1) + fib(n - 2);
};
result
);
LiveCode
-- Iterative, translation of the basic version.
function fibi n
put 0 into aa
put 1 into b
repeat with i = 1 to n
put aa + b into temp
put b into aa
put temp into b
end repeat
return aa
end fibi
-- Recursive
function fibr n
if n <= 1 then
return n
else
return fibr(n-1) + fibr(n-2)
end if
end fibr
LLVM
; This is not strictly LLVM, as it uses the C library function "printf".
; LLVM does not provide a way to print values, so the alternative would be
; to just load the string into memory, and that would be boring.
; Additional comments have been inserted, as well as changes made from the output produced by clang such as putting more meaningful labels for the jumps
$"PRINT_LONG" = comdat any
@"PRINT_LONG" = linkonce_odr unnamed_addr constant [5 x i8] c"%ld\0A\00", comdat, align 1
;--- The declaration for the external C printf function.
declare i32 @printf(i8*, ...)
;--------------------------------------------------------------------
;-- Function for calculating the nth fibonacci numbers
;--------------------------------------------------------------------
define i32 @fibonacci(i32) {
%2 = alloca i32, align 4 ;-- allocate local copy of n
%3 = alloca i32, align 4 ;-- allocate a
%4 = alloca i32, align 4 ;-- allocate b
store i32 %0, i32* %2, align 4 ;-- store copy of n
store i32 0, i32* %3, align 4 ;-- a := 0
store i32 1, i32* %4, align 4 ;-- b := 1
br label %loop
loop:
%5 = load i32, i32* %2, align 4 ;-- load n
%6 = icmp sgt i32 %5, 0 ;-- n > 0
br i1 %6, label %loop_body, label %exit
loop_body:
%7 = load i32, i32* %3, align 4 ;-- load a
%8 = load i32, i32* %4, align 4 ;-- load b
%9 = add nsw i32 %7, %8 ;-- t = a + b
store i32 %8, i32* %3, align 4 ;-- store a = b
store i32 %9, i32* %4, align 4 ;-- store b = t
%10 = load i32, i32* %2, align 4 ;-- load n
%11 = add nsw i32 %10, -1 ;-- decrement n
store i32 %11, i32* %2, align 4 ;-- store n
br label %loop
exit:
%12 = load i32, i32* %3, align 4 ;-- load a
ret i32 %12 ;-- return a
}
;--------------------------------------------------------------------
;-- Main function for printing successive fibonacci numbers
;--------------------------------------------------------------------
define i32 @main() {
%1 = alloca i32, align 4 ;-- allocate index
store i32 0, i32* %1, align 4 ;-- index := 0
br label %loop
loop:
%2 = load i32, i32* %1, align 4 ;-- load index
%3 = icmp sle i32 %2, 12 ;-- index <= 12
br i1 %3, label %loop_body, label %exit
loop_body:
%4 = load i32, i32* %1, align 4 ;-- load index
%5 = call i32 @fibonacci(i32 %4)
%6 = call i32 (i8*, ...) @printf(i8* getelementptr inbounds ([5 x i8], [5 x i8]* @"PRINT_LONG", i32 0, i32 0), i32 %5)
%7 = load i32, i32* %1, align 4 ;-- load index
%8 = add nsw i32 %7, 1 ;-- increment index
store i32 %8, i32* %1, align 4 ;-- store index
br label %loop
exit:
ret i32 0 ;-- return EXIT_SUCCESS
}
0
1
1
2
3
5
8
13
21
34
55
89
144
Logo
to fib :n [:a 0] [:b 1]
if :n < 1 [output :a]
output (fib :n-1 :b :a+:b)
end
LOLCODE
HAI 1.2
HOW DUZ I fibonacci YR N
EITHER OF BOTH SAEM N AN 1 AN BOTH SAEM N AN 0
O RLY?
YA RLY, FOUND YR 1
NO WAI
I HAS A N1
I HAS A N2
N1 R DIFF OF N AN 1
N2 R DIFF OF N AN 2
N1 R fibonacci N1
N2 R fibonacci N2
FOUND YR SUM OF N1 AN N2
OIC
IF U SAY SO
KTHXBYE
Lua
Recursive
--calculates the nth fibonacci number. Breaks for negative or non-integer n.
function fibs(n)
return n < 2 and n or fibs(n - 1) + fibs(n - 2)
end
Pedantic Recursive
--more pedantic version, returns 0 for non-integer n
function pfibs(n)
if n ~= math.floor(n) then return 0
elseif n < 0 then return pfibs(n + 2) - pfibs(n + 1)
elseif n < 2 then return n
else return pfibs(n - 1) + pfibs(n - 2)
end
end
Tail Recursive
function a(n,u,s) if n<2 then return u+s end return a(n-1,u+s,u) end
function trfib(i) return a(i-1,1,0) end
Table Recursive
fib_n = setmetatable({1, 1}, {__index = function(z,n) return n<=0 and 0 or z[n-1] + z[n-2] end})
Table Recursive 2
-- table recursive done properly (values are actually saved into table;
-- also the first element of Fibonacci sequence is 0, so the initial table should be {0, 1}).
fib_n = setmetatable({0, 1}, {
__index = function(t,n)
if n <= 0 then return 0 end
t[n] = t[n-1] + t[n-2]
return t[n]
end
})
Iterative
function ifibs(n)
local p0,p1=0,1
for _=1,n do p0,p1 = p1,p0+p1 end
return p0
end
Luck
function fib(x: int): int = (
let cache = {} in
let fibc x = if x<=1 then x else (
if x not in cache then
cache[x] = fibc(x-1) + fibc(x-2);
cache[x]
) in fibc(x)
);;
for x in range(10) do print(fib(x))
Lush
(de fib-rec (n)
(if (< n 2)
n
(+ (fib-rec (- n 2)) (fib-rec (- n 1)))))
LSL
Rez a box on the ground, and add the following as a New Script.
integer Fibonacci(integer n) {
if(n<2) {
return n;
} else {
return Fibonacci(n-1)+Fibonacci(n-2);
}
}
default {
state_entry() {
integer x = 0;
for(x=0 ; x<35 ; x++) {
llOwnerSay("Fibonacci("+(string)x+")="+(string)Fibonacci(x));
}
}
}
Output:
Fibonacci(0)=0
Fibonacci(1)=1
Fibonacci(2)=1
Fibonacci(3)=2
Fibonacci(4)=3
Fibonacci(5)=5
Fibonacci(6)=8
Fibonacci(7)=13
Fibonacci(8)=21
Fibonacci(9)=34
Fibonacci(10)=55
Fibonacci(11)=89
Fibonacci(12)=144
Fibonacci(13)=233
Fibonacci(14)=377
Fibonacci(15)=610
Fibonacci(16)=987
Fibonacci(17)=1597
Fibonacci(18)=2584
Fibonacci(19)=4181
Fibonacci(20)=6765
Fibonacci(21)=10946
Fibonacci(22)=17711
Fibonacci(23)=28657
Fibonacci(24)=46368
Fibonacci(25)=75025
Fibonacci(26)=121393
Fibonacci(27)=196418
Fibonacci(28)=317811
Fibonacci(29)=514229
Fibonacci(30)=832040
Fibonacci(31)=1346269
Fibonacci(32)=2178309
Fibonacci(33)=3524578
Fibonacci(34)=5702887
M2000 Interpreter
Return decimal type and use an Inventory (as closure) to store known return values. All closures are in scope in every recursive call (we use here lambda(), but we can use fib(), If we make Fib1=fib then we have to use lambda() for recursion.
Inventory K=0:=0,1:=1
fib=Lambda K (x as decimal)-> {
If Exist(K, x) Then =Eval(K) :Exit
Def Ret as Decimal
Ret=If(x>1->Lambda(x-1)+Lambda(x-2), x)
Append K, x:=Ret
=Ret
}
\\ maximum 139
For i=1 to 139 {
Print Fib(i)
}
Here an example where we use a BigNum class to make a Group which hold a stack of values, and take 14 digits per item in stack. We can use inventory to hold groups, so we use the fast fib() function from code above, where we remove the type definition of Ret variable, and set two first items in inventory as groups.
Class BigNum {
a=stack
Function Digits {
=len(.a)*14-(14-len(str$(stackitem(.a,len(.a)) ,"")))
}
Operator "+" (n) {
\\ we get a copy, but .a is pointer
\\ we make a copy, and get a new pointer
.a<=stack(.a)
acc=0
carry=0
const d=100000000000000@
k=min.data(Len(.a), len(n.a))
i=each(.a, 1,k )
j=each(n.a, 1,k)
while i, j {
acc=stackitem(i)+stackitem(j)+carry
carry= acc div d
return .a, i^+1:=acc mod d
}
if len(.a)<len(n.a) Then {
i=each(n.a, k+1, -1)
while i {
acc=stackitem(i)+carry
carry= acc div d
stack .a {data acc mod d}
}
} ELse.if len(.a)>len(n.a) Then {
i=each(.a, k+1, -1)
while i {
acc=stackitem(i)+carry
carry= acc div d
Return .a, i^+1:=acc mod d
if carry else exit
}
}
if carry then stack .a { data carry}
}
Function tostring$ {
if len(.a)=0 then ="0" : Exit
if len(.a)=1 then =str$(Stackitem(.a),"") : Exit
document buf$=str$(Stackitem(.a, len(.a)),"")
for i=len(.a)-1 to 1 {
Stack .a {
buf$=str$(StackItem(i), "00000000000000")
}
}
=buf$
}
class:
Module BigNum (s$) {
s$=filter$(s$,"+-.,")
if s$<>"" Then {
repeat {
If len(s$)<14 then Stack .a { Data val(s$) }: Exit
Stack .a { Data val(Right$(s$, 14)) }
S$=Left$(S$, len(S$)-14)
} Until S$=""
}
}
}
Inventory K=0:=BigNum("0"),1:=BigNum("1")
fib=Lambda K (x as decimal)-> {
If Exist(K, x) Then =Eval(K) :Exit
Ret=If(x>1->Lambda(x-1)+Lambda(x-2), bignum(str$(x,"")))
Append K, x:=Ret
=Ret
}
\\ Using this to handle form refresh by code
Set Fast!
For i=1 to 4000 {
N=Fib(i)
Print i
Print N.tostring$()
Refresh
}
M4
define(`fibo',`ifelse(0,$1,0,`ifelse(1,$1,1,
`eval(fibo(decr($1)) + fibo(decr(decr($1))))')')')dnl
define(`loop',`ifelse($1,$2,,`$3($1) loop(incr($1),$2,`$3')')')dnl
loop(0,15,`fibo')
Maple
> f := n -> ifelse(n<3,1,f(n-1)+f(n-2));
> f(2);
1
> f(3);
2
=={{header|Mathematica}} / {{header|Wolfram Language}}== The Wolfram Language already has a built-in function Fibonacci, but a simple recursive implementation would be
fib[0] = 0
fib[1] = 1
fib[n_Integer] := fib[n - 1] + fib[n - 2]
An optimization is to cache the values already calculated:
fib[0] = 0
fib[1] = 1
fib[n_Integer] := fib[n] = fib[n - 1] + fib[n - 2]
The above implementations may be too simplistic, as the first is incredibly slow for any reasonable range due to nested recursions and while the second is faster it uses an increasing amount of memory. The following uses recursion much more effectively while not using memory:
fibi[prvprv_Integer, prv_Integer, rm_Integer] :=
If[rm < 1, prvprv, fibi[prv, prvprv + prv, rm - 1]]
fib[n_Integer] := fibi[0, 1, n]
However, the recursive approaches in Mathematica are limited by the limit set for recursion depth (default 1024 or 4096 for the above cases), limiting the range for 'n' to about 1000 or 2000. The following using an iterative approach has an extremely high limit (greater than a million):
fib[n_Integer] := Block[{tmp, prvprv = 0, prv = 1},
For[i = 0, i < n, i++, tmp = prv; prv += prvprv; prvprv = tmp];
Return[prvprv]]
If one wanted a list of Fibonacci numbers, the following is quite efficient:
fibi[{prvprv_Integer, prv_Integer}] := {prv, prvprv + prv}
fibList[n_Integer] := Map[Take[#, 1] &, NestList[fibi, {0, 1}, n]] // Flatten
Output from the last with "fibList[100]":
{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, \
1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, \
196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, \
9227465, 14930352, 24157817, 39088169, 63245986, 102334155, \
165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, \
2971215073, 4807526976, 7778742049, 12586269025, 20365011074, \
32951280099, 53316291173, 86267571272, 139583862445, 225851433717, \
365435296162, 591286729879, 956722026041, 1548008755920, \
2504730781961, 4052739537881, 6557470319842, 10610209857723, \
17167680177565, 27777890035288, 44945570212853, 72723460248141, \
117669030460994, 190392490709135, 308061521170129, 498454011879264, \
806515533049393, 1304969544928657, 2111485077978050, \
3416454622906707, 5527939700884757, 8944394323791464, \
14472334024676221, 23416728348467685, 37889062373143906, \
61305790721611591, 99194853094755497, 160500643816367088, \
259695496911122585, 420196140727489673, 679891637638612258, \
1100087778366101931, 1779979416004714189, 2880067194370816120, \
4660046610375530309, 7540113804746346429, 12200160415121876738, \
19740274219868223167, 31940434634990099905, 51680708854858323072, \
83621143489848422977, 135301852344706746049, 218922995834555169026, \
354224848179261915075}
The Wolfram Language can also solve recurrence equations using the built-in function RSolve
fib[n] /. RSolve[{fib[n] == fib[n - 1] + fib[n - 2], fib[0] == 0,
fib[1] == 1}, fib[n], n][[1]]
which evaluates to the built-in function Fibonacci[n]
This function can also be expressed as
Fibonacci[n] // FunctionExpand // FullSimplify
which evaluates to
(2^-n ((1 + Sqrt[5])^n - (-1 + Sqrt[5])^n Cos[n π]))/Sqrt[5]
and is defined for all real or complex values of n.
MATLAB
Matrix
function f = fib(n)
f = [1 1 ; 1 0]^(n-1);
f = f(1,1);
end
Iterative
function F = fibonacci(n)
Fn = [1 0]; %Fn(1) is F_{n-2}, Fn(2) is F_{n-1}
F = 0; %F is F_{n}
for i = (1:abs(n))
Fn(2) = F;
F = sum(Fn);
Fn(1) = Fn(2);
end
if n < 0
F = F*((-1)^(n+1));
end
end
Dramadah Matrix Method
The MATLAB help file suggests an interesting method of generating the Fibonacci numbers. Apparently the determinate of the Dramadah Matrix of type 3 (MATLAB designation) and size n-by-n is the nth Fibonacci number. This method is implimented below.
function number = fibonacci2(n)
if n == 1
number = 1;
elseif n == 0
number = 0;
elseif n < 0
number = ((-1)^(n+1))*fibonacci2(-n);;
else
number = det(gallery('dramadah',n,3));
end
end
Tartaglia/Pascal Triangle Method
function number = fibonacci(n)
%construct the Tartaglia/Pascal Triangle
pt=tril(ones(n));
for r = 3 : n
% Every element is the addition of the two elements
% on top of it. That means the previous row.
for c = 2 : r-1
pt(r, c) = pt(r-1, c-1) + pt(r-1, c);
end
end
number=trace(rot90(pt));
end
Maxima
/* fib(n) is built-in; here is an implementation */
fib2(n) := (matrix([0, 1], [1, 1])^^n)[1, 2]$
fib2(100)-fib(100);
0
fib2(-10);
-55
MAXScript
Iterative
fn fibIter n =
(
if n < 2 then
(
n
)
else
(
fib = 1
fibPrev = 1
for num in 3 to n do
(
temp = fib
fib += fibPrev
fibPrev = temp
)
fib
)
)
Recursive
fn fibRec n =
(
if n < 2 then
(
n
)
else
(
fibRec (n - 1) + fibRec (n - 2)
)
)
Mercury
Mercury is both a logic language and a functional language. As such there are two possible interfaces for calculating a Fibonacci number. This code shows both styles. Note that much of the code here is ceremony put in place to have this be something which can actually compile. The actual Fibonacci number generation is contained in the predicate fib/2 and in the function fib/1. The predicate main/2 illustrates first the unification semantics of the predicate form and the function call semantics of the function form.
The provided code uses a very naive form of generating a Fibonacci number. A more realistic implementation would use memoization to cache previous results, exchanging time for space. Also, in the case of supplying both a function implementation and a predicate implementation, one of the two would be implemented in terms of the other. Examples of this are given as comments below.
fib.m
% The following code is derived from the Mercury Tutorial by Ralph Becket.
% http://www.mercury.csse.unimelb.edu.au/information/papers/book.pdf
:- module fib.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module int.
:- pred fib(int::in, int::out) is det.
fib(N, X) :-
( if N =< 2
then X = 1
else fib(N - 1, A), fib(N - 2, B), X = A + B ).
:- func fib(int) = int is det.
fib(N) = X :- fib(N, X).
main(!IO) :-
fib(40, X),
write_string("fib(40, ", !IO),
write_int(X, !IO),
write_string(")\n", !IO),
write_string("fib(40) = ", !IO),
write_int(fib(40), !IO),
write_string("\n", !IO).
Iterative algorithm
The much faster iterative algorithm can be written as:
:- pred fib_acc(int::in, int::in, int::in, int::in, int::out) is det.
fib_acc(N, Limit, Prev2, Prev1, Res) :-
( N < Limit ->
% limit not reached, continue computation.
( N =< 2 ->
Res0 = 1
;
Res0 = Prev2 + Prev1
),
fib_acc(N+1, Limit, Prev1, Res0, Res)
;
% Limit reached, return the sum of the two previous results.
Res = Prev2 + Prev1
).
This predicate can be called as
fib_acc(1, 40, 1, 1, Result)
It has several inputs which form the loop, the first is the current number, the second is a limit, ie when to stop counting. And the next two are accumulators for the last and next-to-last results.
Memoization
But what if you want the speed of the fib_acc with the recursive (more declarative) definition of fib? Then use memoization, because Mercury is a pure language fib(N, F) will always give the same F for the same N, guaranteed. Therefore memoization asks the compiler to use a table to remember the value for F for any N, and it's a one line change:
:- pragma memo(fib/2).
:- pred fib(int::in, int::out) is det.
fib(N, X) :-
( if N =< 2
then X = 1
else fib(N - 1, A), fib(N - 2, B), X = A + B ).
We've shown the definition of fib/2 again, but the only change here is the memoization pragma (see the reference manual). This is not part of the language specification and different Mercury implementations are allowed to ignore it, however there is only one implementation so in practice memoization is fully supported.
Memoization trades speed for space, a table of results is constructed and kept in memory. So this version of fib consumes more memory than than fib_acc. It is also slightly slower than fib_acc since it must manage its table of results but it is much much faster than without memoization. Memoization works very well for the Fibonacci sequence because in the naive version the same results are calculated over and over again.
Metafont
vardef fibo(expr n) =
if n=0: 0
elseif n=1: 1
else:
fibo(n-1) + fibo(n-2)
fi
enddef;
for i=0 upto 10: show fibo(i); endfor
end
Microsoft Small Basic
Iterative
' Fibonacci sequence - 31/07/2018
n = 139
f1 = 0
f2 = 1
TextWindow.WriteLine("fibo(0)="+f1)
TextWindow.WriteLine("fibo(1)="+f2)
For i = 2 To n
f3 = f1 + f2
TextWindow.WriteLine("fibo("+i+")="+f3)
f1 = f2
f2 = f3
EndFor
fibo(139)=50095301248058391139327916261
===Binet's Formula===
' Fibonacci sequence - Binet's Formula - 31/07/2018
n = 69
sq5=Math.SquareRoot(5)
phi1=(1+sq5)/2
phi2=(1-sq5)/2
phi1n=phi1
phi2n=phi2
For i = 2 To n
phi1n=phi1n*phi1
phi2n=phi2n*phi2
TextWindow.Write(Math.Floor((phi1n-phi2n)/sq5)+" ")
EndFor
1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099 53316291173 86267571272 139583862445 225851433717 365435296162 591286729879 956722026041 1548008755920 2504730781961 4052739537881 6557470319842 10610209857723 17167680177565 27777890035288 44945570212853 72723460248141 117669030460994
min
(
(2 <)
((0 1 (dup rollup +)) dip pred times nip)
unless
) :fib
MiniScript
fibonacci = function(n)
if n < 2 then return n
ans = 0
n1 = 0
n2 = 1
for i in range(n-1, 1)
ans = n1 + n2
n1 = n2
n2 = ans
end for
return ans
end function
print fibonacci(6)
Mirah
def fibonacci(n:int)
return n if n < 2
fibPrev = 1
fib = 1
3.upto(Math.abs(n)) do
oldFib = fib
fib = fib + fibPrev
fibPrev = oldFib
end
fib * (n<0 ? int(Math.pow(n+1, -1)) : 1)
end
puts fibonacci 1
puts fibonacci 2
puts fibonacci 3
puts fibonacci 4
puts fibonacci 5
puts fibonacci 6
puts fibonacci 7
MIPS Assembly
This is the iterative approach to the Fibonacci sequence.
.text
main: li $v0, 5 # read integer from input. The read integer will be stroed in $v0
syscall
beq $v0, 0, is1
beq $v0, 1, is1
li $s4, 1 # the counter which has to equal to $v0
li $s0, 1
li $s1, 1
loop: add $s2, $s0, $s1
addi $s4, $s4, 1
beq $v0, $s4, iss2
add $s0, $s1, $s2
addi $s4, $s4, 1
beq $v0, $s4, iss0
add $s1, $s2, $s0
addi $s4, $s4, 1
beq $v0, $s4, iss1
b loop
iss0: move $a0, $s0
b print
iss1: move $a0, $s1
b print
iss2: move $a0, $s2
b print
is1: li $a0, 1
b print
print: li $v0, 1
syscall
li $v0, 10
syscall
=={{header|MK-61/52}}==
Instruction: ''n'' В/О С/П, where ''n'' is serial number of the number of Fibonacci sequence; С/П for the following numbers.
## ML
=
## Standard ML
=
### =Recursion=
This version is tail recursive.
```sml
fun fib n =
let
fun fib' (0,a,b) = a
| fib' (n,a,b) = fib' (n-1,a+b,a)
in
fib' (n,0,1)
end
=
MLite
=
=Recursion=
Tail recursive.
fun fib
(0, x1, x2) = x2
| (n, x1, x2) = fib (n-1, x2, x1+x2)
| n = fib (n, 0, 1)
ML/I
MCSKIP "WITH" NL
"" Fibonacci - recursive
MCSKIP MT,<>
MCINS %.
MCDEF FIB WITHS ()
AS <MCSET T1=%A1.
MCGO L1 UNLESS 2 GR T1
%T1.<>MCGO L0
%L1.%FIB(%T1.-1)+FIB(%T1.-2).>
fib(0) is FIB(0)
fib(1) is FIB(1)
fib(2) is FIB(2)
fib(3) is FIB(3)
fib(4) is FIB(4)
fib(5) is FIB(5)
=={{header|Modula-2}}==
MODULE Fibonacci;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE Fibonacci(n : LONGINT) : LONGINT;
VAR
a,b,c : LONGINT;
BEGIN
IF n<0 THEN RETURN 0 END;
a:=1;
b:=1;
WHILE n>0 DO
c := a + b;
a := b;
b := c;
DEC(n)
END;
RETURN a
END Fibonacci;
VAR
buf : ARRAY[0..63] OF CHAR;
i : INTEGER;
r : LONGINT;
BEGIN
FOR i:=0 TO 10 DO
r := Fibonacci(i);
FormatString("%l\n", buf, r);
WriteString(buf);
END;
ReadChar
END Fibonacci.
=={{header|Modula-3}}==
Recursive
PROCEDURE Fib(n: INTEGER): INTEGER =
BEGIN
IF n < 2 THEN
RETURN n;
ELSE
RETURN Fib(n-1) + Fib(n-2);
END;
END Fib;
=== Iterative (with negatives) ===
PROCEDURE IterFib(n: INTEGER): INTEGER =
VAR
limit := ABS(n);
prev := 0;
curr, next: INTEGER;
BEGIN
(* trivial case *)
IF n = 0 THEN RETURN 0; END;
IF n > 0 THEN (* positive case *)
curr := 1;
FOR i := 2 TO limit DO
next := prev + curr;
prev := curr;
curr := next;
END;
ELSE (* negative case *)
curr := -1;
FOR i := 2 TO limit DO
next := prev - curr;
prev := curr;
curr := next;
END;
END;
RETURN curr;
END IterFib;
Monicelli
Recursive version. It includes a main that reads a number N from standard input and prints the Nth Fibonacci number.
# Main
Lei ha clacsonato
voglio un nonnulla, Necchi mi porga un nonnulla
il nonnulla come se fosse brematurata la supercazzola bonaccia con il nonnulla o scherziamo?
un nonnulla a posterdati
# Fibonacci function 'bonaccia'
blinda la supercazzola Necchi bonaccia con antani Necchi o scherziamo? che cos'è l'antani?
minore di 3: vaffanzum 1! o tarapia tapioco: voglio unchiamo, Necchi come se fosse brematurata
la supercazzola bonaccia con antani meno 1 o scherziamo? voglio duechiamo,
Necchi come se fosse brematurata la supercazzola bonaccia con antani meno 2 o scherziamo? vaffanzum
unchiamo più duechiamo! e velocità di esecuzione
MontiLang
Reads number from standard input and prints to that number in the fibonacci sequence
0 VAR a .
1 VAR b .
INPUT TOINT
FOR :
a b + VAR c .
a PRINT .
b VAR a .
c VAR b .
ENDFOR
Forth-style solution
def over
swap dup rot swap
enddef
|Enter a number to obtain Fibonacci sequence: | input nip var count .
0 1
FOR count
over out |, | out . + swap
ENDFOR
. print
input
clear
Simpler
|Enter a number to obtain Fibonacci sequence: | input nip 1 - var count .
0 1
FOR count
out |, | out . dup rot +
ENDFOR
print
input /# wait until press ENTER #/
clear /# empties the stack #/
MUMPS
Iterative
FIBOITER(N)
;Iterative version to get the Nth Fibonacci number
;N must be a positive integer
;F is the tree containing the values
;I is a loop variable.
QUIT:(N\1'=N)!(N<0) "Error: "_N_" is not a positive integer."
NEW F,I
SET F(0)=0,F(1)=1
QUIT:N<2 F(N)
FOR I=2:1:N SET F(I)=F(I-1)+F(I-2)
QUIT F(N)
USER>W $$FIBOITER^ROSETTA(30)
832040
Nanoquery
Iterative
def fibIter($n)
if ($n < 2)
return $n
end if
$fib = 1
$fibPrev = 1
for ($num = 2) ($num < $n) ($num = $num+1)
$fib = ($fib + $fibPrev)
$fibPrev = ($fib - $fibPrev)
end for
return $fib
end fibIter
Nemerle
Recursive
using System;
using System.Console;
module Fibonacci
{
Fibonacci(x : long) : long
{
|x when x < 2 => x
|_ => Fibonacci(x - 1) + Fibonacci(x - 2)
}
Main() : void
{
def num = Int64.Parse(ReadLine());
foreach (n in $[0 .. num])
WriteLine("{0}: {1}", n, Fibonacci(n));
}
}
Tail Recursive
Fibonacci(x : long, current : long, next : long) : long
{
match(x)
{
|0 => current
|_ => Fibonacci(x - 1, next, current + next)
}
}
Fibonacci(x : long) : long
{
Fibonacci(x, 0, 1)
}
NESL
Recursive
function fib(n) = if n < 2 then n else fib(n - 2) + fib(n - 1);
NetRexx
/* NetRexx */
options replace format comments java crossref savelog symbols
numeric digits 210000 /*prepare for some big 'uns. */
parse arg x y . /*allow a single number or range.*/
if x == '' then do /*no input? Then assume -30-->+30*/
x = -30
y = -x
end
if y == '' then y = x /*if only one number, show fib(n)*/
loop k = x to y /*process each Fibonacci request.*/
q = fib(k)
w = q.length /*if wider than 25 bytes, tell it*/
say 'Fibonacci' k"="q
if w > 25 then say 'Fibonacci' k "has a length of" w
end k
exit
/*-------------------------------------FIB subroutine (non-recursive)---*/
method fib(arg) private static
parse arg n
na = n.abs
if na < 2 then return na /*handle special cases. */
a = 0
b = 1
loop j = 2 to na
s = a + b
a = b
b = s
end j
if n > 0 | na // 2 == 1 then return s /*if positive or odd negative... */
else return -s /*return a negative Fib number. */
NewLISP
Iterative
(define (fibonacci n)
(let (L '(0 1))
(dotimes (i n)
(setq L (list (L 1) (apply + L))))
(L 1)) )
Recursive
(define (fibonacci n)
(if (< n 2) 1
(+ (fibonacci (- n 1))
(fibonacci (- n 2)))))
Matrix multiplication
(define (fibonacci n)
(letn (f '((0 1) (1 1)) fib f)
(dotimes (i n)
(set 'fib (multiply fib f)))
(fib 0 1)) )
(print(fibonacci 10)) ;;89
NGS
Iterative
F fib(n:Int) {
n < 2 returns n
local a = 1, b = 1
# i is automatically local because of for()
for(i=2; i<n; i=i+1) {
local next = a + b
a = b
b = next
}
b
}
Nial
Iterative
On my machine, about 1.7s for 100,000 iterations, n=92. Maybe a few percent faster than iterative Python. Note that n>92 produces overflow; Python keeps going - single iteration with n=1,000,000 takes it about 15s.
fibi is op n {
if n<2 then
n
else
x1:=0; x2:=1;
for i with tell (n - 1) do
x:=x1+x2;
x1:=x2;
x2:=x;
endfor;
x2
endif};
Iterative using fold. Slightly faster, <1.6s:
fibf is op n {1 pick ((n- 1) fold [1 pick, +] 0 1)};
Tacit verion of above. Slightly faster still, <1.4s:
fibf2 is 1 pick fold [1 pick, +] reverse (0 1 hitch) (-1+);
Recursive
Really slow (over 8s for single iteration, n=33). (Similar to time for recursive python version with n=37.)
fibr is op n {fork [2>, +, + [fibr (-1 +), fibr (-2 +)]] n};
...or tacit version. More than twice as fast (?) but still slow:
, +, + [fibr2 (-1 +), fibr2 (-2 +)]];
Matrix
Matrix inner product (ip). This appears to be the fastest, about 1.0s for 100,000 iterations, n=92: Note that n>92 produces negative result.
fibm is op n {floor (0 1 pick (reduce ip (n reshape [2 2 reshape 1 1 1 0])))};
Could it look a little more like J? (Maybe 5% slower than above.)
$ is reshape;
~ is tr f op a b {b f a}; % Goes before verb, rather than after like in J;
_ is floor; % Not really J, but J-ish? (Cannot redefine "<.".);
fibm2 is _(0 1 pick reduce ip([2 2$1 1 1 0](~$)));
Alternate, not involving replicating matrix n times, but maybe 50% slower than the fastest matrix version above - similar speed to iterative:
fibm3 is op n {a:=2 2$1 1 1 0; _(0 1 pick ((n- 1) fold (a ip) a))};
Nim
Analytic
proc Fibonacci(n: int): int64 =
var fn = float64(n)
var p: float64 = (1.0 + sqrt(5.0)) / 2.0
var q: float64 = 1.0 / p
return int64((pow(p, fn) + pow(q, fn)) / sqrt(5.0))
Iterative
proc Fibonacci(n: int): int =
var
first = 0
second = 1
for i in 0 .. <n:
swap first, second
second += first
result = first
Recursive
proc Fibonacci(n: int): int64 =
if n <= 2:
result = 1
else:
result = Fibonacci(n - 1) + Fibonacci(n - 2)
===Tail-recursive===
proc Fibonacci(n: int, current: int64, next: int64): int64 =
if n == 0:
result = current
else:
result = Fibonacci(n - 1, next, current + next)
proc Fibonacci(n: int): int64 =
result = Fibonacci(n, 0, 1)
Continuations
iterator fib: int {.closure.} =
var a = 0
var b = 1
while true:
yield a
swap a, b
b = a + b
var f = fib
for i in 0.. <10:
echo f()
=={{header|Oberon-2}}==
MODULE Fibonacci;
IMPORT
Out := NPCT:Console;
PROCEDURE Fibs(VAR r: ARRAY OF LONGREAL);
VAR
i: LONGINT;
BEGIN
r[0] := 1.0; r[1] := 1.0;
FOR i := 2 TO LEN(r) - 1 DO
r[i] := r[i - 2] + r[i - 1];
END
END Fibs;
PROCEDURE FibsR(n: LONGREAL): LONGREAL;
BEGIN
IF n < 2. THEN
RETURN n
ELSE
RETURN FibsR(n - 1) + FibsR(n - 2)
END
END FibsR;
PROCEDURE Show(r: ARRAY OF LONGREAL);
VAR
i: LONGINT;
BEGIN
Out.String("First ");Out.Int(LEN(r),0);Out.String(" Fibonacci numbers");Out.Ln;
FOR i := 0 TO LEN(r) - 1 DO
Out.LongRealFix(r[i],8,0)
END;
Out.Ln
END Show;
PROCEDURE Gen(s: LONGINT);
VAR
x: POINTER TO ARRAY OF LONGREAL;
BEGIN
NEW(x,s);
Fibs(x^);
Show(x^)
END Gen;
PROCEDURE GenR(s: LONGINT);
VAR
i: LONGINT;
BEGIN
Out.String("First ");Out.Int(s,0);Out.String(" Fibonacci numbers (Recursive)");Out.Ln;
FOR i := 1 TO s DO
Out.LongRealFix(FibsR(i),8,0)
END;
Out.Ln
END GenR;
BEGIN
Gen(10);
Gen(20);
GenR(10);
GenR(20);
END Fibonacci.
First 10 Fibonacci numbers
1. 1. 2. 3. 5. 8. 13. 21. 34. 55.
First 20 Fibonacci numbers
1. 1. 2. 3. 5. 8. 13. 21. 34. 55. 89. 144. 233. 377. 610. 987. 1597. 2584. 4181. 6765.
First 10 Fibonacci numbers (Recursive)
1. 1. 2. 3. 5. 8. 13. 21. 34. 55.
First 20 Fibonacci numbers (Recursive)
1. 1. 2. 3. 5. 8. 13. 21. 34. 55. 89. 144. 233. 377. 610. 987. 1597. 2584. 4181. 6765.
Objeck
Recursive
bundle Default {
class Fib {
function : Main(args : String[]), Nil {
for(i := 0; i <= 10; i += 1;) {
Fib(i)->PrintLine();
};
}
function : native : Fib(n : Int), Int {
if(n < 2) {
return n;
};
return Fib(n-1) + Fib(n-2);
}
}
}
=={{header|Objective-C}}==
Recursive
-(long)fibonacci:(int)position
{
long result = 0;
if (position < 2) {
result = position;
} else {
result = [self fibonacci:(position -1)] + [self fibonacci:(position -2)];
}
return result;
}
Iterative
+(long)fibonacci:(int)index {
long beforeLast = 0, last = 1;
while (index > 0) {
last += beforeLast;
beforeLast = last - beforeLast;
--index;
}
return last;
}
OCaml
Iterative
let fib_iter n =
if n < 2 then
n
else let fib_prev = ref 1
and fib = ref 1 in
for num = 2 to n - 1 do
let temp = !fib in
fib := !fib + !fib_prev;
fib_prev := temp
done;
!fib
Recursive
let rec fib_rec n =
if n < 2 then
n
else
fib_rec (n - 1) + fib_rec (n - 2)
let rec fib = function
0 -> 0
| 1 -> 1
| n -> if n > 0 then fib (n-1) + fib (n-2)
else fib (n+2) - fib (n+1)
Arbitrary Precision
Using OCaml's [http://caml.inria.fr/pub/docs/manual-ocaml/libref/Num.html Num] module.
open Num
let fib =
let rec fib_aux f0 f1 = function
| 0 -> f0
| 1 -> f1
| n -> fib_aux f1 (f1 +/ f0) (n - 1)
in
fib_aux (num_of_int 0) (num_of_int 1)
(* support for negatives *)
let fib n =
if n < 0 && n mod 2 = 0 then minus_num (fib (abs n))
else fib (abs n)
;;
(* It can be called from the command line with an argument *)
(* Result is send to standart output *)
let n = int_of_string Sys.argv.(1) in
print_endline (string_of_num (fib n))
compile with: ocamlopt nums.cmxa -o fib fib.ml
Output:
$ ./fib 0
0
$ ./fib 10
55
$ ./fib 399
108788617463475645289761992289049744844995705477812699099751202749393926359816304226
$ ./fib -6
-8
=== O(log(n)) with arbitrary precision === This performs log2(N) matrix multiplys. Each multiplication is not constant-time but increases sub-linearly, about O(log(N)).
open Num
let mul (a,b,c) (d,e,f) = let bxe = b*/e in
(a*/d +/ bxe, a*/e +/ b*/f, bxe +/ c*/f)
let id = (Int 1, Int 0, Int 1)
let rec pow a n =
if n=0 then id else
let b = pow a (n/2) in
if (n mod 2) = 0 then mul b b else mul a (mul b b)
let fib n =
let (_,y,_) = (pow (Int 1, Int 1, Int 0) n) in
string_of_num y
;;
Printf.printf "fib %d = %s\n" 300 (fib 300)
Output:
fib 300 = 222232244629420445529739893461909967206666939096499764990979600
Octave
'''Recursive'''
% recursive
function fibo = recfibo(n)
if ( n < 2 )
fibo = n;
else
fibo = recfibo(n-1) + recfibo(n-2);
endif
endfunction
'''Iterative'''
% iterative
function fibo = iterfibo(n)
if ( n < 2 )
fibo = n;
else
f = zeros(2,1);
f(1) = 0;
f(2) = 1;
for i = 2 : n
t = f(2);
f(2) = f(1) + f(2);
f(1) = t;
endfor
fibo = f(2);
endif
endfunction
'''Testing'''
% testing
for i = 0 : 20
printf("%d %d\n", iterfibo(i), recfibo(i));
endfor
Oforth
: fib 0 1 rot #[ tuck + ] times drop ;
OPL
FIBON:
REM Fibonacci sequence is generated to the Organiser II floating point variable limit.
REM CLEAR/ON key quits.
REM Mikesan - http://forum.psion2.org/
LOCAL A,B,C
A=1 :B=1 :C=1
PRINT A,
DO
C=A+B
A=B
B=C
PRINT A,
UNTIL GET=1
Order
Recursive
#include <order/interpreter.h>
#define ORDER_PP_DEF_8fib_rec \
ORDER_PP_FN(8fn(8N, \
8if(8less(8N, 2), \
8N, \
8add(8fib_rec(8sub(8N, 1)), \
8fib_rec(8sub(8N, 2))))))
ORDER_PP(8fib_rec(10))
Tail recursive version (example supplied with language):
#include <order/interpreter.h>
#define ORDER_PP_DEF_8fib \
ORDER_PP_FN(8fn(8N, \
8fib_iter(8N, 0, 1)))
#define ORDER_PP_DEF_8fib_iter \
ORDER_PP_FN(8fn(8N, 8I, 8J, \
8if(8is_0(8N), \
8I, \
8fib_iter(8dec(8N), 8J, 8add(8I, 8J)))))
ORDER_PP(8to_lit(8fib(8nat(5,0,0))))
Memoization
#include <order/interpreter.h>
#define ORDER_PP_DEF_8fib_memo \
ORDER_PP_FN(8fn(8N, \
8tuple_at(0, 8fib_memo_inner(8N, 8seq))))
#define ORDER_PP_DEF_8fib_memo_inner \
ORDER_PP_FN(8fn(8N, 8M, \
8cond((8less(8N, 8seq_size(8M)), 8pair(8seq_at(8N, 8M), 8M)) \
(8equal(8N, 0), 8pair(0, 8seq(0))) \
(8equal(8N, 1), 8pair(1, 8seq(0, 1))) \
(8else, \
8lets((8S, 8fib_memo_inner(8sub(8N, 2), 8M)) \
(8T, 8fib_memo_inner(8dec(8N), 8tuple_at(1, 8S))) \
(8U, 8add(8tuple_at(0, 8S), 8tuple_at(0, 8T))), \
8pair(8U, \
8seq_append(8tuple_at(1, 8T), 8seq(8U))))))))
ORDER_PP(
8for_each_in_range(8fn(8N,
8print(8to_lit(8fib_memo(8N)) (,) 8space)),
1, 21)
)
Oz
Iterative
Using mutable references (cells).
fun{FibI N}
Temp = {NewCell 0}
A = {NewCell 0}
B = {NewCell 1}
in
for I in 1..N do
Temp := @A + @B
A := @B
B := @Temp
end
@A
end
Recursive
Inefficient (blows up the stack).
fun{FibR N}
if N < 2 then N
else {FibR N-1} + {FibR N-2}
end
end
===Tail-recursive=== Using accumulators.
fun{Fib N}
fun{Loop N A B}
if N == 0 then
B
else
{Loop N-1 A+B A}
end
end
in
{Loop N 1 0}
end
===Lazy-recursive===
declare
fun lazy {FiboSeq}
{LazyMap
{Iterate fun {$ [A B]} [B A+B] end [0 1]}
Head}
end
fun {Head A|_} A end
fun lazy {Iterate F I}
I|{Iterate F {F I}}
end
fun lazy {LazyMap Xs F}
case Xs of X|Xr then {F X}|{LazyMap Xr F}
[] nil then nil
end
end
in
{Show {List.take {FiboSeq} 8}}
PARI/GP
===Built-in===
fibonocci(n)
Matrix
fib(n)=([1,1;1,0]^n)[1,2]
Analytic
This uses the Binet form.
fib(n)=my(phi=(1+sqrt(5))/2);round((phi^n-phi^-n)/sqrt(5))
The second term can be dropped since the error is always small enough to be subsumed by the rounding.
fib(n)=round(((1+sqrt(5))/2)^n/sqrt(5))
Algebraic
This is an exact version of the above formula. quadgen(5) represents and the number is stored in the form . imag takes the coefficient of . This uses the relation
:
and hence real(quadgen(5)^n) would give the (n-1)-th Fibonacci number.
fib(n)=imag(quadgen(5)^n)
A more direct translation (note that ) would be
fib(n)=my(phi=quadgen(5));(phi^n-(-1/phi)^n)/(2*phi-1)
Combinatorial
This uses the generating function. It can be trivially adapted to give the first n Fibonacci numbers.
fib(n)=polcoeff(x/(1-x-x^2)+O(x^(n+1)),n)
Binary powering
fib(n)={
if(n<=0,
if(n,(-1)^(n+1)*fib(n),0)
,
my(v=lucas(n-1));
(2*v[1]+v[2])/5
)
};
lucas(n)={
if (!n, return([2,1]));
my(v=lucas(n >> 1), z=v[1], t=v[2], pr=v[1]*v[2]);
n=n%4;
if(n%2,
if(n==3,[v[1]*v[2]+1,v[2]^2-2],[v[1]*v[2]-1,v[2]^2+2])
,
if(n,[v[1]^2+2,v[1]*v[2]+1],[v[1]^2-2,v[1]*v[2]-1])
)
};
Recursive
fib(n)={
if(n<2,
n
,
fib(n-1)+fib(n)
)
};
Anonymous recursion
This uses self() which gives a self-reference.
fib(n)={
if(n<2,
n
,
my(s=self());
s(n-2)+s(n-1)
)
};
It can be used without being named:
apply(n->if(n<2,n,my(s=self());s(n-2)+s(n-1)), [1..10])
gives
%1 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
Memoization
F=[];
fib(n)={
if(n>#F,
F=concat(F, vector(n-#F));
F[n]=fib(n-1)+fib(n-2)
,
if(n<2,
n
,
if(F[n],F[n],F[n]=fib(n-1)+fib(n-2))
)
);
}
Iterative
fib(n)={
if(n<0,return((-1)^(n+1)*fib(n)));
my(a=0,b=1,t);
while(n,
t=a+b;
a=b;
b=t;
n--
);
a
};
Chebyshev
This solution uses Chebyshev polynomials of the second kind (Chyebyshev U-polynomials).
fib(n)=n--;polchebyshev(n,2,I/2)*I^n;
or
fib(n)=abs(polchebyshev(n-1,2,I/2));
===Anti-Hadamard matrix=== All n×n [https://en.wikipedia.org/wiki/Logical_matrix (0,1)] lower [https://en.wikipedia.org/wiki/Hessenberg_matrix Hessenberg matrices] have determinant at most F(n). The n×n anti-Hadamard matrix R. L. Graham and N. J. A. Sloane, [http://www.math.ucsd.edu/~ronspubs/84_03_anti_hadamard.pdf Anti-Hadamard matrices], Linear Algebra Appl. 62 (1984), 113–137. matches this upper bound, and hence can be used as an inefficient method for computing Fibonacci numbers of positive index. These matrices are the same as Matlab's type-3 "Dramadah" matrices, following a naming suggestion of C. L. Mallows according to Graham & Sloane.
matantihadamard(n)={
matrix(n,n,i,j,
my(t=j-i+1);
if(t<1,t%2,t<3)
);
}
fib(n)=matdet(matantihadamard(n))
Testing adjacent bits
The Fibonacci numbers can be characterized (for n > 0) as the number of n-bit strings starting and ending with 1 without adjacent 0s. This inefficient, exponential-time algorithm demonstrates:
fib(n)=
{
my(g=2^(n+1)-1);
sum(i=2^(n-1),2^n-1,
bitor(i,i<<1)==g
);
}
===One-by-one=== This code is purely for amusement and requires n > 1. It tests numbers in order to see if they are Fibonacci numbers, and waits until it has seen ''n'' of them.
fib(n)=my(k=0);while(n--,k++;while(!issquare(5*k^2+4)&&!issquare(5*k^2-4),k++));k
Pascal
Analytic
function fib(n: integer):longInt;
const
Sqrt5 = sqrt(5.0);
C1 = ln((Sqrt5+1.0)*0.5);//ln( 1.618..)
//C2 = ln((1.0-Sqrt5)*0.5);//ln(-0.618 )) tsetsetse
C2 = ln((Sqrt5-1.0)*0.5);//ln(+0.618 ))
begin
IF n>0 then
begin
IF odd(n) then
fib := round((exp(C1*n) + exp(C2*n) )/Sqrt5)
else
fib := round((exp(C1*n) - exp(C2*n) )/Sqrt5)
end
else
Fibdirekt := 0
end;
Recursive
function fib(n: integer): integer;
begin
if (n = 0) or (n = 1)
then
fib := n
else
fib := fib(n-1) + fib(n-2)
end;
Iterative
function fib(n: integer): integer;
var
f0, f1, tmpf0, k: integer;
begin
f1 := n;
IF f1 >1 then
begin
k := f1-1;
f0 := 0;
f1 := 1;
repeat
tmpf0 := f0;
f0 := f1;
f1 := f1+tmpf0;
dec(k);
until k = 0;
end
else
IF f1 < 0 then
f1 := 0;
fib := f1;
end;
Analytic2
function FiboMax(n: integer):Extended; //maXbox
begin
result:= (pow((1+SQRT5)/2,n)-pow((1-SQRT5)/2,n))/SQRT5
end;
function Fibo_BigInt(n: integer): string; //maXbox
var tbig1, tbig2, tbig3: TInteger;
begin
result:= '0'
tbig1:= TInteger.create(1); //temp
tbig2:= TInteger.create(0); //result (a)
tbig3:= Tinteger.create(1); //b
for it:= 1 to n do begin
tbig1.assign(tbig2)
tbig2.assign(tbig3);
tbig1.add(tbig3);
tbig3.assign(tbig1);
end;
result:= tbig2.toString(false)
tbig3.free;
tbig2.free;
tbig1.free;
end;
writeln(floattoStr(FiboMax(555)))
4.3516638122555E115
writeln(Fibo_BigInt(555))
43516638122555047989641805373140394725407202037260729735885664398655775748034950972577909265605502785297675867877570
Perl
Iterative
sub fib_iter {
my $n = shift;
use bigint try => "GMP,Pari";
my ($v2,$v1) = (-1,1);
($v2,$v1) = ($v1,$v2+$v1) for 0..$n;
$v1;
}
Recursive
sub fibRec {
my $n = shift;
$n < 2 ? $n : fibRec($n - 1) + fibRec($n - 2);
}
Modules
Quite a few modules have ways to do this. Performance is not typically an issue with any of these until 100k or so. With GMP available, the first three are ''much'' faster at large values.
# Uses GMP method so very fast
use Math::AnyNum qw/fibonacci/;
say fibonacci(10000);
# Uses GMP method, so also very fast
use Math::GMP;
say Math::GMP::fibonacci(10000);
# Binary ladder, GMP if available, Pure Perl otherwise
use ntheory qw/lucasu/;
say lucasu(1, -1, 10000);
# All Perl
use Math::NumSeq::Fibonacci;
my $seq = Math::NumSeq::Fibonacci->new;
say $seq->ith(10000);
# All Perl
use Math::Big qw/fibonacci/;
say 0+fibonacci(10000); # Force scalar context
# Perl, gives floating point *approximation*
use Math::Fibonacci qw/term/;
say term(10000);
Perl 6
List Generator
This constructs the fibonacci sequence as a lazy infinite list.
constant @fib = 0, 1, *+* ... *;
If you really need a function for it:
sub fib ($n) { @fib[$n] }
To support negative indices:
constant @neg-fib = 0, 1, *-* ... *;
sub fib ($n) { $n >= 0 ?? @fib[$n] !! @neg-fib[-$n] }
Iterative
sub fib (Int $n --> Int) {
$n > 1 or return $n;
my ($prev, $this) = 0, 1;
($prev, $this) = $this, $this + $prev for 1 ..^ $n;
return $this;
}
Recursive
proto fib (Int $n --> Int) {*}
multi fib (0) { 0 }
multi fib (1) { 1 }
multi fib ($n) { fib($n - 1) + fib($n - 2) }
Analytic
sub fib (Int $n --> Int) {
constant φ1 = 1 / constant φ = (1 + sqrt 5)/2;
constant invsqrt5 = 1 / sqrt 5;
floor invsqrt5 * (φ**$n + φ1**$n);
}
Phix
function fibonacci(integer n) -- iterative, works for -ve numbers
atom a=0, b=1
if n=0 then return 0 end if
if abs(n)>=79 then ?9/0 end if -- inaccuracies creep in above 78
for i=1 to abs(n)-1 do
{a,b} = {b,a+b}
end for
if n<0 and remainder(n,2)=0 then return -fcache[absn] end if
return fcache[absn]
end function
for i=0 to 28 do
if i then puts(1,", ") end if
printf(1,"%d", fibonacci(i))
end for
puts(1,"\n")
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811
Using native integers/atoms, errors creep in above 78, so the same program converted to use mpfr:
-- demo\rosetta\fibonacci.exw
include mpfr.e
mpz res = NULL, prev, next
integer lastn
atom t0 = time()
function fibonampz(integer n) -- resumable, works for -ve numbers, yields mpz
integer absn = abs(n)
if res=NULL or absn!=abs(lastn)+1 then
if res=NULL then
prev = mpz_init(0)
res = mpz_init(1)
next = mpz_init()
else
if n==lastn then return res end if
end if
mpz_fib2_ui(res,prev,absn)
else
if lastn<0 and remainder(lastn,2)=0 then
mpz_mul_si(res,res,-1)
end if
mpz_add(next,res,prev)
{prev,res,next} = {res,next,prev}
end if
if n<0 and remainder(n,2)=0 then
mpz_mul_si(res,res,-1)
end if
lastn = n
return res
end function
for i=0 to 28 do
if i then puts(1,", ") end if
printf(1,"%s", {mpz_get_str(fibonampz(i))})
end for
puts(1,"\n")
printf(1,"%s\n", {mpz_get_str(fibonampz(705))})
string s = mpz_get_str(fibonampz(4784969))
integer l = length(s)
s[40..-40] = "..."
?{l,s}
?elapsed(time()-t0)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811
970066202977562212558683426760773016559904631977220423547980211057068777324159443678590358026859129109599109446646966713225742014317926940054191330
{1000000,"107273956418004772293648135962250043219...407167474856539211500699706378405156269"}
"2.1s"
PHP
Iterative
function fibIter($n) {
if ($n < 2) {
return $n;
}
$fibPrev = 0;
$fib = 1;
foreach (range(1, $n-1) as $i) {
list($fibPrev, $fib) = array($fib, $fib + $fibPrev);
}
return $fib;
}
Recursive
function fibRec($n) {
return $n < 2 ? $n : fibRec($n-1) + fibRec($n-2);
}
PicoLisp
Recursive
(de fibo (N)
(if (>= 2 N)
1
(+ (fibo (dec N)) (fibo (- N 2))) ) )
Recursive with Cache
Using a recursive version doesn't need to be slow, as the following shows:
(de fibo (N)
(cache '(NIL) N # Use a cache to accelerate
(if (>= 2 N)
N
(+ (fibo (dec N)) (fibo (- N 2))) ) ) )
(bench (fibo 1000))
Output:
0.012 sec
-> 43466557686937456435688527675040625802564660517371780402481729089536555417949
05189040387984007925516929592259308032263477520968962323987332247116164299644090
6533187938298969649928516003704476137795166849228875
Iterative
(de fib (N)
(let (A 0 B 1)
(do N
(prog1 B (setq B (+ A B) A @)) ) ) )
Coroutines
(co 'fibo
(let (A 0 B 1)
(yield 'ready)
(while
(yield
(swap 'B (+ (swap 'A B) B)) ) ) ) )
(do 15
(printsp (yield 'next 'fibo)) )
(prinl)
(yield NIL 'fibo)
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
PIR
Recursive:
.sub fib
.param int n
.local int nt
.local int ft
if n < 2 goto RETURNN
nt = n - 1
ft = fib( nt )
dec nt
nt = fib(nt)
ft = ft + nt
.return( ft )
RETURNN:
.return( n )
end
.end
.sub main :main
.local int counter
.local int f
counter=0
LOOP:
if counter > 20 goto DONE
f = fib(counter)
print f
print "\n"
inc counter
goto LOOP
DONE:
end
.end
Iterative (stack-based):
.sub fib
.param int n
.local int counter
.local int f
.local pmc fibs
.local int nmo
.local int nmt
fibs = new 'ResizableIntegerArray'
if n == 0 goto RETURN0
if n == 1 goto RETURN1
push fibs, 0
push fibs, 1
counter = 2
FIBLOOP:
if counter > n goto DONE
nmo = pop fibs
nmt = pop fibs
f = nmo + nmt
push fibs, nmt
push fibs, nmo
push fibs, f
inc counter
goto FIBLOOP
RETURN0:
.return( 0 )
end
RETURN1:
.return( 1 )
end
DONE:
f = pop fibs
.return( f )
end
.end
.sub main :main
.local int counter
.local int f
counter=0
LOOP:
if counter > 20 goto DONE
f = fib(counter)
print f
print "\n"
inc counter
goto LOOP
DONE:
end
.end
Pike
Iterative
int
fibIter(int n) {
int fibPrev, fib, i;
if (n < 2) {
return 1;
}
fibPrev = 0;
fib = 1;
for (i = 1; i < n; i++) {
int oldFib = fib;
fib += fibPrev;
fibPrev = oldFib;
}
return fib;
}
Recursive
int
fibRec(int n) {
if (n < 2) {
return(1);
}
return( fib(n-2) + fib(n-1) );
}
PL/I
/* Form the n-th Fibonacci number, n > 1. */
get list(n);
f1 = 0; f2 = 1;
do i = 2 to n;
f3 = f1 + f2;
put skip edit('fibo(',i,')=',f3)(a,f(5),a,f(5));
f1 = f2;
f2 = f3;
end;
PL/pgSQL
Recursive
CREATE OR REPLACE FUNCTION fib(n INTEGER) RETURNS INTEGER AS $$
BEGIN
IF (n < 2) THEN
RETURN n;
END IF;
RETURN fib(n - 1) + fib(n - 2);
END;
$$ LANGUAGE plpgsql;
Calculated
CREATE OR REPLACE FUNCTION fibFormula(n INTEGER) RETURNS INTEGER AS $$
BEGIN
RETURN round(pow((pow(5, .5) + 1) / 2, n) / pow(5, .5));
END;
$$ LANGUAGE plpgsql;
Linear
CREATE OR REPLACE FUNCTION fibLinear(n INTEGER) RETURNS INTEGER AS $$
DECLARE
prevFib INTEGER := 0;
fib INTEGER := 1;
BEGIN
IF (n < 2) THEN
RETURN n;
END IF;
WHILE n > 1 LOOP
SELECT fib, prevFib + fib INTO prevFib, fib;
n := n - 1;
END LOOP;
RETURN fib;
END;
$$ LANGUAGE plpgsql;
Tail recursive
CREATE OR REPLACE FUNCTION fibTailRecursive(n INTEGER, prevFib INTEGER DEFAULT 0, fib INTEGER DEFAULT 1)
RETURNS INTEGER AS $$
BEGIN
IF (n = 0) THEN
RETURN prevFib;
END IF;
RETURN fibTailRecursive(n - 1, fib, prevFib + fib);
END;
$$ LANGUAGE plpgsql;
PL/SQL
Create or replace Function fnu_fibonnaci(p_iNumber integer)
return integer
is
nuFib integer;
nuP integer;
nuQ integer;
Begin
if p_iNumber is not null then
if p_iNumber=0 then
nuFib:=0;
Elsif p_iNumber=1 then
nuFib:=1;
Else
nuP:=0;
nuQ:=1;
For nuI in 2..p_iNumber loop
nuFib:=nuP+nuQ;
nuP:=nuQ;
nuQ:=nuFib;
End loop;
End if;
End if;
return(nuFib);
End fnu_fibonnaci;
Pop11
define fib(x);
lvars a , b;
1 -> a;
1 -> b;
repeat x - 1 times
(a + b, b) -> (b, a);
endrepeat;
a;
enddefine;
PostScript
Enter the desired number for "n" and run through your favorite postscript previewer or send to your postscript printer:
%!PS
% We want the 'n'th fibonacci number
/n 13 def
% Prepare output canvas:
/Helvetica findfont 20 scalefont setfont
100 100 moveto
%define the function recursively:
/fib { dup
3 lt
{ pop 1 }
{ dup 1 sub fib exch 2 sub fib add }
ifelse
} def
(Fib\() show n (....) cvs show (\)=) show n fib (.....) cvs show
showpage
Potion
Recursive
Starts with int and upgrades on-the-fly to doubles.
recursive = (n):
if (n <= 1): 1. else: recursive (n - 1) + recursive (n - 2)..
n = 40
("fib(", n, ")= ", recursive (n), "\n") join print
recursive(40)= 165580141
real 0m2.851s
Iterative
iterative = (n) :
curr = 0
prev = 1
tmp = 0
n times:
tmp = curr
curr = curr + prev
prev = tmp
.
curr
.
Matrix based
sqr = (x): x * x.
# Based on the fact that
# F2n = Fn(2Fn+1 - Fn)
# F2n+1 = Fn ^2 + Fn+1 ^2
matrix = (n) :
algorithm = (n) :
"computes (Fn, Fn+1)"
if (n < 2): return ((0, 1), (1, 1)) at(n).
# n = e + {0, 1}
q = algorithm(n / 2) # q = (Fe/2, Fe/2+1)
q = (q(0) * (2 * q(1) - q(0)), sqr(q(0)) + sqr(q(1))) # q => (Fe, Fe+1)
if (n % 2 == 1) : # q => (Fe+{0, 1}, Fe+1+{0,1}) = (Fn, Fn+1)
q = (q(1), q(1) + q(0))
.
q
.
algorithm(n)(0)
.
Handling negative values
## PowerBASIC
There seems to be a limitation (dare I say, bug?) in PowerBASIC regarding how large numbers are stored. 10E17 and larger get rounded to the nearest 10. For F(n), where ABS(n) > 87, is affected like this:
actual: displayed:
F(88) 1100087778366101931 1100087778366101930
F(89) 1779979416004714189 1779979416004714190
F(90) 2880067194370816120 2880067194370816120
F(91) 4660046610375530309 4660046610375530310
F(92) 7540113804746346429 7540113804746346430
```powerbasic
FUNCTION fibonacci (n AS LONG) AS QUAD
DIM u AS LONG, a AS LONG, L0 AS LONG, outP AS QUAD
STATIC fibNum() AS QUAD
u = UBOUND(fibNum)
a = ABS(n)
IF u < 1 THEN
REDIM fibNum(1)
fibNum(1) = 1
u = 1
END IF
SELECT CASE a
CASE 0 TO 92
IF a > u THEN
REDIM PRESERVE fibNum(a)
FOR L0 = u + 1 TO a
fibNum(L0) = fibNum(L0 - 1) + fibNum(L0 - 2)
IF 88 = L0 THEN fibNum(88) = fibNum(88) + 1
NEXT
END IF
IF n < 0 THEN
fibonacci = fibNum(a) * ((-1)^(a+1))
ELSE
fibonacci = fibNum(a)
END IF
CASE ELSE
'Even without the above-mentioned bug, we're still limited to
'F(+/-92), due to data type limits. (F(93) = &hA94F AD42 221F 2702)
ERROR 6
END SELECT
END FUNCTION
FUNCTION PBMAIN () AS LONG
DIM n AS LONG
#IF NOT %DEF(%PB_CC32)
OPEN "out.txt" FOR OUTPUT AS 1
#ENDIF
FOR n = -92 TO 92
#IF %DEF(%PB_CC32)
PRINT STR$(n); ": "; FORMAT$(fibonacci(n), "#")
#ELSE
PRINT #1, STR$(n) & ": " & FORMAT$(fibonacci(n), "#")
#ENDIF
NEXT
CLOSE
END FUNCTION
PowerShell
Iterative
function FibonacciNumber ( $count )
{
$answer = @(0,1)
while ($answer.Length -le $count)
{
$answer += $answer[-1] + $answer[-2]
}
return $answer
}
An even shorter version that eschews function calls altogether:
$count = 8
$answer = @(0,1)
0..($count - $answer.Length) | Foreach { $answer += $answer[-1] + $answer[-2] }
$answer
Recursive
function fib($n) {
switch ($n) {
0 { return 0 }
1 { return 1 }
{ $_ -lt 0 } { return [Math]::Pow(-1, -$n + 1) * (fib (-$n)) }
default { return (fib ($n - 1)) + (fib ($n - 2)) }
}
}
Processing
void setup() {
size(400, 400);
fill(255, 64);
frameRate(2);
}
void draw() {
int num = fibonacciNum(frameCount);
println(frameCount, num);
rect(0,0,num, num);
if(frameCount==14) frameCount = -1; // restart
}
int fibonacciNum(int n) {
return (n < 2) ? n : fibonacciNum(n - 1) + fibonacciNum(n - 2);
}
On the nth frame, the nth Fibonacci number is printed to the console and a square of that size is drawn on the sketch surface. The sketch restarts to keep drawing within the window size.
1
1
2
3
5
8
13
21
34
55
89
144
233
377
Prolog
fib(1, 1) :- !.
fib(0, 0) :- !.
fib(N, Value) :-
A is N - 1, fib(A, A1),
B is N - 2, fib(B, B1),
Value is A1 + B1.
This naive implementation works, but is very slow for larger values of N. Here are some simple measurements (in SWI-Prolog):
?- time(fib(0,F)).
% 2 inferences, 0.000 CPU in 0.000 seconds (88% CPU, 161943 Lips)
F = 0.
?- time(fib(10,F)).
% 265 inferences, 0.000 CPU in 0.000 seconds (98% CPU, 1458135 Lips)
F = 55.
?- time(fib(20,F)).
% 32,836 inferences, 0.016 CPU in 0.016 seconds (99% CPU, 2086352 Lips)
F = 6765.
?- time(fib(30,F)).
% 4,038,805 inferences, 1.122 CPU in 1.139 seconds (98% CPU, 3599899 Lips)
F = 832040.
?- time(fib(40,F)).
% 496,740,421 inferences, 138.705 CPU in 140.206 seconds (99% CPU, 3581264 Lips)
F = 102334155.
As you can see, the calculation time goes up exponentially as N goes higher.
===Poor man's memoization=== The performance problem can be readily fixed by the addition of two lines of code (the first and last in this version):
%:- dynamic fib/2. % This is ISO, but GNU doesn't like it.
:- dynamic(fib/2). % Not ISO, but works in SWI, YAP and GNU unlike the ISO declaration.
fib(1, 1) :- !.
fib(0, 0) :- !.
fib(N, Value) :-
A is N - 1, fib(A, A1),
B is N - 2, fib(B, B1),
Value is A1 + B1,
asserta((fib(N, Value) :- !)).
Let's take a look at the execution costs now:
?- time(fib(0,F)).
% 2 inferences, 0.000 CPU in 0.000 seconds (90% CPU, 160591 Lips)
F = 0.
?- time(fib(10,F)).
% 37 inferences, 0.000 CPU in 0.000 seconds (96% CPU, 552610 Lips)
F = 55.
?- time(fib(20,F)).
% 41 inferences, 0.000 CPU in 0.000 seconds (96% CPU, 541233 Lips)
F = 6765.
?- time(fib(30,F)).
% 41 inferences, 0.000 CPU in 0.000 seconds (95% CPU, 722722 Lips)
F = 832040.
?- time(fib(40,F)).
% 41 inferences, 0.000 CPU in 0.000 seconds (96% CPU, 543572 Lips)
F = 102334155.
In this case by asserting the new N,Value pairing as a rule in the database we're making the classic time/space tradeoff. Since the space costs are (roughly) linear by N and the time costs are exponential by N, the trade-off is desirable. You can see the poor man's memoizing easily:
?- listing(fib).
:- dynamic fib/2.
fib(40, 102334155) :- !.
fib(39, 63245986) :- !.
fib(38, 39088169) :- !.
fib(37, 24157817) :- !.
fib(36, 14930352) :- !.
fib(35, 9227465) :- !.
fib(34, 5702887) :- !.
fib(33, 3524578) :- !.
fib(32, 2178309) :- !.
fib(31, 1346269) :- !.
fib(30, 832040) :- !.
fib(29, 514229) :- !.
fib(28, 317811) :- !.
fib(27, 196418) :- !.
fib(26, 121393) :- !.
fib(25, 75025) :- !.
fib(24, 46368) :- !.
fib(23, 28657) :- !.
fib(22, 17711) :- !.
fib(21, 10946) :- !.
fib(20, 6765) :- !.
fib(19, 4181) :- !.
fib(18, 2584) :- !.
fib(17, 1597) :- !.
fib(16, 987) :- !.
fib(15, 610) :- !.
fib(14, 377) :- !.
fib(13, 233) :- !.
fib(12, 144) :- !.
fib(11, 89) :- !.
fib(10, 55) :- !.
fib(9, 34) :- !.
fib(8, 21) :- !.
fib(7, 13) :- !.
fib(6, 8) :- !.
fib(5, 5) :- !.
fib(4, 3) :- !.
fib(3, 2) :- !.
fib(2, 1) :- !.
fib(1, 1) :- !.
fib(0, 0) :- !.
fib(A, D) :-
B is A+ -1,
fib(B, E),
C is A+ -2,
fib(C, F),
D is E+F,
asserta((fib(A, D):-!)).
All of the interim N/Value pairs have been asserted as facts for quicker future use, speeding up the generation of the higher Fibonacci numbers.
Continuation passing style
Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl
:- use_module(lambda).
fib(N, FN) :-
cont_fib(N, _, FN, \_^Y^_^U^(U = Y)).
cont_fib(N, FN1, FN, Pred) :-
( N < 2 ->
call(Pred, 0, 1, FN1, FN)
;
N1 is N - 1,
P = \X^Y^Y^U^(U is X + Y),
cont_fib(N1, FNA, FNB, P),
call(Pred, FNA, FNB, FN1, FN)
).
With lazy lists
Works with SWI-Prolog and others that support freeze/2.
fib([0,1|X]) :-
ffib(0,1,X).
ffib(A,B,X) :-
freeze(X, (C is A+B, X=[C|Y], ffib(B,C,Y)) ).
The predicate fib(Xs) unifies Xs with an infinite list whose values are the Fibonacci sequence. The list can be used like this:
?- fib(X), length(A,15), append(A,_,X), writeln(A).
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]
Generators idiom
take( 0, Next, Z-Z, Next).
take( N, Next, [A|B]-Z, NZ):- N>0, !, next( Next, A, Next1),
N1 is N-1,
take( N1, Next1, B-Z, NZ).
next( fib(A,B), A, fib(B,C)):- C is A+B.
%% usage: ?- take(15, fib(0,1), _X-[], G), writeln(_X).
%% [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]
%% G = fib(610, 987)
Yet another implementation
One of my favorites; loosely similar to the first example, but without the performance penalty, and needs nothing special to implement. Not even a dynamic database predicate. Attributed to M.E. for the moment, but simply because I didn't bother to search for the many people who probably did it like this long before I did. If someone knows who came up with it first, please let us know.
% Fibonacci sequence generator
fib(C, [P,S], C, N) :- N is P + S.
fib(C, [P,S], Cv, V) :- succ(C, Cn), N is P + S, !, fib(Cn, [S,N], Cv, V).
fib(0, 0).
fib(1, 1).
fib(C, N) :- fib(2, [0,1], C, N). % Generate from 3rd sequence on
Looking at performance:
?- time(fib(30,X)).
% 86 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 832040
?- time(fib(40,X)).
% 116 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 102334155
?- time(fib(100,X)).
% 296 inferences, 0.000 CPU in 0.001 seconds (0% CPU, Infinite Lips)
X = 354224848179261915075
What I really like about this one, is it is also a generator- i.e. capable of generating all the numbers in sequence needing no bound input variables or special Prolog predicate support (such as freeze/3 in the previous example):
?- time(fib(X,Fib)).
% 0 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = Fib, Fib = 0 ;
% 1 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = Fib, Fib = 1 ;
% 3 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 2,
Fib = 1 ;
% 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 3,
Fib = 2 ;
% 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 4,
Fib = 3 ;
% 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = Fib, Fib = 5 ;
% 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 6,
Fib = 8
...etc.
It stays at 5 inferences per iteration after X=3. Also, quite useful:
?- time(fib(100,354224848179261915075)).
% 296 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
true .
?- time(fib(X,354224848179261915075)).
% 394 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
X = 100 .
Pure
Tail Recursive
fib n = loop 0 1 n with
loop a b n = if n==0 then a else loop b (a+b) (n-1);
end;
PureBasic
Macro based calculation
Macro Fibonacci (n)
Int((Pow(((1+Sqr(5))/2),n)-Pow(((1-Sqr(5))/2),n))/Sqr(5))
EndMacro
Recursive
Procedure FibonacciReq(n)
If n<2
ProcedureReturn n
Else
ProcedureReturn FibonacciReq(n-1)+FibonacciReq(n-2)
EndIf
EndProcedure
===Recursive & optimized with a static hash table=== This will be much faster on larger n's, this as it uses a table to store known parts instead of recalculating them. On my machine the speedup compares to above code is Fib(n) Speedup 20 2 25 23 30 217 40 25847 46 1156741
Procedure Fibonacci(n)
Static NewMap Fib.i()
Protected FirstRecursion
If MapSize(Fib())= 0 ; Init the hash table the first run
Fib("0")=0: Fib("1")=1
FirstRecursion = #True
EndIf
If n >= 2
Protected.s s=Str(n)
If Not FindMapElement(Fib(),s) ; Calculate only needed parts
Fib(s)= Fibonacci(n-1)+Fibonacci(n-2)
EndIf
n = Fib(s)
EndIf
If FirstRecursion ; Free the memory when finalizing the first call
ClearMap(Fib())
EndIf
ProcedureReturn n
EndProcedure
'''Example''' Fibonacci(0)= 0 Fibonacci(1)= 1 Fibonacci(2)= 1 Fibonacci(3)= 2 Fibonacci(4)= 3 Fibonacci(5)= 5
FibonacciReq(0)= 0 FibonacciReq(1)= 1 FibonacciReq(2)= 1 FibonacciReq(3)= 2 FibonacciReq(4)= 3 FibonacciReq(5)= 5
Purity
The following takes a natural number and generates an initial segment of the Fibonacci sequence of that length:
data Fib1 = FoldNat
<
const (Cons One (Cons One Empty)),
(uncurry Cons) . ((uncurry Add) . (Head, Head . Tail), id)
>
This following calculates the Fibonacci sequence as an infinite stream of natural numbers:
type (Stream A?,,,Unfold) = gfix X. A? . X?
data Fib2 = Unfold ((outl, (uncurry Add, outl))) ((curry id) One One)
As a histomorphism:
import Histo
data Fib3 = Histo . Memoize
<
const One,
(p1 =>
<
const One,
(p2 => Add (outl $p1) (outl $p2)). UnmakeCofree
> (outr $p1)) . UnmakeCofree
>
Python
Iterative positive and negative
def fib(n,x=[0,1]):
for i in range(abs(n)-1): x=[x[1],sum(x)]
return x[1]*pow(-1,abs(n)-1) if n<0 else x[1] if n else 0
for i in range(-30,31): print fib(i),
Output:
-832040 514229 -317811 196418 -121393 75025 -46368 28657 -17711 10946 -6765 4181 -2584 1597 -987
610 -377 233 -144 89 -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987
1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
Analytic
Binet's formula:
from math import *
def analytic_fibonacci(n):
sqrt_5 = sqrt(5);
p = (1 + sqrt_5) / 2;
q = 1/p;
return int( (p**n + q**n) / sqrt_5 + 0.5 )
for i in range(1,31):
print analytic_fibonacci(i),
Output:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
Iterative
def fibIter(n):
if n < 2:
return n
fibPrev = 1
fib = 1
for num in xrange(2, n):
fibPrev, fib = fib, fib + fibPrev
return fib
Recursive
def fibRec(n):
if n < 2:
return n
else:
return fibRec(n-1) + fibRec(n-2)
Recursive with Memoization
def fibMemo():
pad = {0:0, 1:1}
def func(n):
if n not in pad:
pad[n] = func(n-1) + func(n-2)
return pad[n]
return func
fm = fibMemo()
for i in range(1,31):
print fm(i),
Output:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040
===Better Recursive doesn't need Memoization===
The recursive code as written two sections above is incredibly slow and inefficient due to the nested recursion calls. Although the memoization above makes the code run faster, it is at the cost of extra memory use. The below code is syntactically recursive but actually encodes the efficient iterative process, and thus doesn't require memoization:
def fibFastRec(n):
def fib(prvprv, prv, c):
if c < 1:
return prvprv
else:
return fib(prv, prvprv + prv, c - 1)
return fib(0, 1, n)
However, although much faster and not requiring memory, the above code can only work to a limited 'n' due to the limit on stack recursion depth by Python; it is better to use the iterative code above or the generative one below.
Generative
def fibGen(n):
a, b = 0, 1
while n>0:
yield a
a, b, n = b, a+b, n-1
=Example use=
>>> [i for i in fibGen(11)]
[0,1,1,2,3,5,8,13,21,34,55]
===Matrix-Based=== Translation of the matrix-based approach used in F#.
def prevPowTwo(n):
'Gets the power of two that is less than or equal to the given input'
if ((n & -n) == n):
return n
else:
n -= 1
n |= n >> 1
n |= n >> 2
n |= n >> 4
n |= n >> 8
n |= n >> 16
n += 1
return (n/2)
def crazyFib(n):
'Crazy fast fibonacci number calculation'
powTwo = prevPowTwo(n)
q = r = i = 1
s = 0
while(i < powTwo):
i *= 2
q, r, s = q*q + r*r, r * (q + s), (r*r + s*s)
while(i < n):
i += 1
q, r, s = q+r, q, r
return q
Large step recurrence
This is much faster for a single, large value of n:
def fib(n, c={0:1, 1:1}):
if n not in c:
x = n // 2
c[n] = fib(x-1) * fib(n-x-1) + fib(x) * fib(n - x)
return c[n]
fib(10000000) # calculating it takes a few seconds, printing it takes eons
Same as above but slightly faster
Putting the dictionary outside the function makes this about 2 seconds faster, could just make a wrapper:
F = {0: 0, 1: 1, 2: 1}
def fib(n):
if n in F:
return F[n]
f1 = fib(n // 2 + 1)
f2 = fib((n - 1) // 2)
F[n] = (f1 * f1 + f2 * f2 if n & 1 else f1 * f1 - f2 * f2)
return F[n]
Generative with Recursion
This can get very slow and uses a lot of memory. Can be sped up by caching the generator results.
def fib():
"""Yield fib[n+1] + fib[n]"""
yield 1 # have to start somewhere
lhs, rhs = fib(), fib()
yield next(lhs) # move lhs one iteration ahead
while True:
yield next(lhs)+next(rhs)
f=fib()
print [next(f) for _ in range(9)]
Output:
[1, 1, 2, 3, 5, 8, 13, 21, 34]
'''Another version of recursive generators solution, starting from 0'''
from itertools import islice
def fib():
yield 0
yield 1
a, b = fib(), fib()
next(b)
while True:
yield next(a)+next(b)
print(tuple(islice(fib(), 10)))
As a scan or a fold
=itertools.accumulate=
The Fibonacci series can be defined quite simply and efficiently as a scan or accumulation, in which the accumulator is a pair of the two last numbers.
'''Fibonacci accumulation'''
from itertools import (accumulate, chain)
# fibs :: Integer :: [Integer]
def fibs(n):
'''An accumulation of the first n integers in
the Fibonacci series. The accumulator is a
pair of the two preceding numbers.
'''
def go(ab, _):
a, b = ab
return (b, a + b)
return [xy[1] for xy in accumulate(
chain(
[(0, 1)],
range(1, n)
),
go
)]
# MAIN ---
if __name__ == '__main__':
print(
'First twenty: ' + repr(
fibs(20)
)
)
First twenty: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]
=functools.reduce=
A fold can be understood as an amnesic scan, and functools.reduce can provide a useful and efficient re-write of the scanning version above, if we only need the Nth term in the series:
'''Nth Fibonacci term (by folding)'''
from functools import (reduce)
# nthFib :: Integer -> Integer
def nthFib(n):
'''Nth integer in the Fibonacci series.'''
def go(ab, _):
a, b = ab
return (b, a + b)
return reduce(go, range(1, n), (0, 1))[1]
# MAIN ---
if __name__ == '__main__':
print(
'1000th term: ' + repr(
nthFib(1000)
)
)
1000th term: 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875
Qi
Recursive
(define fib
0 -> 0
1 -> 1
N -> (+ (fib-r (- N 1))
(fib-r (- N 2))))
Iterative
(define fib-0
V2 V1 0 -> V2
V2 V1 N -> (fib-0 V1 (+ V2 V1) (1- N)))
(define fib
N -> (fib-0 0 1 N))
R
Iterative positive and negative
fib=function(n,x=c(0,1)) {
if (abs(n)>1) for (i in seq(abs(n)-1)) x=c(x[2],sum(x))
if (n<0) return(x[2]*(-1)^(abs(n)-1)) else if (n) return(x[2]) else return(0)
}
sapply(seq(-31,31),fib)
Output:
[1] 1346269 -832040 514229 -317811 196418 -121393 75025 -46368 28657
[10] -17711 10946 -6765 4181 -2584 1597 -987 610 -377
[19] 233 -144 89 -55 34 -21 13 -8 5
[28] -3 2 -1 1 0 1 1 2 3
[37] 5 8 13 21 34 55 89 144 233
[46] 377 610 987 1597 2584 4181 6765 10946 17711
[55] 28657 46368 75025 121393 196418 317811 514229 832040 1346269
Other methods
# recursive
recfibo <- function(n) {
if ( n < 2 ) n
else Recall(n-1) + Recall(n-2)
}
# print the first 21 elements
print.table(lapply(0:20, recfibo))
# iterative
iterfibo <- function(n) {
if ( n < 2 )
n
else {
f <- c(0, 1)
for (i in 2:n) {
t <- f[2]
f[2] <- sum(f)
f[1] <- t
}
f[2]
}
}
print.table(lapply(0:20, iterfibo))
# iterative but looping replaced by map-reduce'ing
funcfibo <- function(n) {
if (n < 2)
n
else {
generator <- function(f, ...) {
c(f[2], sum(f))
}
Reduce(generator, 2:n, c(0,1))[2]
}
}
print.table(lapply(0:20, funcfibo))
Note that an idiomatic way to implement such low level, basic arithmethic operations in R is to implement them C and then call the compiled code.
All three solutions print
[1] 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377
[16] 610 987 1597 2584 4181 6765
Ra
class FibonacciSequence
**Prints the nth fibonacci number**
on start
args := program arguments
if args empty
print .fibonacci(8)
else
try
print .fibonacci(integer.parse(args[0]))
catch FormatException
print to Console.error made !, "Input must be an integer"
exit program with error code
catch OverflowException
print to Console.error made !, "Number too large"
exit program with error code
define fibonacci(n as integer) as integer is shared
**Returns the nth fibonacci number**
test
assert fibonacci(0) = 0
assert fibonacci(1) = 1
assert fibonacci(2) = 1
assert fibonacci(3) = 2
assert fibonacci(4) = 3
assert fibonacci(5) = 5
assert fibonacci(6) = 8
assert fibonacci(7) = 13
assert fibonacci(8) = 21
body
a, b := 0, 1
for n
a, b := b, a + b
return a
Racket
Tail Recursive
(define (fib n)
(let loop ((cnt 0) (a 0) (b 1))
(if (= n cnt)
a
(loop (+ cnt 1) b (+ a b)))))
(define (fib n (a 0) (b 1))
(if (< n 2)
1
(+ a (fib (- n 1) b (+ a b)))))
Matrix Form
#lang racket
(require math/matrix)
(define (fibmat n) (matrix-ref
(matrix-expt (matrix ([1 1]
[1 0]))
n)
1 0))
(fibmat 1000)
Retro
Recursive
: fib ( n-m ) dup [ 0 = ] [ 1 = ] bi or if; [ 1- fib ] sip [ 2 - fib ] do + ;
Iterative
: fib ( n-N )
[ 0 1 ] dip [ over + swap ] times drop ;
REXX
With 210,000 numeric decimal digits, this REXX program can handle Fibonacci numbers past one million.
[Generally speaking, some REXX interpreters can handle up to around eight million decimal digits.]
This version of the REXX program can also handle ''negative'' Fibonacci numbers.
/*REXX program calculates the Nth Fibonacci number, N can be zero or negative. */
numeric digits 210000 /*be able to handle ginormous numbers. */
parse arg x y . /*allow a single number or a range. */
if x=='' | x=="," then do; x=-40; y=+40; end /*No input? Then use range -40 ──► +40*/
if y=='' | y=="," then y=x /*if only one number, display fib(X).*/
w= max(length(x), length(y) ) /*W: used for making formatted output.*/
fw= 10 /*Minimum maximum width. Sounds ka─razy*/
do j=x to y; q= fib(j) /*process all of the Fibonacci requests*/
L= length(q) /*obtain the length (decimal digs) of Q*/
fw= max(fw, L) /*fib number length, or the max so far.*/
say 'Fibonacci('right(j,w)") = " right(q,fw) /*right justify Q.*/
if L>10 then say 'Fibonacci('right(j, w)") has a length of" L
end /*j*/ /* [↑] list a Fib. sequence of x──►y */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
fib: procedure; parse arg n; an= abs(n) /*use │n│ (the absolute value of N).*/
a= 0; b= 1; if an<2 then return an /*handle two special cases: zero & one.*/
/* [↓] this method is non─recursive. */
do k=2 to an; $= a+b; a= b; b= $ /*sum the numbers up to │n│ */
end /*k*/ /* [↑] (only positive Fibs nums used).*/
/* [↓] an//2 [same as] (an//2==1).*/
if n>0 | an//2 then return $ /*Positive or even? Then return sum. */
return -$ /*Negative and odd? Return negative sum*/
Fibonacci(-40) = -102334155
Fibonacci(-39) = 63245986
Fibonacci(-38) = -39088169
Fibonacci(-37) = 24157817
Fibonacci(-36) = -14930352
Fibonacci(-35) = 9227465
Fibonacci(-34) = -5702887
Fibonacci(-33) = 3524578
Fibonacci(-32) = -2178309
Fibonacci(-31) = 1346269
Fibonacci(-30) = -832040
Fibonacci(-29) = 514229
Fibonacci(-28) = -317811
Fibonacci(-27) = 196418
Fibonacci(-26) = -121393
Fibonacci(-25) = 75025
Fibonacci(-24) = -46368
Fibonacci(-23) = 28657
Fibonacci(-22) = -17711
Fibonacci(-21) = 10946
Fibonacci(-20) = -6765
Fibonacci(-19) = 4181
Fibonacci(-18) = -2584
Fibonacci(-17) = 1597
Fibonacci(-16) = -987
Fibonacci(-15) = 610
Fibonacci(-14) = -377
Fibonacci(-13) = 233
Fibonacci(-12) = -144
Fibonacci(-11) = 89
Fibonacci(-10) = -55
Fibonacci( -9) = 34
Fibonacci( -8) = -21
Fibonacci( -7) = 13
Fibonacci( -6) = -8
Fibonacci( -5) = 5
Fibonacci( -4) = -3
Fibonacci( -3) = 2
Fibonacci( -2) = -1
Fibonacci( -1) = 1
Fibonacci( 0) = 0
Fibonacci( 1) = 1
Fibonacci( 2) = 1
Fibonacci( 3) = 2
Fibonacci( 4) = 3
Fibonacci( 5) = 5
Fibonacci( 6) = 8
Fibonacci( 7) = 13
Fibonacci( 8) = 21
Fibonacci( 9) = 34
Fibonacci( 10) = 55
Fibonacci( 11) = 89
Fibonacci( 12) = 144
Fibonacci( 13) = 233
Fibonacci( 14) = 377
Fibonacci( 15) = 610
Fibonacci( 16) = 987
Fibonacci( 17) = 1597
Fibonacci( 18) = 2584
Fibonacci( 19) = 4181
Fibonacci( 20) = 6765
Fibonacci( 21) = 10946
Fibonacci( 22) = 17711
Fibonacci( 23) = 28657
Fibonacci( 24) = 46368
Fibonacci( 25) = 75025
Fibonacci( 26) = 121393
Fibonacci( 27) = 196418
Fibonacci( 28) = 317811
Fibonacci( 29) = 514229
Fibonacci( 30) = 832040
Fibonacci( 31) = 1346269
Fibonacci( 32) = 2178309
Fibonacci( 33) = 3524578
Fibonacci( 34) = 5702887
Fibonacci( 35) = 9227465
Fibonacci( 36) = 14930352
Fibonacci( 37) = 24157817
Fibonacci( 38) = 39088169
Fibonacci( 39) = 63245986
Fibonacci( 40) = 102334155
```
'''output''' when the following was used as input: 10000
```txt
Fibonacci(10000) = 3364476487643178326662161200510754331030214846068006390656476997468008144216666236815559551363373402558206533268083615937373479048386526826304089246305643188735454436955982749160660209988418393386465273130008883026923567361313511757929743785441375213052050434770160226475831890652789085515436615958298727968298751063120057542878345321551510387081829896979161312785626503319548714021428753269818796204693609787990035096230229102636813149319527563022783762844154036058440257211433496118002309120828704608892396232883546150577658327125254609359112820392528539343462090424524892940390
170623388899108584106518317336043747073790855263176432573399371287193758774689747992630583706574283016163740896917842637862421283525811282051637029808933209990570792006436742620238978311147005407499845925036063356093388383192338678305613643535189213327973290813373264265263398976392272340788292817795358057099369104917547080893184105614632233821746563732124822638309210329770164805472624384237486241145309381220656491403275108664339451751216152654536133311131404243685480510676584349352383695965342807176877532834823434555736671973139274627362910821067928078471803532913117677892465908993863545932789
452377767440619224033763867400402133034329749690202832814593341882681768389307200363479562311710310129195316979460763273758925353077255237594378843450406771555577905645044301664011946258097221672975861502696844314695203461493229110597067624326851599283470989128470674086200858713501626031207190317208609408129832158107728207635318662461127824553720853236530577595643007251774431505153960090516860322034916322264088524885243315805153484962243484829938090507048348244932745373262456775587908918719080366205800959474315005240253270974699531877072437682590741993963226598414749819360928522394503970716544
3156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875
Fibonacci(10000) has a length of 2090 decimal digits
```
## Rockstar
===Iterative (minimized)===
```Rockstar
Fibonacci takes Number
FNow is 0
FNext is 1
While FNow is less than Number
Say FNow
Put FNow into Temp
Put FNow into FNext
Put FNext plus Temp into FNext
Say Fibonacci taking 1000 (prints out highest number in Fibonacci sequence less than 1000)
```
===Iterative (idiomatic)===
```Rockstar
Love takes Time
My love was addictions
Put my love into your heart
Build it up
Until my love is as strong as Time
Whisper my love
Put my love into a river
Put your heart into my love
Put it with a river into your heart
Shout; Love taking 1000 (years, years)
```
The semicolon and the comment (years, years) in this version are there only for poetic effect
## Ring
```ring
give n
x = fib(n)
see n + " Fibonacci is : " + x
func fib nr if nr = 0 return 0 ok
if nr = 1 return 1 ok
if nr > 1 return fib(nr-1) + fib(nr-2) ok
```
## Ruby
### Iterative
```ruby
def fib(n, sequence=[1])
n.times do
current_number, last_number = sequence.last(2)
sequence << current_number + (last_number or 0)
end
sequence.last
end
```
### Recursive
```ruby
def fib(n, sequence=[1])
return sequence.last if n == 0
current_number, last_number = sequence.last(2)
sequence << current_number + (last_number or 0)
fib(n-1, sequence)
end
```
### Recursive with Memoization
```ruby
# Use the Hash#default_proc feature to
# lazily calculate the Fibonacci numbers.
fib = Hash.new do |f, n|
f[n] = if n <= -2
(-1)**(n + 1) * f[n.abs]
elsif n <= 1
n.abs
else
f[n - 1] + f[n - 2]
end
end
# examples: fib[10] => 55, fib[-10] => (-55/1)
```
### Matrix
```ruby
require 'matrix'
# To understand why this matrix is useful for Fibonacci numbers, remember
# that the definition of Matrix.**2 for any Matrix[[a, b], [c, d]] is
# is [[a*a + b*c, a*b + b*d], [c*a + d*b, c*b + d*d]]. In other words, the
# lower right element is computing F(k - 2) + F(k - 1) every time M is multiplied
# by itself (it is perhaps easier to understand this by computing M**2, 3, etc, and
# watching the result march up the sequence of Fibonacci numbers).
M = Matrix[[0, 1], [1,1]]
# Matrix exponentiation algorithm to compute Fibonacci numbers.
# Let M be Matrix [[0, 1], [1, 1]]. Then, the lower right element of M**k is
# F(k + 1). In other words, the lower right element of M is F(2) which is 1, and the
# lower right element of M**2 is F(3) which is 2, and the lower right element
# of M**3 is F(4) which is 3, etc.
#
# This is a good way to compute F(n) because the Ruby implementation of Matrix.**(n)
# uses O(log n) rather than O(n) matrix multiplications. It works by squaring squares
# ((m**2)**2)... as far as possible
# and then multiplying that by by M**(the remaining number of times). E.g., to compute
# M**19, compute partial = ((M**2)**2) = M**16, and then compute partial*(M**3) = M**19.
# That's only 5 matrix multiplications of M to compute M*19.
def self.fib_matrix(n)
return 0 if n <= 0 # F(0)
return 1 if n == 1 # F(1)
# To get F(n >= 2), compute M**(n - 1) and extract the lower right element.
return CS::lower_right(M**(n - 1))
end
# Matrix utility to return
# the lower, right-hand element of a given matrix.
def self.lower_right matrix
return nil if matrix.row_size == 0
return matrix[matrix.row_size - 1, matrix.column_size - 1]
end
```
### Generative
```ruby
fib = Enumerator.new do |y|
f0, f1 = 0, 1
loop do
y << f0
f0, f1 = f1, f0 + f1
end
end
```
Usage:
```txt
p fib.lazy.drop(8).next # => 21
```
"Fibers are primitives for implementing light weight cooperative concurrency in Ruby. Basically they are a means of creating code blocks that can be paused and resumed, much like threads. The main difference is that they are never preempted and that the scheduling must be done by the programmer and not the VM." [http://www.ruby-doc.org/ruby-1.9/classes/Fiber.html]
```ruby
fib = Fiber.new do
a,b = 0,1
loop do
Fiber.yield a
a,b = b,a+b
end
end
9.times {puts fib.resume}
```
using a lambda
```ruby
def fib_gen
a, b = 1, 1
lambda {ret, a, b = a, b, a+b; ret}
end
```
```txt
irb(main):034:0> fg = fib_gen
=> #
irb(main):035:0> 9.times { puts fg.call}
1
1
2
3
5
8
13
21
34
=> 9
```
===Binet's Formula===
```ruby
def fib
phi = (1 + Math.sqrt(5)) / 2
((phi**self - (-1 / phi)**self) / Math.sqrt(5)).to_i
end
```
```txt
1.9.3p125 :001 > def fib
1.9.3p125 :002?> phi = (1 + Math.sqrt(5)) / 2
1.9.3p125 :003?> ((phi**self - (-1 / phi)**self) / Math.sqrt(5)).to_i
1.9.3p125 :004?> end
=> nil
1.9.3p125 :005 > (0..10).map(&:fib)
=> [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
```
## Run BASIC
```runbasic
for i = 0 to 10
print i;" ";fibR(i);" ";fibI(i)
next i
end
function fibR(n)
if n < 2 then fibR = n else fibR = fibR(n-1) + fibR(n-2)
end function
function fibI(n)
b = 1
for i = 1 to n
t = a + b
a = b
b = t
next i
fibI = a
end function
```
## Rust
### Iterative
```rust
use std::mem;
fn main() {
let mut prev = 0;
// Rust needs this type hint for the checked_add method
let mut curr = 1usize;
while let Some(n) = curr.checked_add(prev) {
prev = curr;
curr = n;
println!("{}", n);
}
}
```
### Recursive
```rust
use std::mem;
fn main() {
fibonacci(0,1);
}
fn fibonacci(mut prev: usize, mut curr: usize) {
mem::swap(&mut prev, &mut curr);
if let Some(n) = curr.checked_add(prev) {
println!("{}", n);
fibonacci(prev, n);
}
}
```
===Recursive (with pattern matching)===
```rust
fn fib(n: u32) -> u32 {
match n {
0 => 0,
1 => 1,
n => fib(n - 1) + fib(n - 2),
}
}
```
===Tail recursive (with pattern matching)===
```rust
fn fib_tail_recursive(nth: usize) -> usize {
fn fib_tail_iter(n: usize, prev_fib: usize, fib: usize) -> usize {
match n {
0 => prev_fib,
n => fib_tail_iter(n - 1, fib, prev_fib + fib),
}
}
fib_tail_iter(nth, 0, 1)
}
```
### Analytic
This uses a feature from nightly Rust which makes it possible to (cleanly) return an iterator without the additional overhead of putting it on the heap. In stable Rust, we'd need to return a Box> which has the cost of an additional allocation and the overhead of dynamic dispatch. The version below does not require the use of the heap and is done entirely through static dispatch.
```rust
#![feature(conservative_impl_trait)]
fn main() {
for num in fibonacci_gen(10) {
println!("{}", num);
}
}
fn fibonacci_gen(terms: i32) -> impl Iterator {
let sqrt_5 = 5.0f64.sqrt();
let p = (1.0 + sqrt_5) / 2.0;
let q = 1.0/p;
(1..terms).map(move |n| ((p.powi(n) + q.powi(n)) / sqrt_5 + 0.5) as u64)
}
```
### Using an Iterator
Iterators are very idiomatic in rust, though they may be overkill for such a simple problem.
```rust
use std::mem;
struct Fib {
prev: usize,
curr: usize,
}
impl Fib {
fn new() -> Self {
Fib {prev: 0, curr: 1}
}
}
impl Iterator for Fib {
type Item = usize;
fn next(&mut self) -> Option{
mem::swap(&mut self.curr, &mut self.prev);
self.curr.checked_add(self.prev).map(|n| {
self.curr = n;
n
})
}
}
fn main() {
for num in Fib::new() {
println!("{}", num);
}
}
```
## SAS
### Iterative
This code builds a table fib holding the first few values of the Fibonacci sequence.
```sas
data fib;
a=0;
b=1;
do n=0 to 20;
f=a;
output;
a=b;
b=f+a;
end;
keep n f;
run;
```
### Naive recursive
This code provides a simple example of defining a function and using it recursively. One of the members of the sequence is written to the log.
```sas
options cmplib=work.f;
proc fcmp outlib=work.f.p;
function fib(n);
if n = 0 or n = 1
then return(1);
else return(fib(n - 2) + fib(n - 1));
endsub;
run;
data _null_;
x = fib(5);
put 'fib(5) = ' x;
run;
```
## Sather
The implementations use the arbitrary precision class INTI.
```sather
class MAIN is
-- RECURSIVE --
fibo(n :INTI):INTI
pre n >= 0
is
if n < 2.inti then return n; end;
return fibo(n - 2.inti) + fibo(n - 1.inti);
end;
-- ITERATIVE --
fibo_iter(n :INTI):INTI
pre n >= 0
is
n3w :INTI;
if n < 2.inti then return n; end;
last ::= 0.inti; this ::= 1.inti;
loop (n - 1.inti).times!;
n3w := last + this;
last := this;
this := n3w;
end;
return this;
end;
main is
loop i ::= 0.upto!(16);
#OUT + fibo(i.inti) + " ";
#OUT + fibo_iter(i.inti) + "\n";
end;
end;
end;
```
=={{header|S-BASIC}}==
Note that the 23rd Fibonacci number (=28657) is the largest that can be generated without overflowing S-BASIC's integer data type.
```basic
rem - iterative function to calculate nth fibonacci number
function fibonacci(n = integer) = integer
var f, i, p1, p2 = integer
p1 = 0
p2 = 1
if n = 0 then
f = 0
else
for i = 1 to n
f = p1 + p2
p2 = p1
p1 = f
next i
end = f
rem - exercise the function
var i = integer
for i = 0 to 10
print fibonacci(i);
next i
end
```
```txt
0 1 1 2 3 5 8 13 21 34 55
```
## Scala
### Recursive
```scala
def fib(i:Int):Int = i match{
case 0 => 0
case 1 => 1
case _ => fib(i-1) + fib(i-2)
}
```
### Lazy sequence
```scala
lazy val fib: Stream[Int] = 0 #:: 1 #:: fib.zip(fib.tail).map{case (a,b) => a + b}
```
### Tail recursive
```scala
def fib(x:Int, prev: BigInt = 0, next: BigInt = 1):BigInt = x match {
case 0 => prev
case _ => fib(x-1, next, next + prev)
}
```
### foldLeft
```scala
// Fibonacci using BigInt with Stream.foldLeft optimized for GC (Scala v2.9 and above)
// Does not run out of memory for very large Fibonacci numbers
def fib(n:Int) = {
def series(i:BigInt,j:BigInt):Stream[BigInt] = i #:: series(j, i+j)
series(1,0).take(n).foldLeft(BigInt("0"))(_+_)
}
// Small test
(0 to 13) foreach {n => print(fib(n).toString + " ")}
// result: 0 1 1 2 3 5 8 13 21 34 55 89 144 233
```
### Iterator
```scala
val it = Iterator.iterate((0,1)){case (a,b) => (b,a+b)}.map(_._1)
//example:
println(it.take(13).mkString(",")) //prints: 0,1,1,2,3,5,8,13,21,34,55,89,144
```
## Scheme
### Iterative
```scheme
(define (fib-iter n)
(do ((num 2 (+ num 1))
(fib-prev 1 fib)
(fib 1 (+ fib fib-prev)))
((>= num n) fib)))
```
### Recursive
```scheme
(define (fib-rec n)
(if (< n 2)
n
(+ (fib-rec (- n 1))
(fib-rec (- n 2)))))
```
This version is tail recursive:
```scheme
(define (fib n)
(let loop ((a 0) (b 1) (n n))
(if (= n 0) a
(loop b (+ a b) (- n 1)))))
```
### Recursive Sequence Generator
Although the tail recursive version above is quite efficient, it only generates the final nth Fibonacci number and not the sequence up to that number without wasteful repeated calls to the procedure/function.
The following procedure generates the sequence of Fibonacci numbers using a simplified version of a lazy list/stream - since no memoization is requried, it just implements future values by using a zero parameter lambda "thunk" with a closure containing the last and the pre-calculated next value of the sequence; in this way it uses almost no memory during the sequence generation other than as required for the last and the next values of the sequence (note that the test procedure does not generate a linked list to contain the elements of the sequence to show, but rather displays each one by one in sequence):
```scheme
(define (fib)
(define (nxt lv nv) (cons nv (lambda () (nxt nv (+ lv nv)))))
(cons 0 (lambda () (nxt 0 1))))
;;; test...
(define (show-stream-take n strm)
(define (shw-nxt n strm) (begin (display (car strm))
(if (> n 1) (begin (display " ") (shw-nxt (- n 1) ((cdr strm)))) (display ")"))))
(begin (display "(") (shw-nxt n strm)))
(show-stream-take 30 (fib))
```
```txt
(0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229)
```
### Dijkstra Algorithm
```scheme
;;; Fibonacci numbers using Edsger Dijkstra's algorithm
;;; http://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF
(define (fib n)
(define (fib-aux a b p q count)
(cond ((= count 0) b)
((even? count)
(fib-aux a
b
(+ (* p p) (* q q))
(+ (* q q) (* 2 p q))
(/ count 2)))
(else
(fib-aux (+ (* b q) (* a q) (* a p))
(+ (* b p) (* a q))
p
q
(- count 1)))))
(fib-aux 1 0 0 1 n))
```
## Scilab
clear
n=46
f1=0; f2=1
printf("fibo(%d)=%d\n",0,f1)
printf("fibo(%d)=%d\n",1,f2)
for i=2:n
f3=f1+f2
printf("fibo(%d)=%d\n",i,f3)
f1=f2
f2=f3
end
```
```txt
...
fibo(43)=433494437
fibo(44)=701408733
fibo(45)=1134903170
fibo(46)=1836311903
```
## sed
```sed
#!/bin/sed -f
# First we need to convert each number into the right number of ticks
# Start by marking digits
s/[0-9]/<&/g
# We have to do the digits manually.
s/0//g; s/1/|/g; s/2/||/g; s/3/|||/g; s/4/||||/g; s/5/|||||/g
s/6/||||||/g; s/7/|||||||/g; s/8/||||||||/g; s/9/|||||||||/g
# Multiply by ten for each digit from the front.
:tens
s/|= 2 ticks into two of n-1, with a mark between
s/|\(|\+\)/\1-\1/g
# Convert the previous mark and the first tick after it to a different mark
# giving us n-1+n-2 marks.
s/-|/+/g
# Jump back unless we're done.
t split
# Get rid of the pluses, we're done with them.
s/+//g
# Convert back to digits
:back
s/||||||||||/ 2 then
result := fib(pred(number)) + fib(number - 2);
elsif number = 0 then
result := 0;
end if;
end func;
```
Original source: [http://seed7.sourceforge.net/algorith/math.htm#fib]
### Iterative
This funtion uses a bigInteger result:
```seed7
const func bigInteger: fib (in integer: number) is func
result
var bigInteger: result is 1_;
local
var integer: i is 0;
var bigInteger: a is 0_;
var bigInteger: c is 0_;
begin
for i range 1 to pred(number) do
c := a;
a := result;
result +:= c;
end for;
end func;
```
Original source: [http://seed7.sourceforge.net/algorith/math.htm#iterative_fib]
## SequenceL
### Recursive
```sequencel
fibonacci(n) :=
n when n < 2
else
fibonacci(n - 1) + fibonacci(n - 2);
```
Based on: [https://www.youtube.com/watch?v=5JVC5dDtnyg]
### Tail Recursive
```sequencel
fibonacci(n) := fibonacciHelper(0, 1, n);
fibonacciHelper(prev, next, n) :=
prev when n < 1
else
next when n = 1
else
fibonacciHelper(next, next + prev, n - 1);
```
### Matrix
```sequencel
fibonacci(n) := fibonacciHelper([[1,0],[0,1]], n);
fibonacciHelper(M(2), n) :=
let
N := [[1,1],[1,0]];
in
M[1,1] when n <= 1
else
fibonacciHelper(matmul(M, N), n - 1);
matmul(A(2), B(2)) [i,j] := sum( A[i,all] * B[all,j] );
```
Based on the C# version: [http://rosettacode.org/wiki/Fibonacci_sequence#C.23]
Using the SequenceL Matrix Multiply solution: [http://rosettacode.org/wiki/Matrix_multiplication#SequenceL]
## SETL
```setl
$ Print out the first ten Fibonacci numbers
$ This uses Set Builder Notation, it roughly means
$ 'collect fib(n) forall n in {0,1,2,3,4,5,6,7,8,9,10}'
print({fib(n) : n in {0..10}});
$ Iterative Fibonacci function
proc fib(n);
A := 0; B := 1; C := n;
for i in {0..n} loop
C := A + B;
A := B;
B := C;
end loop;
return C;
end proc;
```
## Shen
```Shen
(define fib
0 -> 0
1 -> 1
N -> (+ (fib (+ N 1)) (fib (+ N 2)))
where (< N 0)
N -> (+ (fib (- N 1)) (fib (- N 2))))
```
## Sidef
### Iterative
```ruby
func fib_iter(n) {
var (a, b) = (0, 1)
{ (a, b) = (b, a+b) } * n
return a
}
```
### Recursive
```ruby
func fib_rec(n) {
n < 2 ? n : (__FUNC__(n-1) + __FUNC__(n-2))
}
```
### Recursive with memoization
```ruby
func fib_mem (n) is cached {
n < 2 ? n : (__FUNC__(n-1) + __FUNC__(n-2))
}
```
===Closed-form===
```ruby
func fib_closed(n) {
define S = (1.25.sqrt + 0.5)
define T = (-S + 1)
(S**n - T**n) / (-T + S) -> round
}
```
===Built-in===
```ruby
say fib(12) #=> 144
```
## Simula
Straightforward iterative implementation.
```simula
INTEGER PROCEDURE fibonacci(n);
INTEGER n;
BEGIN
INTEGER lo, hi, temp, i;
lo := 0;
hi := 1;
FOR i := 1 STEP 1 UNTIL n - 1 DO
BEGIN
temp := hi;
hi := hi + lo;
lo := temp
END;
fibonacci := hi
END;
```
## SkookumScript
===Built-in===
SkookumScript's Integer class has a fast built-in fibonnaci() method.
```javascript>42.fibonacci 42.fibonacci form as built-in form above.
```javascript
// Assuming code is in Integer.fibonacci() method
() Integer
[
if this < 2 [this] else [[this - 1].fibonacci + [this - 2].fibonacci]
]
```
Recursive procedure in fibonacci(42) form.
```javascript
// Assuming in fibonacci(n) procedure
(Integer n) Integer
[
if n < 2 [n] else [fibonacci(n - 1) + fibonacci(n - 2)]
]
```
### Iterative
Iterative method in 42.fibonacci form.
```javascript
// Assuming code is in Integer.fibonacci() method
() Integer
[
if this < 2
[this]
else
[
!prev: 1
!next: 1
2.to_pre this
[
!sum : prev + next
prev := next
next := sum
]
next
]
]
```
Optimized iterative method in 42.fibonacci form.
Though the best optimiation is to write it in C++ as with the built-in form that comes with SkookumScript.
```javascript
// Bind : is faster than assignment :=
// loop is faster than to_pre (which uses a closure)
() Integer
[
if this < 2
[this]
else
[
!prev: 1
!next: 1
!sum
!count: this - 2
loop
[
if count = 0 [exit]
count--
sum : prev + next
prev : next
next : sum
]
next
]
]
```
## Slate
```slate
n@(Integer traits) fib
[
n <= 0 ifTrue: [^ 0].
n = 1 ifTrue: [^ 1].
(n - 1) fib + (n - 2) fib
].
slate[15]> 10 fib = 55.
True
```
## Smalltalk
```smalltalk
|fibo|
fibo := [ :i |
|ac t|
ac := Array new: 2.
ac at: 1 put: 0 ; at: 2 put: 1.
( i < 2 )
ifTrue: [ ac at: (i+1) ]
ifFalse: [
2 to: i do: [ :l |
t := (ac at: 2).
ac at: 2 put: ( (ac at: 1) + (ac at: 2) ).
ac at: 1 put: t
].
ac at: 2.
]
].
0 to: 10 do: [ :i |
(fibo value: i) displayNl
]
```
## smart BASIC
The Iterative method is slow (relatively) and the Recursive method doubly so since it references the Iterative function twice.
The N-th Term (fibN) function is much faster as it utilizes Binet's Formula.
- fibR: Fibonacci Recursive
- fibI: Fibonacci Iterative
- fibN: Fibonacci N-th Term
```qbasic
FOR i = 0 TO 15
PRINT fibR(i),fibI(i),fibN(i)
NEXT i
/* Recursive Method */
DEF fibR(n)
IF n <= 1 THEN
fibR = n
ELSE
fibR = fibR(n-1) + fibR(n-2)
ENDIF
END DEF
/* Iterative Method */
DEF fibI(n)
a = 0
b = 1
FOR i = 1 TO n
temp = a + b
a = b
b = temp
NEXT i
fibI = a
END DEF
/* N-th Term Method */
DEF fibN(n)
uphi = .5 + SQR(5)/2
lphi = .5 - SQR(5)/2
fibN = (uphi^n-lphi^n)/SQR(5)
END DEF
```
## SNOBOL4
### Recursive
```snobol
define("fib(a)") :(fib_end)
fib fib = lt(a,2) a :s(return)
fib = fib(a - 1) + fib(a - 2) :(return)
fib_end
while a = trim(input) :f(end)
output = a " " fib(a) :(while)
end
```
===Tail-recursive===
```SNOBOL4
define('trfib(n,a,b)') :(trfib_end)
trfib trfib = eq(n,0) a :s(return)
trfib = trfib(n - 1, a + b, a) :(return)
trfib_end
```
### Iterative
```SNOBOL4
define('ifib(n)f1,f2') :(ifib_end)
ifib ifib = le(n,2) 1 :s(return)
f1 = 1; f2 = 1
if1 ifib = gt(n,2) f1 + f2 :f(return)
f1 = f2; f2 = ifib; n = n - 1 :(if1)
ifib_end
```
### Analytic
Note: Snobol4+ lacks built-in sqrt( ) function.
```SNOBOL4
define('afib(n)s5') :(afib_end)
afib s5 = sqrt(5)
afib = (((1 + s5) / 2) ^ n - ((1 - s5) / 2) ^ n) / s5
afib = convert(afib,'integer') :(return)
afib_end
```
Test and display all, Fib 1 .. 10
```SNOBOL4
loop i = lt(i,10) i + 1 :f(show)
s1 = s1 fib(i) ' ' ; s2 = s2 trfib(i,0,1) ' '
s3 = s3 ifib(i) ' '; s4 = s4 afib(i) ' ' :(loop)
show output = s1; output = s2; output = s3; output = s4
end
```
Output:
```txt
1 1 2 3 5 8 13 21 34 55
1 1 2 3 5 8 13 21 34 55
1 1 2 3 5 8 13 21 34 55
1 1 2 3 5 8 13 21 34 55
```
## SNUSP
This is modular SNUSP (which introduces @ and # for threading).
### Iterative
```snusp
@!\+++++++++# /<<+>+>-\
fib\==>>+<!\ ?/\
#</@>\?-<</@>/>+<-\
\-/ \ !\ !\ !\ ?/#
```
### Recursive
```snusp> /
### ==
\ />
+<<-\ />+<-\
fib==!/?!\-?!\->+>+<>-@\
### ==?/<@\
?/<#
| #+==/ fib(n-2)|+fib(n-1)|
\
### ==recursion===
/!
### ==
/
```
## Softbridge BASIC
### Iterative
```basic
Function Fibonacci(n)
x = 0
y = 1
i = 0
n = ABS(n)
If n < 2 Then
Fibonacci = n
Else
Do Until (i = n)
sum = x+y
x=y
y=sum
i=i+1
Loop
Fibonacci = x
End If
End Function
```
## Spin
### Iterative
```spin
con
_clkmode = xtal1 + pll16x
_clkfreq = 80_000_000
obj
ser : "FullDuplexSerial.spin"
pub main | i
ser.start(31, 30, 0, 115200)
repeat i from 0 to 10
ser.dec(fib(i))
ser.tx(32)
waitcnt(_clkfreq + cnt)
ser.stop
cogstop(0)
pub fib(i) : b | a
b := a := 1
repeat i
a := b + (b := a)
```
```txt
1 1 2 3 5 8 13 21 34 55 89
```
## SPL
### Analytic
```spl
fibo(n)=
s5 = #.sqrt(5)
<= (((1+s5)/2)^n-((1-s5)/2)^n)/s5
.
```
### Iterative
```spl
fibo(n)=
? n<2, <= n
f2 = 0
f1 = 1
> i, 2..n
f = f1+f2
f2 = f1
f1 = f
<
<= f
.
```
### Recursive
```spl
fibo(n)=
? n<2, <= n
<= fibo(n-1)+fibo(n-2)
.
```
## SQL
### Analytic
As a running sum:
```SQL
select round ( exp ( sum (ln ( ( 1 + sqrt( 5 ) ) / 2)
) over ( order by level ) ) / sqrt( 5 ) ) fibo
from dual
connect by level <= 10;
```
```txt
FIB
----------
1
1
2
3
5
8
13
21
34
55
10 rows selected.
```
As a power:
```SQL
select round ( power( ( 1 + sqrt( 5 ) ) / 2, level ) / sqrt( 5 ) ) fib
from dual
connect by level <= 10;
```
```txt
FIB
----------
1
1
2
3
5
8
13
21
34
55
10 rows selected.
```
### Recursive
Oracle 12c required
```sql
SQL> with fib(e,f) as (select 1, 1 from dual union all select e+f,e from fib where e <= 55) select f from fib;
F
----------
1
1
2
3
5
8
13
21
34
55
10 rows selected.
```
```postgresql
CREATE FUNCTION fib(n int) RETURNS numeric AS $$
-- This recursive with generates endless list of Fibonacci numbers.
WITH RECURSIVE fibonacci(current, previous) AS (
-- Initialize the current with 0, so the first value will be 0.
-- The previous value is set to 1, because its only goal is not
-- special casing the zero case, and providing 1 as the second
-- number in the sequence.
--
-- The numbers end with dots to make them numeric type in
-- Postgres. Numeric type has almost arbitrary precision
-- (technically just 131,072 digits, but that's good enough for
-- most purposes, including calculating huge Fibonacci numbers)
SELECT 0., 1.
UNION ALL
-- To generate Fibonacci number, we need to add together two
-- previous Fibonacci numbers. Current number is saved in order
-- to be accessed in the next iteration of recursive function.
SELECT previous + current, current FROM fibonacci
)
-- The user is only interested in current number, not previous.
SELECT current FROM fibonacci
-- We only need one number, so limit to 1
LIMIT 1
-- Offset the query by the requested argument to get the correct
-- position in the list.
OFFSET n
$$ LANGUAGE SQL RETURNS NULL ON NULL INPUT IMMUTABLE;
```
## SSEM
Calculates the tenth Fibonacci number. To calculate the nth, change the initial value of the counter to n-1 (subject to the restriction that the answer must be small enough to fit in a signed 32-bit integer, the SSEM's only data type). The algorithm is basically straightforward, but the absence of an Add instruction makes the implementation a little more complicated than it would otherwise be.
```ssem
10101000000000100000000000000000 0. -21 to c acc = -n
01101000000001100000000000000000 1. c to 22 temp = acc
00101000000001010000000000000000 2. Sub. 20 acc -= m
10101000000001100000000000000000 3. c to 21 n = acc
10101000000000100000000000000000 4. -21 to c acc = -n
10101000000001100000000000000000 5. c to 21 n = acc
01101000000000100000000000000000 6. -22 to c acc = -temp
00101000000001100000000000000000 7. c to 20 m = acc
11101000000000100000000000000000 8. -23 to c acc = -count
00011000000001010000000000000000 9. Sub. 24 acc -= -1
00000000000000110000000000000000 10. Test skip next if acc<0
10011000000000000000000000000000 11. 25 to CI goto (15 + 1)
11101000000001100000000000000000 12. c to 23 count = acc
11101000000000100000000000000000 13. -23 to c acc = -count
11101000000001100000000000000000 14. c to 23 count = acc
00011000000000000000000000000000 15. 24 to CI goto (-1 + 1)
10101000000000100000000000000000 16. -21 to c acc = -n
10101000000001100000000000000000 17. c to 21 n = acc
10101000000000100000000000000000 18. -21 to c acc = -n
00000000000001110000000000000000 19. Stop
00000000000000000000000000000000 20. 0 var m = 0
10000000000000000000000000000000 21. 1 var n = 1
00000000000000000000000000000000 22. 0 var temp = 0
10010000000000000000000000000000 23. 9 var count = 9
11111111111111111111111111111111 24. -1 const -1
11110000000000000000000000000000 25. 15 const 15
```
## Stata
```stata
program fib
args n
clear
qui set obs `n'
qui gen a=1
qui replace a=a[_n-1]+a[_n-2] in 3/l
end
```
An implementation using '''[https://www.stata.com/help.cgi?dyngen dyngen]'''.
```stata
program fib
args n
clear
qui set obs `n'
qui gen a=.
dyngen {
update a=a[_n-1]+a[_n-2], missval(1)
}
end
fib 10
list
```
'''Output'''
```txt
+----+
| a |
|----|
1. | 1 |
2. | 1 |
3. | 2 |
4. | 3 |
5. | 5 |
|----|
6. | 8 |
7. | 13 |
8. | 21 |
9. | 34 |
10. | 55 |
+----+
```
### Mata
```stata
. mata
: function fib(n) {
return((((1+sqrt(5))/2):^n-((1-sqrt(5))/2):^n)/sqrt(5))
}
: fib(0..10)
1 2 3 4 5 6 7 8 9 10 11
+--------------------------------------------------------+
1 | 0 1 1 2 3 5 8 13 21 34 55 |
+--------------------------------------------------------+
: end
```
## StreamIt
```streamit
void->int feedbackloop Fib {
join roundrobin(0,1);
body in->int filter {
work pop 1 push 1 peek 2 { push(peek(0) + peek(1)); pop(); }
};
loop Identity;
split duplicate;
enqueue(0);
enqueue(1);
}
```
## SuperCollider
### Recursive
nth fibonacci term for positive n
```SuperCollider
f = { |n| if(n < 2) { n } { f.(n-1) + f.(n-2) } };
(0..20).collect(f)
```
nth fibonacci term for positive and negative n.
```SuperCollider
f = { |n| var u = neg(sign(n)); if(abs(n) < 2) { n } { f.(2 * u + n) + f.(u + n) } };
(-20..20).collect(f)
```
### Analytic
```SuperCollider
(
f = { |n|
var sqrt5 = sqrt(5);
var p = (1 + sqrt5) / 2;
var q = reciprocal(p);
((p ** n) + (q ** n) / sqrt5 + 0.5).trunc
};
(0..20).collect(f)
)
```
### Iterative
```SuperCollider
f = { |n| var a = [1, 1]; n.do { a = a.addFirst(a[0] + a[1]) }; a.reverse };
f.(18)
```
## Swift
### Analytic
```Swift
import Cocoa
func fibonacci(n: Int) -> Int {
let square_root_of_5 = sqrt(5.0)
let p = (1 + square_root_of_5) / 2
let q = 1 / p
return Int((pow(p,CDouble(n)) + pow(q,CDouble(n))) / square_root_of_5 + 0.5)
}
for i in 1...30 {
println(fibonacci(i))
}
```
### Iterative
```Swift
func fibonacci(n: Int) -> Int {
if n < 2 {
return n
}
var fibPrev = 1
var fib = 1
for num in 2...n {
(fibPrev, fib) = (fib, fib + fibPrev)
}
return fib
}
```
Sequence:
```swift
func fibonacci() -> SequenceOf {
return SequenceOf {() -> GeneratorOf in
var window: (UInt, UInt, UInt) = (0, 0, 1)
return GeneratorOf {
window = (window.1, window.2, window.1 + window.2)
return window.0
}
}
}
```
### Recursive
```Swift
func fibonacci(n: Int) -> Int {
if n < 2 {
return n
} else {
return fibonacci(n-1) + fibonacci(n-2)
}
}
println(fibonacci(30))
```
## Tailspin
### Recursive simple
The simplest exponential-time recursive algorithm only handling positive N. Note that the "#" is the tailspin internal recursion which sends the value to the matchers. In this case where there is no initial block and no templates state, we could equivalently write the templates name "nthFibonacci" in place of the "#" to do a normal recursion.
```tailspin
templates nthFibonacci
<0|1> $ !
<> ($ - 1 -> #) + ($ - 2 -> #) !
end nthFibonacci
```
===Iterative, mutable state===
We could use the templates internal mutable state, still only positive N.
```tailspin
templates nthFibonacci
@: {n0: 0, n1: 1};
1..$ -> @: {n0: $@.n1, n1: $@.n0 + $@.n1};
$@.n0!
end nthFibonacci
```
To handle negatives, we can keep track of the sign and send it to the matchers.
```tailspin
templates nthFibonacci
@: {n0: 0, n1: 1};
def sign: $ -> (<0..> 1! <> -1!);
1..$*$sign -> $sign -> #
$@.n0!
<1>
@: {n0: $@.n1, n1: $@.n0 + $@.n1};
<-1>
@: {n0: $@.n1 - $@.n0, n1: $@.n0};
end nthFibonacci
```
### State machine
Instead of mutating state, we could just recurse internally on a state structure.
```tailspin
templates nthFibonacci
{ N: $, n0: 0, n1: 1 } -> #
<{ N: <0> }>
$.n0 !
<{ N: <1..>}>
{ N: $.N - 1, n0: $.n1, n1: $.n0 + $.n1} -> #
<>
{ N: $.N + 1, n1: $.n0, n0: $.n1 - $.n0} -> #
end nthFibonacci
8 -> nthFibonacci -> '$;
' -> !OUT::write
-5 -> nthFibonacci -> '$;
' -> !OUT::write
-6 -> nthFibonacci -> '$;
' -> !OUT::write
```
```txt
21
5
-8
```
## Tcl
### Simple Version
These simple versions do not handle negative numbers -- they will return N for N < 2
### =Iterative=
```tcl
proc fibiter n {
if {$n < 2} {return $n}
set prev 1
set fib 1
for {set i 2} {$i < $n} {incr i} {
lassign [list $fib [incr fib $prev]] prev fib
}
return $fib
}
```
### =Recursive=
```tcl
proc fib {n} {
if {$n < 2} then {expr {$n}} else {expr {[fib [expr {$n-1}]]+[fib [expr {$n-2}]]} }
}
```
The following {{works with|Tcl|8.5}}: defining a procedure in the ::tcl::mathfunc namespace allows that proc to be used as a function in [http://www.tcl.tk/man/tcl8.5/TclCmd/expr.htm expr] expressions.
```tcl
proc tcl::mathfunc::fib {n} {
if { $n < 2 } {
return $n
} else {
return [expr {fib($n-1) + fib($n-2)}]
}
}
# or, more tersely
proc tcl::mathfunc::fib {n} {expr {$n<2 ? $n : fib($n-1) + fib($n-2)}}
```
E.g.:
```tcl
expr {fib(7)} ;# ==> 13
namespace path tcl::mathfunc #; or, interp alias {} fib {} tcl::mathfunc::fib
fib 7 ;# ==> 13
```
====Tail-Recursive====
In Tcl 8.6 a ''tailcall'' function is available to permit writing tail-recursive functions in Tcl. This makes deeply recursive functions practical. The availability of large integers also means no truncation of larger numbers.
```tcl
proc fib-tailrec {n} {
proc fib:inner {a b n} {
if {$n < 1} {
return $a
} elseif {$n == 1} {
return $b
} else {
tailcall fib:inner $b [expr {$a + $b}] [expr {$n - 1}]
}
}
return [fib:inner 0 1 $n]
}
```
% fib-tailrec 100
354224848179261915075
### Handling Negative Numbers
### =Iterative=
```tcl
proc fibiter n {
if {$n < 0} {
set n [expr {abs($n)}]
set sign [expr {-1**($n+1)}]
} else {
set sign 1
}
if {$n < 2} {return $n}
set prev 1
set fib 1
for {set i 2} {$i < $n} {incr i} {
lassign [list $fib [incr fib $prev]] prev fib
}
return [expr {$sign * $fib}]
}
fibiter -5 ;# ==> 5
fibiter -6 ;# ==> -8
```
### =Recursive=
```tcl
proc tcl::mathfunc::fib {n} {expr {$n<-1 ? -1**($n+1) * fib(abs($n)) : $n<2 ? $n : fib($n-1) + fib($n-2)}}
expr {fib(-5)} ;# ==> 5
expr {fib(-6)} ;# ==> -8
```
### For the Mathematically Inclined
This works up to , after which the limited precision of IEEE double precision floating point arithmetic starts to show.
```tcl
proc fib n {expr {round((.5 + .5*sqrt(5)) ** $n / sqrt(5))}}
```
## Tern
### Recursive
```tern
func fib(n) {
if (n < 2) {
return 1;
}
return fib(n - 1) + fib(n - 2);
}
```
### Coroutine
```tern
func fib(n) {
let a = 1;
let b = 2;
until(n-- <= 0) {
yield a;
(a, b) = (b, a + b);
}
}
```
=={{header|TI-83 BASIC}}==
Iterative:
```ti83b
{0,1
While 1
Disp Ans(1
{Ans(2),sum(Ans
End
```
Binet's formula:
```ti83b
Prompt N
.5(1+√(5 //golden ratio
(Ans^N–(-Ans)^-N)/√(5
```
=={{header|TI-89 BASIC}}==
### Recursive
Optimized implementation (too slow to be usable for ''n'' higher than about 12).
```ti89b
fib(n)
when(n<2, n, fib(n-1) + fib(n-2))
```
### Iterative
Unoptimized implementation (I think the for loop can be eliminated, but I'm not sure).
```ti89b
fib(n)
Func
Local a,b,c,i
0→a
1→b
For i,1,n
a→c
b→a
c+b→b
EndFor
a
EndFunc
```
## TSE SAL
```TSE SAL
// library: math: get: series: fibonacci 1.0.0.0.3 ] [] [kn, ri, su, 20-01-2013 22:04:02]
INTEGER PROC FNMathGetSeriesFibonacciI( INTEGER nI )
//
// Method:
//
// 1. Take the sum of the last 2 terms
//
// 2. Let the sum be the last term
// and goto step 1
//
INTEGER I = 0
INTEGER minI = 1
INTEGER maxI = nI
INTEGER term1I = 0
INTEGER term2I = 1
INTEGER term3I = 0
//
FOR I = minI TO maxI
//
// make value 3 equal to sum of two previous values 1 and 2
//
term3I = term1I + term2I
//
// make value 1 equal to next value 2
//
term1I = term2I
//
// make value 2 equal to next value 3
//
term2I = term3I
//
ENDFOR
//
RETURN( term3I )
//
END
PROC Main()
STRING s1[255] = "3"
REPEAT
IF ( NOT ( Ask( " = ", s1, _EDIT_HISTORY_ ) ) AND ( Length( s1 ) > 0 ) ) RETURN() ENDIF
Warn( FNMathGetSeriesFibonacciI( Val( s1 ) ) ) // gives e.g. 3
UNTIL FALSE
END
```
## TUSCRIPT
```tuscript
$$ MODE TUSCRIPT
ASK "What fibionacci number do you want?": searchfib=""
IF (searchfib!='digits') STOP
Loop n=0,{searchfib}
IF (n==0) THEN
fib=fiba=n
ELSEIF (n==1) THEN
fib=fibb=n
ELSE
fib=fiba+fibb, fiba=fibb, fibb=fib
ENDIF
IF (n!=searchfib) CYCLE
PRINT "fibionacci number ",n,"=",fib
ENDLOOP
```
Output:
```txt
What fibionacci number do you want? >12
fibionacci number 12=144
```
Output:
```txt
What fibionacci number do you want? >31
fibionacci number 31=1346269
```
Output:
```txt
What fibionacci number do you want? >46
fibionacci 46=1836311903
```
## UnixPipes
```bash
echo 1 |tee last fib ; tail -f fib | while read x
do
cat last | tee -a fib | xargs -n 1 expr $x + |tee last
done
```
## UNIX Shell
```bash
#!/bin/bash
a=0
b=1
max=$1
for (( n=1; "$n" <= "$max"; $((n++)) ))
do
a=$(($a + $b))
echo "F($n): $a"
b=$(($a - $b))
done
```
Recursive:
```bash
fib() {
local n=$1
[ $n -lt 2 ] && echo -n $n || echo -n $(( $( fib $(( n - 1 )) ) + $( fib $(( n - 2 )) ) ))
}
```
## Ursa
### Iterative
```ursa
def fibIter (int n)
if (< n 2)
return n
end if
decl int fib fibPrev num
set fib (set fibPrev 1)
for (set num 2) (< num n) (inc num)
set fib (+ fib fibPrev)
set fibPrev (- fib fibPrev)
end for
return fib
end
```
## Ursala
All three methods are shown here, and all have unlimited precision.
```Ursala
#import std
#import nat
iterative_fib = ~&/(0,1); ~&r->ll ^|\predecessor ^/~&r sum
recursive_fib = {0,1}^?* iota20
```
output:
```txt
<
<0,0,0>,
<1,1,1>,
<1,1,1>,
<2,2,2>,
<3,3,3>,
<5,5,5>,
<8,8,8>,
<13,13,13>,
<21,21,21>,
<34,34,34>,
<55,55,55>,
<89,89,89>,
<144,144,144>,
<233,233,233>,
<377,377,377>,
<610,610,610>,
<987,987,987>,
<1597,1597,1597>,
<2584,2584,2584>,
<4181,4181,4181>>
```
## V
Generate n'th fib by using binary recursion
```v
[fib
[small?] []
[pred dup pred]
[+]
binrec].
```
## Vala
### Recursive
Using int, but could easily replace with double, long, ulong, etc.
```vala
int fibRec(int n){
if (n < 2)
return n;
else
return fibRec(n - 1) + fibRec(n - 2);
}
```
### Iterative
Using int, but could easily replace with double, long, ulong, etc.
```vala
int fibIter(int n){
if (n < 2)
return n;
int last = 0;
int cur = 1;
int next;
for (int i = 1; i < n; ++i){
next = last + cur;
last = cur;
cur = next;
}
return cur;
}
```
## VAX Assembly
```VAX Assembly
0000 0000 1 .entry main,0
7E 7CFD 0002 2 clro -(sp) ;result buffer
5E DD 0005 3 pushl sp ;pointer to buffer
10 DD 0007 4 pushl #16 ;descriptor: len of buffer
5B 5E D0 0009 5 movl sp, r11 ;-> descriptor
000C 6
7E 01 7D 000C 7 movq #1, -(sp) ;init 0,1
000F 8 loop:
7E 6E 04 AE C1 000F 9 addl3 4(sp), (sp), -(sp) ;next element on stack
17 1D 0014 10 bvs ret ;vs - overflow set, exit
0016 11
5B DD 0016 12 pushl r11 ;-> descriptor by ref
04 AE DF 0018 13 pushal 4(sp) ;-> fib on stack by ref
00000000'GF 02 FB 001B 14 calls #2, g^ots$cvt_l_ti ;convert integer to string
5B DD 0022 15 pushl r11 ;
00000000'GF 01 FB 0024 16 calls #1, g^lib$put_output ;show result
E2 11 002B 17 brb loop
002D 18 ret:
04 002D 19 ret
002E 20 .end main
$ run fib
...
14930352
24157817
39088169
63245986
102334155
165580141
267914296
433494437
701408733
1134903170
1836311903
$
```
## VBA
Like Visual Basic .NET, but with keyword "Public" and type Variant (subtype Currency) instead of Decimal:
```vb
Public Function Fib(ByVal n As Integer) As Variant
Dim fib0 As Variant, fib1 As Variant, sum As Variant
Dim i As Integer
fib0 = 0
fib1 = 1
For i = 1 To n
sum = fib0 + fib1
fib0 = fib1
fib1 = sum
Next i
Fib = fib0
End Function
```
With Currency type, maximum value is fibo(73).
The (slow) recursive version:
```VBA
Public Function RFib(Term As Integer) As Long
If Term < 2 Then RFib = Term Else RFib = RFib(Term - 1) + RFib(Term - 2)
End Function
```
With Long type, maximum value is fibo(46).
## VBScript
===Non-recursive, object oriented, generator===
Defines a generator class, with a default Get property. Uses Currency for larger-than-Long values. Tests for overflow and switches to Double. Overflow information also available from class.
### =Class Definition:=
```vb
class generator
dim t1
dim t2
dim tn
dim cur_overflow
Private Sub Class_Initialize
cur_overflow = false
t1 = ccur(0)
t2 = ccur(1)
tn = ccur(t1 + t2)
end sub
public default property get generated
on error resume next
generated = ccur(tn)
if err.number <> 0 then
generated = cdbl(tn)
cur_overflow = true
end if
t1 = ccur(t2)
if err.number <> 0 then
t1 = cdbl(t2)
cur_overflow = true
end if
t2 = ccur(tn)
if err.number <> 0 then
t2 = cdbl(tn)
cur_overflow = true
end if
tn = ccur(t1+ t2)
if err.number <> 0 then
tn = cdbl(t1) + cdbl(t2)
cur_overflow = true
end if
on error goto 0
end property
public property get overflow
overflow = cur_overflow
end property
end class
```
### =Invocation:=
```vb
dim fib
set fib = new generator
dim i
for i = 1 to 100
wscript.stdout.write " " & fib
if fib.overflow then
wscript.echo
exit for
end if
next
```
### =Output:=
```vbscript> 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099 53316291173 86267571272 139583862445 225851433717 365435296162 591286729879 956722026041 1548008755920 2504730781961 4052739537881 6557470319842 10610209857723 17167680177565 27777890035288 44945570212853 72723460248141 117669030460994 190392490709135 308061521170129 498454011879264 806515533049393let memo fib n =
n { > 1 => + (fib (- n 1)) (fib (- n 2)) };
```
### Iterative
```WDTE>let s =
import 'stream';
let a => import 'arrays';
let fib n => (
let reducer p n => [a.at p 1; + (a.at p 0) (a.at p 1)];
s.range 1 n
-> s.reduce [0; 1] reducer
-> a.at 1
;
);
```
## Whitespace
### Iterative
This program generates Fibonacci numbers until it is [http://ideone.com/VBDLzk forced to terminate].
```Whitespace
```
It was generated from the following pseudo-Assembly.
```asm
push 0
push 1
0:
swap
dup
onum
push 10
ochr
copy 1
add
jump 0
```
```txt
$ wspace fib.ws | head -n 6
0
1
1
2
3
5
```
### Recursive
This program takes a number ''n'' on standard input and outputs the ''n''th member of the Fibonacci sequence.
```Whitespace
```
```asm
; Read n.
push 0
dup
inum
load
; Call fib(n), ouput the result and a newline, then exit.
call 0
onum
push 10
ochr
exit
0:
dup
push 2
sub
jn 1 ; Return if n < 2.
dup
push 1
sub
call 0 ; Call fib(n - 1).
swap ; Get n back into place.
push 2
sub
call 0 ; Call fib(n - 2).
add ; Leave the sum on the stack.
1:
ret
```
```txt
$ echo 10 | wspace fibrec.ws
55
```
## Wrapl
### Generator
```wrapl
DEF fib() (
VAR seq <- [0, 1]; EVERY SUSP seq:values;
REP SUSP seq:put(seq:pop + seq[1])[-1];
);
```
To get the 17th number:
```wrapl
16 SKIP fib();
```
To get the list of all 17 numbers:
```wrapl
ALL 17 OF fib();
```
### Iterator
Using type match signature to ensure integer argument:
```wrapl
TO fib(n @ Integer.T) (
VAR seq <- [0, 1];
EVERY 3:to(n) DO seq:put(seq:pop + seq[1]);
RET seq[-1];
);
```
## x86 Assembly
```asm
TITLE i hate visual studio 4 (Fibs.asm)
; __ __/--------\
; >__ \ / | |\
; \ \___/ @ \ / \__________________
; \____ \ / \\\
; \____ Coded with love by: |||
; \ Alexander Alvonellos |||
; | 9/29/2011 / ||
; | | MM
; | |--------------| |
; |< | |< |
; | | | |
; |mmmmmm| |mmmmm|
;; Epic Win.
INCLUDE Irvine32.inc
.data
BEERCOUNT = 48;
Fibs dd 0, 1, BEERCOUNT DUP(0);
.code
main PROC
; I am not responsible for this code.
; They made me write it, against my will.
;Here be dragons
mov esi, offset Fibs; offset array; ;;were to start (start)
mov ecx, BEERCOUNT; ;;count of items (how many)
mov ebx, 4; ;;size (in number of bytes)
call DumpMem;
mov ecx, BEERCOUNT; ;//http://www.wolframalpha.com/input/?i=F ib%5B47%5D+%3E+4294967295
mov esi, offset Fibs
NextPlease:;
mov eax, [esi]; ;//Get me the data from location at ESI
add eax, [esi+4]; ;//add into the eax the data at esi + another double (next mem loc)
mov [esi+8], eax; ;//Move that data into the memory location after the second number
add esi, 4; ;//Update the pointer
loop NextPlease; ;//Thank you sir, may I have another?
;Here be dragons
mov esi, offset Fibs; offset array; ;;were to start (start)
mov ecx, BEERCOUNT; ;;count of items (how many)
mov ebx, 4; ;;size (in number of bytes)
call DumpMem;
exit ; exit to operating system
main ENDP
END main
```
## xEec
This will display the first 93 numbers of the sequence.
```xEec
h#1 h#1 h#1 o#
h#10 o$ p
>f
o# h#10 o$ p
ma h? jnext p
t
jnf
```
## XLISP
### Analytic
Uses Binet's method, based on the golden ratio, which almost feels like cheating—but the task specification doesn't require any particular algorithm, and this one is straightforward and fast.
```lisp
(DEFUN FIBONACCI (N)
(FLOOR (+ (/ (EXPT (/ (+ (SQRT 5) 1) 2) N) (SQRT 5)) 0.5)))
```
To test it, we'll define a RANGE function and ask for the first 50 numbers in the sequence:
```lisp
(DEFUN RANGE (X Y)
(IF (<= X Y)
(CONS X (RANGE (+ X 1) Y))))
(PRINT (MAPCAR FIBONACCI (RANGE 1 50)))
```
```txt
(1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025)
```
### Tail recursive
Alternatively, this approach is reasonably efficient:
```lisp
(defun fibonacci (x)
(defun fib (a b n)
(if (= n 2)
b
(fib b (+ a b) (- n 1)) ) )
(if (< x 2)
x
(fib 1 1 x) ) )
```
## Xojo
Pass n to this function where n is the desired number of iterations. This example uses the UInt64 datatype which is as unsigned 64 bit integer. As such, it overflows after the 93rd iteration.
```vb
Function fibo(n As Integer) As UInt64
Dim noOne As UInt64 = 1
Dim noTwo As UInt64 = 1
Dim sum As UInt64
For i As Integer = 3 To n
sum = noOne + noTwo
noTwo = noOne
noOne = sum
Next
Return noOne
End Function
```
## XQuery
```xquery
declare function local:fib($n as xs:integer) as xs:integer {
if($n < 2)
then $n
else local:fib($n - 1) + local:fib($n - 2)
};
```
## zkl
A slight tweak to the task; creates a function that continuously generates fib numbers
```zkl
var fibShift=fcn(ab){ab.append(ab.sum()).pop(0)}.fp(L(0,1));
```
```txt
zkl: do(15){ fibShift().print(",") }
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,
zkl: do(5){ fibShift().print(",") }
610,987,1597,2584,4181,
```
## ZX Spectrum Basic
### =Iterative=
```zxbasic
10 REM Only positive numbers
20 LET n=10
30 LET n1=0: LET n2=1
40 FOR k=1 TO n
50 LET sum=n1+n2
60 LET n1=n2
70 LET n2=sum
80 NEXT k
90 PRINT n1
```
### =Analytic=
```zxbasic
10 DEF FN f(x)=INT (0.5+(((SQR 5+1)/2)^x)/SQR 5)
```