The Fibonacci sequence

Implementing the Fibonacci sequence is considered the "Hello, world!" of Haskell programming. This page collects Haskell implementations of the sequence.

Naive definition

fib 0 = 0
fib 1 = 1
fib n = fib (n-1) + fib (n-2)

Linear-time implementations

One can compute the first n Fibonacci numbers with O(n) additions. If fibs is the infinite list of Fibonacci numbers, one can define

fib n = fibs!!n

Canonical zipWith implementation

fibs = 0 : 1 : zipWith (+) fibs (tail fibs)

With scanl

fibs = fix ((0:) . scanl (+) 1)

With unfoldr

fibs = unfoldr (\(f1,f2) -> Just (f1,(f2,f1+f2))) (0,1)

With iterate

fibs = map fst $ iterate (\(f1,f2) -> (f2,f1+f2)) (0,1)

Log-time implementations

Using 2x2 matrices

The argument of iterate above is a linear transformation, so we can represent it as matrix and compute the nth power of this matrix with O(log n) multiplications and additions. For example, using the simple matrix implementation in Prelude extensions,

fib n = head (apply (Matrix [[0,1], [1,1]] ^ n) [0,1])

This technique works for any linear recurrence.

A fairly fast version, using some identities

fib 0 = 0
fib 1 = 1
fib n | even n         = f1 * (f1 + 2 * f2)
      | n `mod` 4 == 1 = (2 * f1 + f2) * (2 * f1 - f2) + 2
      | otherwise      = (2 * f1 + f2) * (2 * f1 - f2) - 2
   where k = n `div` 2
         f1 = fib k
         f2 = fib (k-1)

Another fast fib

fib = fst . fib2

-- | Return (fib n, fib (n + 1))
fib2 0 = (1, 1)
fib2 1 = (1, 2)
fib2 n
 | even n    = (a*a + b*b, c*c - a*a)
 | otherwise = (c*c - a*a, b*b + c*c)
 where (a,b) = fib2 (n `div` 2 - 1)
       c     = a + b

Fastest Fib in the West

This was contributed by wli (It assumes that the sequence starts with 1.)

import Data.List

fib1 n = snd . foldl fib' (1, 0) . map (toEnum . fromIntegral) $ unfoldl divs n
    where
        unfoldl f x = case f x of
                Nothing     -> []
                Just (u, v) -> unfoldl f v ++ [u]

        divs 0 = Nothing
        divs k = Just (uncurry (flip (,)) (k `divMod` 2))

        fib' (f, g) p
            | p         = (f*(f+2*g), f^2 + g^2)
            | otherwise = (f^2+g^2,   g*(2*f-g))