# The Fibonacci sequence

(Difference between revisions)

## 1 Naive solution

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

## 2 Canonical zipWith implementation

fib = 1 : 1 : zipWith (+) fib (tail fib)

## 3 With scanl

fib = fix ((1:) . scanl (+) 1)

## 4 With unfoldr

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

## 5 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)

## 6 Fastest Fib in the West

This was contributed by wli

import System.Environment
import Data.List

fib 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))

main = getArgs >>= mapM_ (print . fib . read)