fib implementation in supposed log(n)

fib.hs

{-# LANGUAGE BangPatterns #-}
module Main where

import Test.QuickCheck
import System (getArgs)

data M = M !Integer !Integer !Integer !Integer

M a b c d <*> M a' b' c' d' =
  M (a*a'+b*c') (a*b'+b*d')
    (c*a'+d*c') (c*b'+d*d')

mpow :: M -> Int -> M
m `mpow` 0 = m
m `mpow` (n + 1)| odd n    = m' <*> m'
                | otherwise = (m `mpow` n) <*> M 1 1 1 0
              where m' = m `mpow` div n 2
fib :: Int -> Integer
fib n = let M x _ _ _ = identity `mpow` n in x
  where identity = M 1 0 0 1

fibs :: [Integer]
fibs = 1:1:zipWith (+) fibs (tail fibs)

test :: IO ()
test = let theorem n = fibs !! n == fib n
       in quickCheck $ forAll (choose (1,200)) theorem

main :: IO ()
main =  getArgs >>= print . fib . read . head