digits-of-e1.hs

{-
Compute the digits of "e" using continued fractions.
Original program due to Dale Thurston, Aug 2001
-}

import System.Environment

type ContFrac = [Integer]

{-
Compute the decimal representation of e progressively.

A continued fraction expansion for e is

[2,1,2,1,1,4,1,1,6,1,...]
-}

eContFrac :: ContFrac
eContFrac = 2:aux 2 where aux n = 1:n:1:aux (n+2)

{-
We need a general function that applies an arbitrary linear fractional
transformation to a legal continued fraction, represented as a list of
positive integers.  The complicated guard is to see if we can output a
digit regardless of what the input is; i.e., to see if the interval
[1,infinity) is mapped into [k,k+1) for some k.
-}

-- ratTrans (a,b,c,d) x: compute (a + bx)/(c+dx) as a continued fraction
ratTrans :: (Integer,Integer,Integer,Integer) -> ContFrac -> ContFrac
-- Output a digit if we can
ratTrans (a,b,c,d) xs |
  ((signum c == signum d) || (abs c < abs d)) && -- No pole in range
  (c+d)*q <= a+b && (c+d)*q + (c+d) > a+b       -- Next digit is determined
     = q:ratTrans (c,d,a-q*c,b-q*d) xs
  where q = b `div` d
ratTrans (a,b,c,d) (x:xs) = ratTrans (b,a+x*b,d,c+x*d) xs

-- Finally, we convert a continued fraction to digits by repeatedly multiplying by 10.

toDigits :: ContFrac -> [Integer]
toDigits (x:xs) = x:toDigits (ratTrans (10,0,0,1) xs)

e :: [Integer]
e = toDigits eContFrac

main = do
    [digits] <- getArgs
    print (take (read digits) e)