{-# LANGUAGE BangPatterns #-}

import Data.List 
import Data.Ratio

facs :: [Integer]
facs = 1 : zipWith (*) [1..] facs

tangents :: [Integer]
tangents = go 1 facs
  where go k (!x : xs) = x : go (k + 1) (go' 0 x xs)
        go' !j !x (!y : ys) = let z = j * x + (j + 2) * y in z `seq` z : go' (j + 1) z ys

tangent = genericIndex tangents

bernoulli :: Integer -> Rational
bernoulli 0 = 1
bernoulli 1 = -0.5
bernoulli n | odd n = 0
bernoulli n = negateIfOdd (n `div` 2) $ (n * tangent (n `div` 2 - 1)) % (2 ^ n - 4 ^ n)
  where negateIfOdd n = if even n then id else negate