Compute formal power series for functional square root of sin function

comb.hs

import Data.Ratio

(^+) a b = zipWith (+) a b
(^-) a b = zipWith (-) a b

(a : as) `convolve` (b : bs) = (a * b) :
    ((map (a *) bs) ^+ (as `convolve` (b : bs)))

compose (f : fs) (0 : gs) = f : (gs `convolve` (compose fs (0 : gs)))

integrate x = 0 : zipWith (/) x (map fromInteger [1..])

sin' = integrate cos'
cos' = let _ : cos'' = map negate (integrate sin') in 1 : cos''

delta (g : gs) h = let g' = delta gs h
                   in (0 : ((1 : h) `convolve` g')) ^+ gs

fsqrt (0 : 1 : fs) =
    let gs = map (/ 2) $ fs ^- (0 : gs `convolve`
                                ((0 : delta gs gs) ^+
                                 ((2 : gs) `convolve` (gs `compose` g))))
        g = 0 : 1 : gs
    in g

type Q = Ratio Integer
type R = Double

main = mapM_ print $ zip [0..] (fsqrt sin' :: [Q])