taylor series representation of a function

taylorseries.hs

taylor :: (Floating a, Eq a) => Expr a -> [Expr a]
taylor expr = fmap fullSimplify (fmap series exprs)
  where indices = fmap fromIntegral [1..]
        derivs  = fmap (changeVars 'a') (ddxs expr)
          where changeVars c = mapVar (\_ -> Var c)
        facts   = fmap Const $ scanl1 (*) indices
        exprs   = zip (zipWith (:/:) derivs facts) indices -- f^(n)(a)/n!
        series (expr, n) = 
          expr :*: ((Var 'x' :+: (negate' $ Var 'a')) :^: Const n) -- f^(n)(a)/n! * (x - a)^n

maclaurin = fmap (fullSimplify . plugIn 'a' 0) . taylor 

evalTaylorWithPrecision a x prec =  
  sum . map (evalExpr' . plugIn 'x' x . plugIn 'a' a) . take prec . taylor

evalTaylor a x = evalTaylorWithPrecision a x 100
        
evalMaclaurin = evalTaylor 0
evalMacLaurinWithPrecision = evalTaylorWithPrecision 0