```{-# LANGUAGE CPP #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE BangPatterns #-}
-----------------------------------------------------------------------------
-- |
-- Copyright   : (c) Edward Kmett 2010-2015
-- Maintainer  : ekmett@gmail.com
-- Stability   : experimental
-- Portability : GHC only
--
-- Higher order derivatives via a \"dual number tower\".
--
-----------------------------------------------------------------------------

( Tower
, auto
-- * Taylor Series
, taylor
, taylor0
-- * Maclaurin Series
, maclaurin
, maclaurin0
-- * Derivatives
, diff    -- first derivative of (a -> a)
, diff'   -- answer and first derivative of (a -> a)
, diffs   -- answer and all derivatives of (a -> a)
, diffs0  -- zero padded derivatives of (a -> a)
, diffsF  -- answer and all derivatives of (a -> f a)
, diffs0F -- zero padded derivatives of (a -> f a)
-- * Directional Derivatives
, du      -- directional derivative of (f a -> a)
, du'     -- answer and directional derivative of (f a -> a)
, dus     -- answer and all directional derivatives of (f a -> a)
, dus0    -- answer and all zero padded directional derivatives of (f a -> a)
, duF     -- directional derivative of (f a -> g a)
, duF'    -- answer and directional derivative of (f a -> g a)
, dusF    -- answer and all directional derivatives of (f a -> g a)
, dus0F   -- answer and all zero padded directional derivatives of (f a -> g a)
) where

import Data.Functor ((<\$>))
#endif

-- | Compute the answer and all derivatives of a function @(a -> a)@
diffs :: Num a => (Tower a -> Tower a) -> a -> [a]
diffs f a = getADTower \$ apply f a
{-# INLINE diffs #-}

-- | Compute the zero-padded derivatives of a function @(a -> a)@
diffs0 :: Num a => (Tower a -> Tower a) -> a -> [a]
diffs0 f a = zeroPad (diffs f a)
{-# INLINE diffs0 #-}

-- | Compute the answer and all derivatives of a function @(a -> f a)@
diffsF :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
diffsF f a = getADTower <\$> apply f a
{-# INLINE diffsF #-}

-- | Compute the zero-padded derivatives of a function @(a -> f a)@
diffs0F :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
{-# INLINE diffs0F #-}

-- | @taylor f x@ compute the Taylor series of @f@ around @x@.
taylor :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
taylor f x dx = go 1 1 (diffs f x) where
go !n !acc (a:as) = a * acc : go (n + 1) (acc * dx / n) as
go _ _ [] = []

-- | @taylor0 f x@ compute the Taylor series of @f@ around @x@, zero-padded.
taylor0 :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
taylor0 f x dx = zeroPad (taylor f x dx)
{-# INLINE taylor0 #-}

-- | @maclaurin f@ compute the Maclaurin series of @f@
maclaurin :: Fractional a => (Tower a -> Tower a) -> a -> [a]
maclaurin f = taylor f 0
{-# INLINE maclaurin #-}

-- | @maclaurin f@ compute the Maclaurin series of @f@, zero-padded
maclaurin0 :: Fractional a => (Tower a -> Tower a) -> a -> [a]
maclaurin0 f = taylor0 f 0
{-# INLINE maclaurin0 #-}

-- | Compute the first derivative of a function @(a -> a)@
diff :: Num a => (Tower a -> Tower a) -> a -> a
diff f = d . diffs f
{-# INLINE diff #-}

-- | Compute the answer and first derivative of a function @(a -> a)@
diff' :: Num a => (Tower a -> Tower a) -> a -> (a, a)
diff' f = d' . diffs f
{-# INLINE diff' #-}

-- | Compute a directional derivative of a function @(f a -> a)@
du :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> a
du f = d . getADTower . f . fmap withD
{-# INLINE du #-}

-- | Compute the answer and a directional derivative of a function @(f a -> a)@
du' :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> (a, a)
du' f = d' . getADTower . f . fmap withD
{-# INLINE du' #-}

-- | Compute a directional derivative of a function @(f a -> g a)@
duF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g a
duF f = fmap (d . getADTower) . f . fmap withD
{-# INLINE duF #-}

-- | Compute the answer and a directional derivative of a function @(f a -> g a)@
duF' :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g (a, a)
duF' f = fmap (d' . getADTower) . f . fmap withD
{-# INLINE duF' #-}

-- | Given a function @(f a -> a)@, and a tower of derivatives, compute the corresponding directional derivatives.
dus :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
dus f = getADTower . f . fmap tower
{-# INLINE dus #-}

-- | Given a function @(f a -> a)@, and a tower of derivatives, compute the corresponding directional derivatives, zero-padded
dus0 :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]