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

(

-- * Sparse Jacobians (synonyms)
, jacobian
, jacobian'
, jacobianWith
, jacobianWith'
, jacobians

-- * Sparse Hessians
, hessian
, hessian'

, hessianF
, hessianF'

-- * Exposed Types
, Mode(..)
) where

import Control.Applicative ((<\$>))
import Data.Traversable

second :: (a -> b) -> (c, a) -> (c, b)
second g (a,b) = (a, g b)
{-# INLINE second #-}

grad :: (Traversable f, Num a) => FU f a -> f a -> f a
grad f as = d as \$ apply f as

grad' :: (Traversable f, Num a) => FU f a -> f a -> (a, f a)
grad' f as = d' as \$ apply f as

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> f b
gradWith g f as = zipWithT g as \$ grad f as

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> (a, f b)
gradWith' g f as = second (zipWithT g as) \$ grad' f as

jacobian :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (f a)
jacobian f as = d as <\$> apply f as
{-# INLINE jacobian #-}

jacobian' :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (a, f a)
jacobian' f as = d' as <\$> apply f as
{-# INLINE jacobian' #-}

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)
jacobianWith g f as = zipWithT g as <\$> jacobian f as
{-# INLINE jacobianWith #-}

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)
jacobianWith' g f as = second (zipWithT g as) <\$> jacobian' f as
{-# INLINE jacobianWith' #-}

grads :: (Traversable f, Num a) => FU f a -> f a -> Cofree f a
grads f as = ds as \$ apply f as

jacobians :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (Cofree f a)
jacobians f as = ds as <\$> apply f as
{-# INLINE jacobians #-}

d2 :: Functor f => Cofree f a -> f (f a)
d2 = headT . tailT . tailT . tensors
{-# INLINE d2 #-}

d2' :: Functor f => Cofree f a -> (a, f (a, f a))
d2' (a :< as) = (a, fmap (\(da :< das) -> (da, extract <\$> das)) as)
{-# INLINE d2' #-}

hessian :: (Traversable f, Num a) => FU f a -> f a -> f (f a)
hessian f as = d2 \$ grads f as
{-# INLINE hessian #-}

hessian' :: (Traversable f, Num a) => FU f a -> f a -> (a, f (a, f a))
hessian' f as = d2' \$ grads f as
{-# INLINE hessian' #-}

hessianF :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (f (f a))
hessianF f as = d2 <\$> jacobians f as
{-# INLINE hessianF #-}

hessianF' :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (a, f (a, f a))
hessianF' f as = d2' <\$> jacobians f as
{-# INLINE hessianF' #-}

```