```{-# LANGUAGE Rank2Types #-}
-----------------------------------------------------------------------------
-- |
-- Copyright   :  (c) Edward Kmett 2010
-- Maintainer  :  ekmett@gmail.com
-- Stability   :  experimental
-- Portability :  GHC only
--
-- Forward mode automatic differentiation
--
-----------------------------------------------------------------------------

(
-- * Jacobian
, jacobian
, jacobian'
, jacobianWith
, jacobianWith'
-- * Transposed Jacobian
, jacobianT
, jacobianWithT
-- * Hessian Product
, hessianProduct
, hessianProduct'
-- * Derivatives
, diff
, diff'
, diffF
, diffF'
-- * Directional Derivatives
, du
, du'
, duF
, duF'
-- * Exposed Types
, UU, UF, FU, FF
, Mode(..)
) where

import Data.Traversable (Traversable)
import Control.Applicative

du :: (Functor f, Num a) => FU f a -> f (a, a) -> a
du f = tangent . f . fmap (uncurry bundle)
{-# INLINE du #-}

du' :: (Functor f, Num a) => FU f a -> f (a, a) -> (a, a)
du' f = unbundle . f . fmap (uncurry bundle)
{-# INLINE du' #-}

duF :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g a
duF f = fmap tangent . f . fmap (uncurry bundle)
{-# INLINE duF #-}

duF' :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g (a, a)
duF' f = fmap unbundle . f . fmap (uncurry bundle)
{-# INLINE duF' #-}

-- | The 'diff' function calculates the first derivative of a scalar-to-scalar function by forward-mode 'AD'
--
-- > diff sin == cos
diff :: Num a => UU a -> a -> a
diff f a = tangent \$ apply f a
{-# INLINE diff #-}

-- | The 'd'UU' function calculates the result and first derivative of scalar-to-scalar function by F'orward' 'AD'
--
-- > d' sin == sin &&& cos
-- > d' f = f &&& d f
diff' :: Num a => UU a -> a -> (a, a)
diff' f a = unbundle \$ apply f a
{-# INLINE diff' #-}

-- | The 'diffF' function calculates the first derivative of scalar-to-nonscalar function by F'orward' 'AD'
diffF :: (Functor f, Num a) => UF f a -> a -> f a
diffF f a = tangent <\$> apply f a
{-# INLINE diffF #-}

-- | The 'diffF'' function calculates the result and first derivative of a scalar-to-non-scalar function by F'orward' 'AD'
diffF' :: (Functor f, Num a) => UF f a -> a -> f (a, a)
diffF' f a = unbundle <\$> apply f a
{-# INLINE diffF' #-}

-- | A fast, simple transposed Jacobian computed with forward-mode AD.
jacobianT :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> f (g a)
jacobianT f = bind (fmap tangent . f)
{-# INLINE jacobianT #-}

-- | A fast, simple transposed Jacobian computed with forward-mode AD.
jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> f (g b)
jacobianWithT g f = bindWith g' f
where g' a ga = g a . tangent <\$> ga
{-# INLINE jacobianWithT #-}

jacobian :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (f a)
jacobian f as = transposeWith (const id) t p
where
(p, t) = bind' (fmap tangent . f) as
{-# INLINE jacobian #-}

jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)
jacobianWith g f as = transposeWith (const id) t p
where
(p, t) = bindWith' g' f as
g' a ga = g a . tangent <\$> ga
{-# INLINE jacobianWith #-}

jacobian' :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (a, f a)
jacobian' f as = transposeWith row t p
where
(p, t) = bind' f as
row x as' = (primal x, tangent <\$> as')
{-# INLINE jacobian' #-}

jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)
jacobianWith' g f as = transposeWith row t p
where
(p, t) = bindWith' g' f as
row x as' = (primal x, as')
g' a ga = g a . tangent <\$> ga
{-# INLINE jacobianWith' #-}

grad :: (Traversable f, Num a) => FU f a -> f a -> f a
grad f = bind (tangent . f)

grad' :: (Traversable f, Num a) => FU f a -> f a -> (a, f a)
grad' f as = (primal b, tangent <\$> bs)
where
(b, bs) = bind' f as

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> f b
gradWith g f = bindWith g (tangent . f)

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> (a, f b)
gradWith' g f = bindWith' g (tangent . f)

-- | Compute the product of a vector with the Hessian using forward-on-forward-mode AD.
hessianProduct :: (Traversable f, Num a) => FU f a -> f (a, a) -> f a
hessianProduct f = duF \$ grad \$ decomposeMode . f . fmap composeMode