{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE ScopedTypeVariables #-}
-----------------------------------------------------------------------------
-- |
-- Copyright   :  (c) Edward Kmett 2010-2021
-- License     :  BSD3
-- Maintainer  :  ekmett@gmail.com
-- Stability   :  experimental
-- Portability :  GHC only
--
-- Reverse-mode automatic differentiation using Wengert lists and
-- Data.Reflection
--
-- This version is specialized to `Double` enabling the entire
-- structure to be unboxed.
--
-----------------------------------------------------------------------------

module Numeric.AD.Mode.Reverse.Double
  ( ReverseDouble, auto
  -- * Gradient
  , grad
  , grad'
  , gradWith
  , gradWith'

  -- * Jacobian
  , jacobian
  , jacobian'
  , jacobianWith
  , jacobianWith'

  -- * Hessian
  , hessian
  , hessianF

  -- * Derivatives
  , diff
  , diff'
  , diffF
  , diffF'
  ) where

import Data.Typeable
import Data.Functor.Compose
import Data.Reflection (Reifies)
import Numeric.AD.Internal.On
import qualified Numeric.AD.Internal.Reverse as R
import qualified Numeric.AD.Mode.Reverse as M
import Numeric.AD.Internal.Reverse.Double
import Numeric.AD.Mode

-- $setup
--
-- >>> import Numeric.AD.Internal.Doctest

-- | The 'grad' function calculates the gradient of a non-scalar-to-scalar function with reverse-mode AD in a single pass.
--
--
-- >>> grad (\[x,y,z] -> x*y+z) [1,2,3]
-- [2.0,1.0,1.0]
--
-- >>> grad (\[x,y] -> x**y) [0,2]
-- [0.0,NaN]
grad
  :: Traversable f
  => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s)
  -> f Double
  -> f Double
grad :: forall (f :: * -> *).
Traversable f =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (ReverseDouble s) -> ReverseDouble s)
-> f Double -> f Double
grad forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall (f :: * -> *) s.
Functor f =>
f (ReverseDouble s) -> Array Int Double -> f Double
unbind forall {s}. f (ReverseDouble s)
vs forall a b. (a -> b) -> a -> b
$! forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall a b. (a -> b) -> a -> b
$! forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f forall {s}. f (ReverseDouble s)
vs where
  (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE grad #-}

-- | The 'grad'' function calculates the result and gradient of a non-scalar-to-scalar function with reverse-mode AD ƒin a single pass.
--
-- >>> grad' (\[x,y,z] -> x*y+z) [1,2,3]
-- (5.0,[2.0,1.0,1.0])
grad'
  :: Traversable f
  => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s)
  -> f Double
  -> (Double, f Double)
grad' :: forall (f :: * -> *).
Traversable f =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (ReverseDouble s) -> ReverseDouble s)
-> f Double -> (Double, f Double)
grad' forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> case forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f forall {s}. f (ReverseDouble s)
vs of
   ReverseDouble s
r -> (forall s. ReverseDouble s -> Double
primal ReverseDouble s
r, forall (f :: * -> *) s.
Functor f =>
f (ReverseDouble s) -> Array Int Double -> f Double
unbind forall {s}. f (ReverseDouble s)
vs forall a b. (a -> b) -> a -> b
$! forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall a b. (a -> b) -> a -> b
$! ReverseDouble s
r)
  where (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE grad' #-}

-- | @'grad' g f@ function calculates the gradient of a non-scalar-to-scalar function @f@ with reverse-mode AD in a single pass.
-- The gradient is combined element-wise with the argument using the function @g@.
--
-- @
-- 'grad' == 'gradWith' (\_ dx -> dx)
-- 'id' == 'gradWith' 'const'
-- @
gradWith
  :: Traversable f
  => (Double -> Double -> b)
  -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s)
  -> f Double
  -> f b
gradWith :: forall (f :: * -> *) b.
Traversable f =>
(Double -> Double -> b)
-> (forall s.
    (Reifies s Tape, Typeable s) =>
    f (ReverseDouble s) -> ReverseDouble s)
-> f Double
-> f b
gradWith Double -> Double -> b
g forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall (f :: * -> *) b c s.
Functor f =>
(Double -> b -> c) -> f (ReverseDouble s) -> Array Int b -> f c
unbindWith Double -> Double -> b
g forall {s}. f (ReverseDouble s)
vs forall a b. (a -> b) -> a -> b
$! forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall a b. (a -> b) -> a -> b
$! forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f forall {s}. f (ReverseDouble s)
vs
  where (f (ReverseDouble s)
vs,(Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE gradWith #-}

-- | @'grad'' g f@ calculates the result and gradient of a non-scalar-to-scalar function @f@ with reverse-mode AD in a single pass
-- the gradient is combined element-wise with the argument using the function @g@.
--
-- @
-- 'grad'' == 'gradWith'' (\_ dx -> dx)
-- @
gradWith'
  :: Traversable f
  => (Double -> Double -> b)
  -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s)
  -> f Double
  -> (Double, f b)
gradWith' :: forall (f :: * -> *) b.
Traversable f =>
(Double -> Double -> b)
-> (forall s.
    (Reifies s Tape, Typeable s) =>
    f (ReverseDouble s) -> ReverseDouble s)
-> f Double
-> (Double, f b)
gradWith' Double -> Double -> b
g forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> case forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> ReverseDouble s
f forall {s}. f (ReverseDouble s)
vs of
   ReverseDouble s
r -> (forall s. ReverseDouble s -> Double
primal ReverseDouble s
r, forall (f :: * -> *) b c s.
Functor f =>
(Double -> b -> c) -> f (ReverseDouble s) -> Array Int b -> f c
unbindWith Double -> Double -> b
g forall {s}. f (ReverseDouble s)
vs forall a b. (a -> b) -> a -> b
$! forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall a b. (a -> b) -> a -> b
$! ReverseDouble s
r)
  where (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE gradWith' #-}

-- | The 'jacobian' function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in @m@ passes for @m@ outputs.
--
-- >>> jacobian (\[x,y] -> [y,x,x*y]) [2,1]
-- [[0.0,1.0],[1.0,0.0],[1.0,2.0]]
jacobian
  :: (Traversable f, Functor g)
  => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s))
  -> f Double
  -> g (f Double)
jacobian :: forall (f :: * -> *) (g :: * -> *).
(Traversable f, Functor g) =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (ReverseDouble s) -> g (ReverseDouble s))
-> f Double -> g (f Double)
jacobian forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall (f :: * -> *) s.
Functor f =>
f (ReverseDouble s) -> Array Int Double -> f Double
unbind forall {s}. f (ReverseDouble s)
vs forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f forall {s}. f (ReverseDouble s)
vs where
  (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE jacobian #-}

-- | The 'jacobian'' function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using @m@ invocations of reverse AD,
-- where @m@ is the output dimensionality. Applying @fmap snd@ to the result will recover the result of 'jacobian'
-- | An alias for 'gradF''
--
-- >>> jacobian' (\[x,y] -> [y,x,x*y]) [2,1]
-- [(1.0,[0.0,1.0]),(2.0,[1.0,0.0]),(2.0,[1.0,2.0])]
jacobian'
  :: (Traversable f, Functor g)
  => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s))
  -> f Double
  -> g (Double, f Double)
jacobian' :: forall (f :: * -> *) (g :: * -> *).
(Traversable f, Functor g) =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (ReverseDouble s) -> g (ReverseDouble s))
-> f Double -> g (Double, f Double)
jacobian' forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p ->
  let row :: ReverseDouble s -> (Double, f Double)
row ReverseDouble s
a = (forall s. ReverseDouble s -> Double
primal ReverseDouble s
a, forall (f :: * -> *) s.
Functor f =>
f (ReverseDouble s) -> Array Int Double -> f Double
unbind forall {s}. f (ReverseDouble s)
vs forall a b. (a -> b) -> a -> b
$! forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall a b. (a -> b) -> a -> b
$! ReverseDouble s
a)
  in ReverseDouble s -> (Double, f Double)
row forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f forall {s}. f (ReverseDouble s)
vs
  where (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE jacobian' #-}

-- | 'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function @f@ with reverse AD lazily in @m@ passes for @m@ outputs.
--
-- Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the @g@.
--
-- @
-- 'jacobian' == 'jacobianWith' (\_ dx -> dx)
-- 'jacobianWith' 'const' == (\f x -> 'const' x '<$>' f x)
-- @
jacobianWith
  :: (Traversable f, Functor g)
  => (Double -> Double -> b)
  -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s))
  -> f Double
  -> g (f b)
jacobianWith :: forall (f :: * -> *) (g :: * -> *) b.
(Traversable f, Functor g) =>
(Double -> Double -> b)
-> (forall s.
    (Reifies s Tape, Typeable s) =>
    f (ReverseDouble s) -> g (ReverseDouble s))
-> f Double
-> g (f b)
jacobianWith Double -> Double -> b
g forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall (f :: * -> *) b c s.
Functor f =>
(Double -> b -> c) -> f (ReverseDouble s) -> Array Int b -> f c
unbindWith Double -> Double -> b
g forall {s}. f (ReverseDouble s)
vs forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f forall {s}. f (ReverseDouble s)
vs where
  (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE jacobianWith #-}

-- | 'jacobianWith' g f' calculates both the result and the Jacobian of a nonscalar-to-nonscalar function @f@, using @m@ invocations of reverse AD,
-- where @m@ is the output dimensionality. Applying @fmap snd@ to the result will recover the result of 'jacobianWith'
--
-- Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the @g@.
--
-- @'jacobian'' == 'jacobianWith'' (\_ dx -> dx)@
--
jacobianWith'
  :: (Traversable f, Functor g)
  => (Double -> Double -> b)
  -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s))
  -> f Double
  -> g (Double, f b)
jacobianWith' :: forall (f :: * -> *) (g :: * -> *) b.
(Traversable f, Functor g) =>
(Double -> Double -> b)
-> (forall s.
    (Reifies s Tape, Typeable s) =>
    f (ReverseDouble s) -> g (ReverseDouble s))
-> f Double
-> g (Double, f b)
jacobianWith' Double -> Double -> b
g forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f f Double
as = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape (forall a b. (a, b) -> b
snd (Int, Int)
bds) forall a b. (a -> b) -> a -> b
$ \Proxy s
p ->
  let row :: ReverseDouble s -> (Double, f b)
row ReverseDouble s
a = (forall s. ReverseDouble s -> Double
primal ReverseDouble s
a, forall (f :: * -> *) b c s.
Functor f =>
(Double -> b -> c) -> f (ReverseDouble s) -> Array Int b -> f c
unbindWith Double -> Double -> b
g forall {s}. f (ReverseDouble s)
vs forall a b. (a -> b) -> a -> b
$! forall s.
Reifies s Tape =>
Proxy s -> (Int, Int) -> ReverseDouble s -> Array Int Double
partialArrayOf Proxy s
p (Int, Int)
bds forall a b. (a -> b) -> a -> b
$! ReverseDouble s
a)
  in ReverseDouble s -> (Double, f b)
row forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall s.
(Reifies s Tape, Typeable s) =>
f (ReverseDouble s) -> g (ReverseDouble s)
f forall {s}. f (ReverseDouble s)
vs
  where (f (ReverseDouble s)
vs, (Int, Int)
bds) = forall (f :: * -> *) s.
Traversable f =>
f Double -> (f (ReverseDouble s), (Int, Int))
bind f Double
as
{-# INLINE jacobianWith' #-}

-- | Compute the derivative of a function.
--
-- >>> diff sin 0
-- 1.0
diff
  :: (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> ReverseDouble s)
  -> Double
  -> Double
diff :: (forall s.
 (Reifies s Tape, Typeable s) =>
 ReverseDouble s -> ReverseDouble s)
-> Double -> Double
diff forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> ReverseDouble s
f Double
a = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape Int
1 forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall s. Reifies s Tape => Proxy s -> ReverseDouble s -> Double
derivativeOf Proxy s
p forall a b. (a -> b) -> a -> b
$! forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> ReverseDouble s
f (forall s. Double -> Int -> ReverseDouble s
var Double
a Int
0)
{-# INLINE diff #-}

-- | The 'diff'' function calculates the result and derivative, as a pair, of a scalar-to-scalar function.
--
-- >>> diff' sin 0
-- (0.0,1.0)
--
-- >>> diff' exp 0
-- (1.0,1.0)
diff'
  :: (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> ReverseDouble s)
  -> Double
  -> (Double, Double)
diff' :: (forall s.
 (Reifies s Tape, Typeable s) =>
 ReverseDouble s -> ReverseDouble s)
-> Double -> (Double, Double)
diff' forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> ReverseDouble s
f Double
a = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape Int
1 forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall s.
Reifies s Tape =>
Proxy s -> ReverseDouble s -> (Double, Double)
derivativeOf' Proxy s
p forall a b. (a -> b) -> a -> b
$! forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> ReverseDouble s
f (forall s. Double -> Int -> ReverseDouble s
var Double
a Int
0)
{-# INLINE diff' #-}

-- | Compute the derivatives of each result of a scalar-to-vector function with regards to its input.
--
-- >>> diffF (\a -> [sin a, cos a]) 0
-- [1.0,0.0]
--
diffF
  :: Functor f
  => (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> f (ReverseDouble s))
  -> Double
  -> f Double
diffF :: forall (f :: * -> *).
Functor f =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 ReverseDouble s -> f (ReverseDouble s))
-> Double -> f Double
diffF forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> f (ReverseDouble s)
f Double
a = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape Int
1 forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall s. Reifies s Tape => Proxy s -> ReverseDouble s -> Double
derivativeOf Proxy s
p forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> f (ReverseDouble s)
f (forall s. Double -> Int -> ReverseDouble s
var Double
a Int
0)
{-# INLINE diffF #-}

-- | Compute the derivatives of each result of a scalar-to-vector function with regards to its input along with the answer.
--
-- >>> diffF' (\a -> [sin a, cos a]) 0
-- [(0.0,1.0),(1.0,0.0)]
diffF'
  :: Functor f
  => (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> f (ReverseDouble s))
  -> Double
  -> f (Double, Double)
diffF' :: forall (f :: * -> *).
Functor f =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 ReverseDouble s -> f (ReverseDouble s))
-> Double -> f (Double, Double)
diffF' forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> f (ReverseDouble s)
f Double
a = forall r.
Int
-> (forall s. (Reifies s Tape, Typeable s) => Proxy s -> r) -> r
reifyTypeableTape Int
1 forall a b. (a -> b) -> a -> b
$ \Proxy s
p -> forall s.
Reifies s Tape =>
Proxy s -> ReverseDouble s -> (Double, Double)
derivativeOf' Proxy s
p forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall s.
(Reifies s Tape, Typeable s) =>
ReverseDouble s -> f (ReverseDouble s)
f (forall s. Double -> Int -> ReverseDouble s
var Double
a Int
0)
{-# INLINE diffF' #-}

-- | Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in reverse mode.
--
-- However, since the @'grad' f :: f a -> f a@ is square this is not as fast as using the forward-mode Jacobian of a reverse mode gradient provided by 'Numeric.AD.hessian'.
--
-- >>> hessian (\[x,y] -> x*y) [1,2]
-- [[0.0,1.0],[1.0,0.0]]
hessian
  :: Traversable f
  => (forall s s'.
       (Reifies s R.Tape, Typeable s, Reifies s' Tape, Typeable s') =>
       f (On (R.Reverse s (ReverseDouble s'))) -> On (R.Reverse s (ReverseDouble s')))
  -> f Double
  -> f (f Double)
hessian :: forall (f :: * -> *).
Traversable f =>
(forall s s'.
 (Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') =>
 f (On (Reverse s (ReverseDouble s')))
 -> On (Reverse s (ReverseDouble s')))
-> f Double -> f (f Double)
hessian forall s s'.
(Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') =>
f (On (Reverse s (ReverseDouble s')))
-> On (Reverse s (ReverseDouble s'))
f = forall (f :: * -> *) (g :: * -> *).
(Traversable f, Functor g) =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (ReverseDouble s) -> g (ReverseDouble s))
-> f Double -> g (f Double)
jacobian (forall (f :: * -> *) a.
(Traversable f, Num a) =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (Reverse s a) -> Reverse s a)
-> f a -> f a
M.grad (forall t. On t -> t
off forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall s s'.
(Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') =>
f (On (Reverse s (ReverseDouble s')))
-> On (Reverse s (ReverseDouble s'))
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall t. t -> On t
On))
{-# INLINE hessian #-}

-- | Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the reverse-mode Jacobian of the reverse-mode Jacobian of the function.
--
-- Less efficient than 'Numeric.AD.Mode.Mixed.hessianF'.
--
-- >>> hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2 :: Double]
-- [[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]
hessianF
  :: (Traversable f, Functor g)
  => (forall s s'.
        (Reifies s R.Tape, Typeable s, Reifies s' Tape, Typeable s') =>
        f (On (R.Reverse s (ReverseDouble s'))) -> g (On (R.Reverse s (ReverseDouble s'))))
  -> f Double
  -> g (f (f Double))
hessianF :: forall (f :: * -> *) (g :: * -> *).
(Traversable f, Functor g) =>
(forall s s'.
 (Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') =>
 f (On (Reverse s (ReverseDouble s')))
 -> g (On (Reverse s (ReverseDouble s'))))
-> f Double -> g (f (f Double))
hessianF forall s s'.
(Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') =>
f (On (Reverse s (ReverseDouble s')))
-> g (On (Reverse s (ReverseDouble s')))
f = forall {k1} {k2} (f :: k1 -> *) (g :: k2 -> k1) (a :: k2).
Compose f g a -> f (g a)
getCompose forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) (g :: * -> *).
(Traversable f, Functor g) =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (ReverseDouble s) -> g (ReverseDouble s))
-> f Double -> g (f Double)
jacobian (forall {k} {k1} (f :: k -> *) (g :: k1 -> k) (a :: k1).
f (g a) -> Compose f g a
Compose forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) (g :: * -> *) a.
(Traversable f, Functor g, Num a) =>
(forall s.
 (Reifies s Tape, Typeable s) =>
 f (Reverse s a) -> g (Reverse s a))
-> f a -> g (f a)
M.jacobian (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall t. On t -> t
off forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall s s'.
(Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') =>
f (On (Reverse s (ReverseDouble s')))
-> g (On (Reverse s (ReverseDouble s')))
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall t. t -> On t
On))
{-# INLINE hessianF #-}