,_p      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijkl m n o !(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone9:;<=?DIORTjallowed to return False for items with a zero derivative, but we'll give more NaNs than strictly necessaryUallowed to return False for zero, but we give more NaN's than strictly necessary thenEmbed a constantScalar-vector multiplicationVector-scalar multiplicationScalar division  = lift 0    777(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlySafe/0:QRTp3Used to sidestep the need for UndecidableInstances.A C is a tower of all (higher order) partial derivatives of a functionAt each step, a  f& is wrapped in another layer worth of f. )a :- f a :- f (f a) :- f (f (f a)) :- ...Take the tail of a .Take the head of a . Construct a  by unzipping the layers of a q Comonad. prs !" prs !"3(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone69:;<=?DIORT##C is useful for defining new AD primitives in a fairly generic way.#$%&'()*#$(%&')*#$%&'()*#$%&'()*None0DIR+,-.+,-+,-+,-.(c) Edward Kmett 2015BSD3ekmett@gmail.com experimentalGHC onlyNone %&0:DR=>The choice between two AD modes is an AD mode in its own right=>?@ABCDEFGHtIJKLMNOPQRSTUVW =?>@ABCDEFGH =>?CBDE@AFGH=>?@ABCDEFGHtIJKLMNOPQRSTUVW(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone /0:DIQRX>The composition of two AD modes is an AD mode in its own rightXYZ[XYZXYZXYZ[(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone 09:;<=DIR hijkluvmnohijklmnhijklmnhijkluvmno!(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNonewZip a x f with a y g assuming f! has at least as many entries as g.zZip a x f with a y g assuming f, using a default value after f is exhausted.{"Used internally to define various | combinators.}"Used internally to define various | combinators.~aUsed internally to implement functions which truncate lists after the stream of results convergewz{}~wz{}~wz{}~(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone09:;<=DORT (c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone09;<=DORT mode ADCalculate the  using forward mode AD. (c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneCCompute the directional derivative of a function given a zipped up * of the input values and their derivativesNCompute the answer and directional derivative of a function given a zipped up * of the input values and their derivativesMCompute a vector of directional derivatives for a function given a zipped up + of the input values and their derivatives.YCompute a vector of answers and directional derivatives for a function given a zipped up + of the input values and their derivatives.The Y function calculates the first derivative of a scalar-to-scalar function by forward-mode AD diff sin 01.0The U function calculates the result and first derivative of scalar-to-scalar function by  mode AD   ==  "#   f = f "# d f  diff' sin 0 (0.0,1.0) diff' exp 0 (1.0,1.0)The N function calculates the first derivatives of scalar-to-nonscalar function by  mode ADdiffF (\a -> [sin a, cos a]) 0 [1.0,-0.0]The \ function calculates the result and first derivatives of a scalar-to-non-scalar function by  mode ADdiffF' (\a -> [sin a, cos a]) 0[(0.0,1.0),(1.0,-0.0)]BA fast, simple, transposed Jacobian computed with forward-mode AD.2A fast, simple, transposed Jacobian computed with  mode AD) that combines the output with the input.Compute the Jacobian using  mode AD%. This must transpose the result, so ( is faster and allows more result types.7jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]_[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]Compute the Jacobian using  mode ADK and combine the output with the input. This must transpose the result, so ) is faster, and allows more result types.Compute the Jacobian using  mode AD along with the actual answer.Compute the Jacobian using  mode ADW combined with the input using a user specified function, along with the actual answer.9Compute the gradient of a function using forward mode AD.Note, this performs O(n) worse than $% for n2 inputs, in exchange for better space utilization.DCompute the gradient and answer to a function using forward mode AD.Note, this performs O(n) worse than $& for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $' for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and the answer, and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $( for n2 inputs, in exchange for better space utilization.gradWith' (,) sum [0..4]$(10,[(0,1),(1,1),(2,1),(3,1),(4,1)])RCompute the product of a vector with the Hessian using forward-on-forward-mode AD.JCompute the gradient and hessian product using forward-on-forward-mode AD. (c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneOTCCompute the directional derivative of a function given a zipped up * of the input values and their derivativesNCompute the answer and directional derivative of a function given a zipped up * of the input values and their derivativesMCompute a vector of directional derivatives for a function given a zipped up + of the input values and their derivatives.YCompute a vector of answers and directional derivatives for a function given a zipped up + of the input values and their derivatives.The Y function calculates the first derivative of a scalar-to-scalar function by forward-mode + diff sin 01.0The U function calculates the result and first derivative of scalar-to-scalar function by  mode +   ==  "#   f = f "# d f  diff' sin 0 (0.0,1.0) diff' exp 0 (1.0,1.0)The N function calculates the first derivatives of scalar-to-nonscalar function by  mode +diffF (\a -> [sin a, cos a]) 0 [1.0,-0.0]The \ function calculates the result and first derivatives of a scalar-to-non-scalar function by  mode +diffF' (\a -> [sin a, cos a]) 0[(0.0,1.0),(1.0,-0.0)]BA fast, simple, transposed Jacobian computed with forward-mode AD.2A fast, simple, transposed Jacobian computed with  mode +) that combines the output with the input.Compute the Jacobian using  mode +%. This must transpose the result, so ( is faster and allows more result types.7jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]_[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]Compute the Jacobian using  mode +K and combine the output with the input. This must transpose the result, so ) is faster, and allows more result types.Compute the Jacobian using  mode + along with the actual answer.Compute the Jacobian using  mode +W combined with the input using a user specified function, along with the actual answer.9Compute the gradient of a function using forward mode AD.Note, this performs O(n) worse than $% for n2 inputs, in exchange for better space utilization.DCompute the gradient and answer to a function using forward mode AD.Note, this performs O(n) worse than $& for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $' for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and the answer, and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $( for n2 inputs, in exchange for better space utilization.gradWith' (,) sum [0..4]$(10,[(0,1),(1,1),(2,1),(3,1),(4,1)])RCompute the product of a vector with the Hessian using forward-on-forward-mode AD.JCompute the gradient and hessian product using forward-on-forward-mode AD.++ None*9:;<=DR  NoneCCompute the directional derivative of a function given a zipped up * of the input values and their derivativesNCompute the answer and directional derivative of a function given a zipped up * of the input values and their derivativesMCompute a vector of directional derivatives for a function given a zipped up + of the input values and their derivatives.YCompute a vector of answers and directional derivatives for a function given a zipped up + of the input values and their derivatives.The Y function calculates the first derivative of a scalar-to-scalar function by forward-mode AD diff sin 01.0The U function calculates the result and first derivative of scalar-to-scalar function by Forward mode AD   ==  "#   f = f "# d f  diff' sin 0 (0.0,1.0) diff' exp 0 (1.0,1.0)The N function calculates the first derivatives of scalar-to-nonscalar function by Forward mode ADdiffF (\a -> [sin a, cos a]) 0 [1.0,-0.0]The \ function calculates the result and first derivatives of a scalar-to-non-scalar function by Forward mode ADdiffF' (\a -> [sin a, cos a]) 0[(0.0,1.0),(1.0,-0.0)]BA fast, simple, transposed Jacobian computed with forward-mode AD.2A fast, simple, transposed Jacobian computed with Forward mode AD) that combines the output with the input.Compute the Jacobian using Forward mode AD%. This must transpose the result, so ( is faster and allows more result types.7jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]_[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]Compute the Jacobian using Forward mode ADK and combine the output with the input. This must transpose the result, so ) is faster, and allows more result types.Compute the Jacobian using Forward mode AD along with the actual answer.Compute the Jacobian using Forward mode ADW combined with the input using a user specified function, along with the actual answer.9Compute the gradient of a function using forward mode AD.Note, this performs O(n) worse than $% for n2 inputs, in exchange for better space utilization.DCompute the gradient and answer to a function using forward mode AD.Note, this performs O(n) worse than $& for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $' for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and the answer, and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $( for n2 inputs, in exchange for better space utilization.gradWith' (,) sum [0..4]:(10.0,[(0.0,1.0),(1.0,1.0),(2.0,1.0),(3.0,1.0),(4.0,1.0)])(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneOT CCompute the directional derivative of a function given a zipped up * of the input values and their derivatives NCompute the answer and directional derivative of a function given a zipped up * of the input values and their derivatives MCompute a vector of directional derivatives for a function given a zipped up + of the input values and their derivatives. YCompute a vector of answers and directional derivatives for a function given a zipped up + of the input values and their derivatives. The  Y function calculates the first derivative of a scalar-to-scalar function by forward-mode + diff sin 01.0The U function calculates the result and first derivative of scalar-to-scalar function by Forward mode +   ==  "#   f = f "# d f  diff' sin 0 (0.0,1.0) diff' exp 0 (1.0,1.0)The N function calculates the first derivatives of scalar-to-nonscalar function by Forward mode +diffF (\a -> [sin a, cos a]) 0 [1.0,-0.0]The \ function calculates the result and first derivatives of a scalar-to-non-scalar function by Forward mode +diffF' (\a -> [sin a, cos a]) 0[(0.0,1.0),(1.0,-0.0)]BA fast, simple, transposed Jacobian computed with forward-mode AD.2A fast, simple, transposed Jacobian computed with Forward mode +) that combines the output with the input.Compute the Jacobian using Forward mode +%. This must transpose the result, so ( is faster and allows more result types.7jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]_[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]Compute the Jacobian using Forward mode +K and combine the output with the input. This must transpose the result, so ) is faster, and allows more result types.Compute the Jacobian using Forward mode + along with the actual answer.Compute the Jacobian using Forward mode +W combined with the input using a user specified function, along with the actual answer.9Compute the gradient of a function using forward mode AD.Note, this performs O(n) worse than $% for n2 inputs, in exchange for better space utilization.DCompute the gradient and answer to a function using forward mode AD.Note, this performs O(n) worse than $& for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $' for n2 inputs, in exchange for better space utilization.Compute the gradient of a function using forward mode AD and the answer, and combine the result with the input using a user-specified function.Note, this performs O(n) worse than $( for n2 inputs, in exchange for better space utilization.gradWith' (,) sum [0..4]:(10.0,[(0.0,1.0),(1.0,1.0),(2.0,1.0),(3.0,1.0),(4.0,1.0)])     +     +          (c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone09;<=?DORTKahn is a A using reverse-mode automatic differentiation that provides fast diffFU, diff2FU, grad, grad2 and a fast jacobianE when you have a significantly smaller number of outputs than inputs.!A Tape\ records the information needed back propagate from the output to each input during reverse  AD.*back propagate sensitivities along a tape.*hThis returns a list of contributions to the partials. The variable ids returned in the list are likely not unique!+ Return an  of *# given bounds for the variable IDs., Return an  of sparse partials1 !"#$%&'()*+,-./0123456789:;<=>?@ABCDEF !$%#"&'()*+,-./012345 !"#$%&*+,()-.12435'/0( !"#$%&'()*+,-./0123456789:;<=>?@ABCDEF(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone 9;<=?OTJThe Jh function calculates the gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass. grad (\[x,y,z] -> x*y+z) [1,2,3][2,1,1]KThe Ks function calculates the result and gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass.+grad' (\[x,y,z] -> 4*x*exp y+cos z) [1,2,3]L(28.566231899122155,[29.5562243957226,29.5562243957226,-0.1411200080598672])LJ g fE function calculates the gradient of a non-scalar-to-scalar function fq with kahn-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g. J = L (_ dx -> dx)  = L const MK g fG calculates the result and gradient of a non-scalar-to-scalar function fp with kahn-mode AD in a single pass the gradient is combined element-wise with the argument using the function g. K == M (_ dx -> dx)NThe N` function calculates the jacobian of a non-scalar-to-non-scalar function with kahn AD lazily in m passes for m outputs.$jacobian (\[x,y] -> [y,x,x*y]) [2,1][[0,1],[1,0],[1,2]],jacobian (\[x,y] -> [exp y,cos x,x+y]) [1,2]<[[0.0,7.38905609893065],[-0.8414709848078965,0.0],[1.0,1.0]]OThe Ob function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using m invocations of kahn AD, where m( is the output dimensionality. Applying fmap snd* to the result will recover the result of N | An alias for gradF'Kghci> jacobian' ([x,y] -> [y,x,x*y]) [2,1] [(1,[0,1]),(2,[1,0]),(2,[1,2])]PR'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function f with kahn AD lazily in m passes for m outputs.kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g. N = P (_ dx -> dx) P  = (f x ->  x  f x) QPW g f' calculates both the result and the Jacobian of a nonscalar-to-nonscalar function f, using m invocations of kahn AD, where m( is the output dimensionality. Applying fmap snd* to the result will recover the result of PkInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g. O == Q (_ dx -> dx)R%Compute the derivative of a function. diff sin 01.0cos 01.0SThe SZ function calculates the value and derivative, as a pair, of a scalar-to-scalar function. diff' sin 0 (0.0,1.0)T]Compute the derivatives of a function that returns a vector with regards to its single input.diffF (\a -> [sin a, cos a]) 0 [1.0,0.0]U{Compute the derivatives of a function that returns a vector with regards to its single input as well as the primal answer.diffF' (\a -> [sin a, cos a]) 0[(0.0,1.0),(1.0,0.0)]V Compute the V via the N* of the gradient. gradient is computed in  mode and then the N is computed in  mode.However, since the J f :: f a -> f a9 is square this is not as fast as using the forward-mode N( of a reverse mode gradient provided by  ).hessian (\[x,y] -> x*y) [1,2] [[0,1],[1,0]]WRCompute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the -mode Jacobian of the -mode Jacobian of the function.Less efficient than *+.0hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2][[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]JKLMNOPQRSTUVW-.JKLMNOPQRSTUVWJKLMNOPQVWRSTU-.JKLMNOPQRSTUVW(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone 9;<=?OTXThe Xh function calculates the gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass. grad (\[x,y,z] -> x*y+z) [1,2,3][2,1,1]YThe Ys function calculates the result and gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass.+grad' (\[x,y,z] -> 4*x*exp y+cos z) [1,2,3]L(28.566231899122155,[29.5562243957226,29.5562243957226,-0.1411200080598672])ZX g fE function calculates the gradient of a non-scalar-to-scalar function fq with kahn-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g. X = Z (_ dx -> dx)  = Z const [Y g fG calculates the result and gradient of a non-scalar-to-scalar function fp with kahn-mode AD in a single pass the gradient is combined element-wise with the argument using the function g. Y == [ (_ dx -> dx)\The \` function calculates the jacobian of a non-scalar-to-non-scalar function with kahn AD lazily in m passes for m outputs.$jacobian (\[x,y] -> [y,x,x*y]) [2,1][[0,1],[1,0],[1,2]],jacobian (\[x,y] -> [exp y,cos x,x+y]) [1,2]<[[0.0,7.38905609893065],[-0.8414709848078965,0.0],[1.0,1.0]]]The ]b function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using m invocations of kahn AD, where m( is the output dimensionality. Applying fmap snd* to the result will recover the result of \ | An alias for gradF'Kghci> jacobian' ([x,y] -> [y,x,x*y]) [2,1] [(1,[0,1]),(2,[1,0]),(2,[1,2])]^R'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function f with kahn AD lazily in m passes for m outputs.kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g. \ = ^ (_ dx -> dx) ^  = (f x ->  x  f x) _^W g f' calculates both the result and the Jacobian of a nonscalar-to-nonscalar function f, using m invocations of kahn AD, where m( is the output dimensionality. Applying fmap snd* to the result will recover the result of ^kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g. ] == _ (_ dx -> dx)`%Compute the derivative of a function. diff sin 01.0cos 01.0aThe aZ function calculates the value and derivative, as a pair, of a scalar-to-scalar function. diff' sin 0 (0.0,1.0)b]Compute the derivatives of a function that returns a vector with regards to its single input.diffF (\a -> [sin a, cos a]) 0 [1.0,0.0]c{Compute the derivatives of a function that returns a vector with regards to its single input as well as the primal answer.diffF' (\a -> [sin a, cos a]) 0[(0.0,1.0),(1.0,0.0)]d Compute the d via the \* of the gradient. gradient is computed in  mode and then the \ is computed in  mode.However, since the X f :: f a -> f a9 is square this is not as fast as using the forward-mode \( of a reverse mode gradient provided by  ).hessian (\[x,y] -> x*y) [1,2] [[0,1],[1,0]]eRCompute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the -mode Jacobian of the -mode Jacobian of the function.Less efficient than *+.0hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2][[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]XYZ[\]^_`abcde+XYZ[\]^_`abcde+XYZ[\]^_de`abcXYZ[\]^_`abcde(c) Edward Kmett 2012-2015BSD3ekmett@gmail.com experimentalGHC onlyNone%&09:;<=DORT This is used to create a new entry on the chain given a unary function, its derivative with respect to its input, the variable ID of its input, and the value of its input. Used by % and ( internally.This is used to create a new entry on the chain given a binary function, its derivatives with respect to its inputs, their variable IDs and values. Used by ( internally.t^Helper that extracts the derivative of a chain when the chain was constructed with 1 variable.unHelper that extracts both the primal and derivative of a chain when the chain was constructed with 1 variable.5Used internally to push sensitivities down the chain.vDExtract the partials from the current chain for a given AD variable.w Return an  of v# given bounds for the variable IDs.x Return an  of sparse partialsy"Construct a tape that starts with n variables.3fghijklmnopqrstuvwxyz{|}~fhgijklmnorpqstuvwxyz{|}~fghijklmnopqryvwxtu|}~z{s*fghijklmnopqrstuvwxyz{|}~(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone :<=?OTThe k 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][2,1,1]The v function calculates the result and 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] (5,[2,1,1]) g fE function calculates the gradient of a non-scalar-to-scalar function ft with reverse-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g.  ==  (_ dx -> dx)  ==    g fG calculates the result and gradient of a non-scalar-to-scalar function fs with reverse-mode AD in a single pass the gradient is combined element-wise with the argument using the function g.  ==  (_ dx -> dx) The c 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][[0,1],[1,0],[1,2]]The b 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  | An alias for gradF'%jacobian' (\[x,y] -> [y,x,x*y]) [2,1][(1,[0,1]),(2,[1,0]),(2,[1,2])]R'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.kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.  ==  (_ dx -> dx)   == (f x ->  x  f x) W 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 kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.  ==  (_ dx -> dx)%Compute the derivative of a function. diff sin 01.0The Z 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)`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]vCompute 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)]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  f :: f a -> f ai is square this is not as fast as using the forward-mode Jacobian of a reverse mode gradient provided by  ).hessian (\[x,y] -> x*y) [1,2] [[0,1],[1,0]]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 *+.0hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2][[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]ff(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone09:;<=?DQRSWe only store partials in sorted order, so the map contained in a partial will only contain partials with equal or greater keys to that of the map in which it was found. This should be key for efficiently computing sparse hessians. there are only (n + k - 1) choose (k - 1) distinct nth partial derivatives of a function with k inputs.3The value of the derivative of (f*g) of order mi is  [a *  ( ( b) f) *  ( ( c) g) | (a,b,c) <-  mi ]  It is a bit more complicated in V below, since we build the whole tree of derivatives and want to prune the tree with _s as much as possible. The number of terms in the sum for order mi as of differentiation has  ( (+1) as)X terms, so this is *much* more efficient than the naive recursive differentiation with 2^ as terms. The coefficients aX, which collect equivalent derivatives, are suitable products of binomial coefficients.3+6(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone09:;<=DORTTower is an AD B that calculates a tangent tower by forward AD, and provides fast diffsUU, diffsUF" 6(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneOT5Compute the answer and all derivatives of a function (a -> a)2Compute the zero-padded derivatives of a function (a -> a)5Compute the answer and all derivatives of a function  (a -> f a)2Compute the zero-padded derivatives of a function  (a -> f a) taylor f x compute the Taylor series of f around x. taylor0 f x compute the Taylor series of f around x, zero-padded. maclaurin f! compute the Maclaurin series of f maclaurin f! compute the Maclaurin series of f , zero-padded+Compute the first derivative of a function (a -> a)6Compute the answer and first derivative of a function (a -> a)/Compute a directional derivative of a function  (f a -> a)>Compute the answer and a directional derivative of a function  (f a -> a)/Compute a directional derivative of a function  (f a -> g a)>Compute the answer and a directional derivative of a function  (f a -> g a)Given a function  (f a -> a)P, and a tower of derivatives, compute the corresponding directional derivatives.Given a function  (f a -> a)\, and a tower of derivatives, compute the corresponding directional derivatives, zero-paddedGiven a function  (f a -> g a)O, and a tower of derivatives, compute the corresponding directional derivativesGiven a function  (f a -> g a)\, and a tower of derivatives, compute the corresponding directional derivatives, zero-padded(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneOT     +     +          (c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneThe  function finds a zero of a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Examples: take 10 $ findZero (\x->x^2-4) 1B[1.0,1.8571428571428572,1.9997967892704736,1.9999999999994755,2.0].last $ take 10 $ findZero ((+1).(^2)) (1 :+ 1) 0.0 :+ 1.0The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.The  function inverts a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Note: the "take 10 $ inverse sqrt 1 (sqrt 10)j example that works for Newton's method fails with Halley's method because the preconditions do not hold!The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.The  function find a fixedpoint of a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)SIf the stream becomes constant ("it converges"), no further elements are returned.!last $ take 10 $ fixedPoint cos 10.7390851332151607The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.The  function finds an extremum of a scalar function using Halley's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.take 10 $ extremum cos 1G[1.0,0.29616942658570555,4.59979519460002e-3,1.6220740159042513e-8,0.0]The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone :DRT The  function finds a zero of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Examples: take 10 $ findZero (\x->x^2-4) 1I[1.0,2.5,2.05,2.000609756097561,2.0000000929222947,2.000000000000002,2.0].last $ take 10 $ findZero ((+1).(^2)) (1 :+ 1) 0.0 :+ 1.0The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.The  function inverts a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.Example:)last $ take 10 $ inverse sqrt 1 (sqrt 10)10.0The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.The  function find a fixedpoint of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)SIf the stream becomes constant ("it converges"), no further elements are returned.!last $ take 10 $ fixedPoint cos 10.7390851332151607The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.The  function finds an extremum of a scalar function using Newton's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.last $ take 10 $ extremum cos 10.0The  function behaves the same as ~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance. The  b function performs a multivariate optimization, based on the naive-gradient-descent in the file 1stalingrad/examples/flow-tests/pre-saddle-1a.vlad from the VLAD compiler Stalingrad sources. Its output is a stream of increasingly accurate results. (Modulo the usual caveats.)GIt uses reverse mode automatic differentiation to compute the gradient.!`Perform a gradient descent using reverse mode automatic differentiation to compute the gradient.  !  !  !  !(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone%&234:DFORT"FConvex constraint, CC, is a GADT wrapper that hides the existential (sn) which is so prevalent in the rest of the API. This is an engineering convenience for managing the skolems.$The $ function finds a zero of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Examples: take 10 $ findZero (\x->x^2-4) 1I[1.0,2.5,2.05,2.000609756097561,2.0000000929222947,2.000000000000002,2.0].last $ take 10 $ findZero ((+1).(^2)) (1 :+ 1) 0.0 :+ 1.0%The % function behaves the same as $~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.&The & function inverts a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.Example:)last $ take 10 $ inverse sqrt 1 (sqrt 10)10.0'The ' function behaves the same as &~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.(The ( function find a fixedpoint of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)SIf the stream becomes constant ("it converges"), no further elements are returned.!last $ take 10 $ fixedPoint cos 10.7390851332151607)The ) function behaves the same as (~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.*The * function finds an extremum of a scalar function using Newton's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.last $ take 10 $ extremum cos 10.0+The + function behaves the same as *~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.,The ,b function performs a multivariate optimization, based on the naive-gradient-descent in the file 1stalingrad/examples/flow-tests/pre-saddle-1a.vlad from the VLAD compiler Stalingrad sources. Its output is a stream of increasingly accurate results. (Modulo the usual caveats.)GIt uses reverse mode automatic differentiation to compute the gradient.-constrainedDescent obj fs env optimizes the convex function obj$ subject to the convex constraints f <= 0 where f  fs. This is done using a log barrier to model constraints (i.e. Boyd, Chapter 11.3). The returned optimal point for the objective function must satisfy fs , but the initial environment, env, needn't be feasible.Like -+ except the initial point must be feasible./The / function approximates the true gradient of the constFunction by a gradient at a single example. As the algorithm sweeps through the training set, it performs the update for each training example.It uses reverse mode automatic differentiation to compute the gradient The learning rate is constant through out, and is set to 0.0010`Perform a gradient descent using reverse mode automatic differentiation to compute the gradient.1Perform a conjugate gradient descent using reverse mode automatic differentiation to compute the gradient, and using forward-on-forward mode for computing extrema.let sq x = x * x7let rosenbrock [x,y] = sq (1 - x) + 100 * sq (y - sq x)rosenbrock [0,0]1Brosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1True2iPerform a conjugate gradient ascent using reverse mode automatic differentiation to compute the gradient."#$%&'()*+,-./012"#$%&'()*+,-./012$%&'()*+,-"#.012/"#$%&'()*+,-./012(c) Edward Kmett 2015BSD3ekmett@gmail.com experimentalGHC onlyNone :DRT6The 6 function finds a zero of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Examples: take 10 $ findZero (\x->x^2-4) 1I[1.0,2.5,2.05,2.000609756097561,2.0000000929222947,2.000000000000002,2.0]7The 7 function behaves the same as 6L except that it doesn't truncate the list once the results become constant.8The 8 function inverts a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.Example:)last $ take 10 $ inverse sqrt 1 (sqrt 10)10.09The 9 function behaves the same as 8L except that it doesn't truncate the list once the results become constant.:The : function find a fixedpoint of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)SIf the stream becomes constant ("it converges"), no further elements are returned.!last $ take 10 $ fixedPoint cos 10.7390851332151607;The ; function behaves the same as :I except that doesn't truncate the list once the results become constant.<The < function finds an extremum of a scalar function using Newton's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.last $ take 10 $ extremum cos 10.0=The = function behaves the same as <L except that it doesn't truncate the list once the results become constant.6789:;<=6789:;<=6789:;<=6789:;<=(c) Edward Kmett 2015BSD3ekmett@gmail.com experimentalGHC onlyNone :DORT >The > function finds a zero of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Examples: take 10 $ findZero (\x->x^2-4) 1I[1.0,2.5,2.05,2.000609756097561,2.0000000929222947,2.000000000000002,2.0]?The ? function behaves the same as >L except that it doesn't truncate the list once the results become constant.@The @ function inverts a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.Example:)last $ take 10 $ inverse sqrt 1 (sqrt 10)10.0AThe A function behaves the same as @L except that it doesn't truncate the list once the results become constant.BThe B function find a fixedpoint of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)SIf the stream becomes constant ("it converges"), no further elements are returned.!last $ take 10 $ fixedPoint cos 10.7390851332151607CThe C function behaves the same as BI except that doesn't truncate the list once the results become constant.DThe D function finds an extremum of a scalar function using Newton's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.last $ take 10 $ extremum cos 10.0EThe E function behaves the same as DL except that it doesn't truncate the list once the results become constant.FPerform a conjugate gradient descent using reverse mode automatic differentiation to compute the gradient, and using forward-on-forward mode for computing extrema.let sq x = x * x7let rosenbrock [x,y] = sq (1 - x) + 100 * sq (y - sq x)rosenbrock [0,0]1Brosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1TrueGiPerform a conjugate gradient ascent using reverse mode automatic differentiation to compute the gradient. >?@ABCDEFG >?@ABCDEFG >?@ABCDEFG >?@ABCDEFG(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneHIJKLMNOPQRSTUHIJKLMNOPQRSTUHIJKPLMNOQRSTUHIJKLMNOPQRSTU(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneOTVWXYZ[\]^_`abc+VWXYZ[\]^_`abc+VW^XYZ[\]_`abcVWXYZ[\]^_`abc(c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNoneOTdThe d function finds a zero of a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Examples: take 10 $ findZero (\x->x^2-4) 1B[1.0,1.8571428571428572,1.9997967892704736,1.9999999999994755,2.0].last $ take 10 $ findZero ((+1).(^2)) (1 :+ 1) 0.0 :+ 1.0eThe e function behaves the same as d~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.fThe f function inverts a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned. Note: the "take 10 $ inverse sqrt 1 (sqrt 10)j example that works for Newton's method fails with Halley's method because the preconditions do not hold!gThe g function behaves the same as f~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.hThe h function find a fixedpoint of a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)SIf the stream becomes constant ("it converges"), no further elements are returned.!last $ take 10 $ fixedPoint cos 10.7390851332151607iThe i function behaves the same as h~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.jThe j function finds an extremum of a scalar function using Halley's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.take 10 $ extremum cos 1G[1.0,0.29616942658570555,4.59979519460002e-3,1.6220740159042513e-8,0.0]kThe k function behaves the same as j~ except that it doesn't truncate the list once the results become constant. This means it can be used with types without an  instance.defghijkdefghijkdefghijkdefghijk (c) Edward Kmett 2010-2015BSD3ekmett@gmail.com experimentalGHC onlyNone:DORTll f wv% computes the product of the hessian H$ of a non-scalar-to-scalar function f at w =   $ wv with a vector v = snd  $ wv# using "Pearlmutter's method" from  ?http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6143, which states: %H v = (d/dr) grad_w (w + r v) | r = 0Or in other words, we take the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.mm f wv6 computes both the gradient of a non-scalar-to-scalar f at w =   $ wv and the product of the hessian H at w with a vector v = snd  $ wvT using "Pearlmutter's method". The outputs are returned wrapped in the same functor. %H v = (d/dr) grad_w (w + r v) | r = 0Or in other words, we return the gradient and the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.nCompute the Hessian via the Jacobian of the gradient. gradient is computed in reverse mode and then the Jacobian is computed in sparse (forward) mode.hessian (\[x,y] -> x*y) [1,2] [[0,1],[1,0]]oPCompute the order 3 Hessian tensor on a non-scalar-to-non-scalar function using 'Sparse'-on-'Reverse'0hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2][[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]lmno4+-.  ,/012^_aclmno4+^-._naoclm  ,012/lmno,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWWXYZ[\]^_`abcdefghijklmnopqQTrstuvwxyz{|}~     % & ' ( % & ' (   % & ' (%&'(     %&'()+%&'()+  !""#$%&'()   *+,-./012345678%&'()+9:;<<==>?@ABCDEFGHI JKLMNOPQRSTUVWXYZ[[\]^_BC`abcdefghijklmnopqrstuvwxyz{|}~tuvwxyz{|}~%&'()+%&'()+ ) +!!!!!    ad-4.3.4-5Hu7UMFsO80Cajr3FbdVjMNumeric.AD.ModeNumeric.AD.JetNumeric.AD.JacobianNumeric.AD.Internal.TypeNumeric.AD.Internal.OrNumeric.AD.Internal.OnNumeric.AD.Internal.IdentityNumeric.AD.Internal.DenseNumeric.AD.Rank1.ForwardNumeric.AD.Internal.ForwardNumeric.AD.Mode.ForwardNumeric.AD.Rank1.Forward.Double"Numeric.AD.Internal.Forward.DoubleNumeric.AD.Mode.Forward.DoubleNumeric.AD.Rank1.KahnNumeric.AD.Internal.KahnNumeric.AD.Mode.KahnNumeric.AD.Mode.ReverseNumeric.AD.Internal.ReverseNumeric.AD.Rank1.SparseNumeric.AD.Internal.SparseNumeric.AD.Rank1.TowerNumeric.AD.Internal.TowerNumeric.AD.Mode.TowerNumeric.AD.Rank1.HalleyNumeric.AD.Rank1.NewtonNumeric.AD.NewtonNumeric.AD.Rank1.Newton.DoubleNumeric.AD.Newton.DoubleNumeric.AD.Mode.SparseNumeric.AD.Halley Numeric.ADNumeric.AD.Internal.Combinators Control.Arrow&&&Numeric.AD.Mode.Wengertgradgrad'gradWith gradWith'hessianNumeric.AD.Mode.MixedhessianFModeScalarisKnownConstant isKnownZeroauto*^^*^/zero $fModeRatio $fModeComplex $fModeWord64 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backPropagateGHC.ArrArraycontainers-0.5.7.1Data.IntMap.BaseIntMaptopSortAcyclicidconst Data.Functor<$>unarilybinarily dropCellsunbin modifyTapesumpartialSmapisZero truncatedghc-prim GHC.ClassesEqelemconstrainedConvex'SEnvsValueorigEnvmkOptiHatlfurfusecondd2d2' Data.Tuplefst