Add dual number–based second derivatives for ForwardDiff (#310)

* Add dual number-based implementations for ForwardDiff scalar derivatives

* Use functions from utils.jl

* Fixup

* Reorder

* Add preparation to see error

* Add derivative method

* Add derivative! and value_and_derivative! methods

* Fixup

* Drop derivative and derivative!

* Add prepare_second_derivative and value_derivative_and_second_derivative!

* Fixup

* Drop derivative stuff

* Add second_derivative and second_derivative! methods

* Fixup

* Fixup 2

---------

Co-authored-by: Guillaume Dalle <22795598+gdalle@users.noreply.github.com>

Author: gerlero

DifferentiationInterface/ext/DifferentiationInterfaceForwardDiffExt/DifferentiationInterfaceForwardDiffExt.jl

@@ -10,6 +10,7 @@ using DifferentiationInterface:
     HessianExtras,
     JacobianExtras,
     NoDerivativeExtras,
+    NoSecondDerivativeExtras,
     PushforwardExtras
 using ForwardDiff.DiffResults: DiffResults, DiffResult, GradientResult, MutableDiffResult
 using ForwardDiff:

DifferentiationInterface/ext/DifferentiationInterfaceForwardDiffExt/onearg.jl

@@ -61,6 +61,58 @@ function DI.pushforward!(
     return dy
 end
 
+## Second derivative
+
+function DI.prepare_second_derivative(f::F, backend::AutoForwardDiff, x) where {F}
+    return NoSecondDerivativeExtras()
+end
+
+function DI.second_derivative(
+    f::F, backend::AutoForwardDiff, x, ::NoSecondDerivativeExtras
+) where {F}
+    T = tag_type(f, backend, x)
+    xdual = make_dual(T, x, one(x))
+    T2 = tag_type(f, backend, xdual)
+    ydual = f(make_dual(T2, xdual, one(xdual)))
+    return myderivative(T, myderivative(T2, ydual))
+end
+
+function DI.second_derivative!(
+    f::F, der2, backend::AutoForwardDiff, x, ::NoSecondDerivativeExtras
+) where {F}
+    T = tag_type(f, backend, x)
+    xdual = make_dual(T, x, one(x))
+    T2 = tag_type(f, backend, xdual)
+    ydual = f(make_dual(T2, xdual, one(xdual)))
+    return myderivative!(T, der2, myderivative(T2, ydual))
+end
+
+function DI.value_derivative_and_second_derivative(
+    f::F, backend::AutoForwardDiff, x, ::NoSecondDerivativeExtras
+) where {F}
+    T = tag_type(f, backend, x)
+    xdual = make_dual(T, x, one(x))
+    T2 = tag_type(f, backend, xdual)
+    ydual = f(make_dual(T2, xdual, one(xdual)))
+    y = myvalue(T, myvalue(T2, ydual))
+    der = myderivative(T, myvalue(T2, ydual))
+    der2 = myderivative(T, myderivative(T2, ydual))
+    return y, der, der2
+end
+
+function DI.value_derivative_and_second_derivative!(
+    f::F, der, der2, backend::AutoForwardDiff, x, ::NoSecondDerivativeExtras
+) where {F}
+    T = tag_type(f, backend, x)
+    xdual = make_dual(T, x, one(x))
+    T2 = tag_type(f, backend, xdual)
+    ydual = f(make_dual(T2, xdual, one(xdual)))
+    y = myvalue(T, myvalue(T2, ydual))
+    myderivative!(T, der, myvalue(T2, ydual))
+    myderivative!(T, der2, myderivative(T2, ydual))
+    return y, der, der2
+end
+
 ## Gradient
 
 struct ForwardDiffGradientExtras{C} <: GradientExtras