Evaluate symbolic derivatives at specific points

[johnmyleswhite]

It would be useful to provide a tool like differentiate(:(exp(x)), :x)) that returns a compiled function that is immediately evaluable at specific values of x.

[jckt]

So sorry, this is literally the first time I've ever used git or github, and I'm not too sure how pulls work (but I'll figure it out in a moment). Anyway this seems to work:

# differentiate a function f(x) at given x
function symbolic_difference(ex::Expr, wrt::SymbolicVariable, atx::Float64)
    ex = differentiate(ex)
    eval(:($wrt = $atx))
    return eval(ex)
end

The only problem I guess is that here you're messing around with global variables, but I'm still figuring out Julia.

julia> expression = differentiate(:(exp(x)), :x)
:(exp(x))

julia> symbolic_difference(expression, :x, 1.0)
2.7182818284590455

julia> symbolic_difference(expression, :x, 0.0)
1.0

Hmm, also, using eval like this seems dangerous.

[johnmyleswhite]

Yeah, using eval isn't ideal here. Let me ponder this for a bit.

[Luthaf]

Hey there !

I'd really like to do something like this in my code. What is the advancement of this point ? There was a PR (#12) some time ago, is there anything I can do to help with this ?

I am very novice in metaprograming and symbolic computing, but I can try.

[johnmyleswhite]

I have a pretty clear sense how to implement this, but am very busy for the next few weeks. You might try working on it yourself for a while and submitting a PR, which I'll try to make time to review.

[johnmyleswhite]

Here's an overview:

(1) Understand the existing function that takes in a symbolic expression and produces its derivative.

(2) Write a macro that, given a symbolic expression, a) generates its derivative using (1), b) defines a function whose body is the derivative, and c) calls that function on the input.

Thus, you'd end up with a macro like the following:

@differentiate(5.0, sin(x) + cos(x))

[mlubin]

This is implemented efficiently in https://github.com/JuliaDiff/ReverseDiffSource.jl

[Luthaf]

Well, it would be a good thing to have it in this package too : you can fall back to finite differences if there is no way to differentiate analytically the function.

Could the two packages being merged into one ? Or is it worth implementing the ReverseDiffSource method into this one ?

[mlubin]

The methods in ReverseDiffSource/ReverseDiffSparse are way too complex to be merged into the basic Calculus package. If you just need to compile derivatives of a simple symbolic expression, John's approach is pretty straightforward and would be a good addition to Calculus.jl.

[johnmyleswhite]

One thing that would be nice at some point is to try to unify interfaces between the packages so that it's easier to transition from Calculus to ReverseDiffSparse.

[Luthaf]

One thing that would be nice at some point is to try to unify interfaces between the packages so that it's easier to transition from Calculus to ReverseDiffSparse.

Sure !

If you just need to compile derivatives of a simple symbolic expression, John's approach is pretty straightforward and would be a good addition to Calculus.jl.

I may try to do this if I find some time for hacking around. But I am pretty busy right now too ...