Sage Days 24 Projects: Algorithmic Differentiation

Examples of what you could imagine doing: 

sage: f(x,y,z)=sqrt(x^3+y)/(x-y^3*z)^(3/2)
sage: a = fast_callable(f,CDF)
sage: timeit('a(2,3,4)')
625 loops, best of 3: 24.2 µs per loop
sage: g = f.derivative(x,10)
sage: a = fast_callable(g,CDF)
sage: timeit('a(2,3,4)')
625 loops, best of 3: 1.21 ms per loop

I imagine that one could instead do: 

sage: f(x,y,z)=sqrt(x^3+y)/(x-y^3*z)^(3/2)
sage: a = fast_callable(g,CDF)
sage-fantasy: b = a.derivative(x,10)
sage-fantasy: timeit('b(2,3,4)')
625 loops, best of 3: 24.2 µs per loop

This is just a random example I made up, but it's the sort of massive performance one might hope for from AD; the literature should have better examples. There are *tons* of potential Python libraries to consider, e.g., algopy and CppAD (which has Python bindings).

Obviously, differentiating short functions that use a restricted set of functions in sage would make sense.