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> nor does it expand the program into a mathematical formula.

Meh, this is not entirely accurate. Sure it doesn't expand the program into analytic functions whose derivatives are easy to compute; but it still handles the program symbolically. So, in a way, it still transforms the program into a known set of mathematical primitives of which it can then construct a program that computes the derivative in compile time.

It seems that way to me too, but without more details about implementation I'm not sure.

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