Preparing primal result in addition to derivatives

[lassepe]

Some backends, like Symbolics.jl and FastDifferentiation.jl (and maybe some day Reactant.jl) can accelerate not only the derivative computation but also the computation of the primal result. Currently, the prepare_* functions don't seem to account for that so that value_and_* still use the original function. Would it be conceivable for DI users to also opt into primal preparation or is it such a feature considered out of scope?

[gdalle]

Hi @lassepe,
Indeed that is very much in scope, I just haven't done it yet for symbolic backends because there was never any demand. Joint computation of the function and derivative would definitely be more clever than using the original function, and in fact that's what we're looking into for Reactant (#918). PR welcome to do this for Symbolics and FastDifferentiation, I think if you understand the current code the adaptation should be fairly straightforward (and the tests don't need to change)!