Add rewrite validator

[Keno]

The following invariant must hold:

isapprox(f_original(f_reconstruct(u_compressed))), `f_expand_resid(f_compressed(u_compressed)))` 

where f_compressed is the optimized DAE/ODE function we have right now, f_reconstruct is as in DAECompiler v1 and takes a state vector and expands it into a variable vector, f_expand_resid maps from state numbers to original equation numbers and makes all other entries zero, and f_original is the original function without any optimization (but with variable numbers the same as after structural analysis, so there's a little bit of work to do to hook that up right).

Validating this invariant would be a powerful check on our code generation. We need to write code generators for:

• f_reconstruct
• f_expand_resid
• f_original

However, all of these should be relatively straightforward. f_reconstruct codegen can largely be copied from v1. f_expand_resid, is just a bunch of setindex and f_original is largely unmodified IR, with the exception of encoding the variable numbers.

Bonus points for a version of this that can do this on a partition-by-partition basis to narrow down which function is being miscompiled, but that can be follow on PR.

[Keno]

Another thing to mention is that f_original is also the result of the ordinary codegen process with all equations unassigned, no linear opt and all apparently differentiated states differentiated. That's a trivial thing to do, so probably a good place to start. However, since that still goes through our scheduling code, I think I'd rather have a separate codegen that does minimal modification on the post-structural IR. But we can start there and then compare all three.

[serenity4]

Am I understanding correctly that we will not want to do state selection for f_original, and that a separate codegen would mainly forward the post-structural IR to ODE/DAE codegen without performing state selection nor tearing?

[Keno]

Yes