Can rules make decisions based on which primal method is used?

Can we safely detect if a primal function is going to hit a specific default and use that to change the logic in the frule or rrule?

For example, the rrule for det(A) calls inv(A). If we know that the primal function that would be hit is the det(A::AbstractMatrix) definition in generic.jl, then we know that the primal is using the lu decomposition to compute the determinant, and we can reuse that to compute the inverse faster. But if a specialized primal method was being hit, then we probably just want to call the primal and invert separately since that primal is probably more efficient for that type than lu.

Start by reading generic.jl and the existing det(A) rrule and inv-related logic, then investigate whether Julia method dispatch can safely expose which primal method will run. Done would require a decided, tested approach for distinguishing the generic det path from specialized primal methods.