Caching computations in forward mode

Suppose we have a function f: ℝᵐ → ℝⁿ that for some reason we want to differentiate in forward mode, which will require calling all frules m times. This seems wasteful, as the pushforwards often depend on intermediates of the primal function that don't change. In the current implementation of frules, where the output of the pushforward is computed at the same time as the output of the primal, these intermediates would need to be recomputed m times. An example is symmetric eigendecomposition, where the eigendecomposition really only needs to be computed once but will instead be computed m times.

I'm sure there are good reasons for implementing this way. One I can think of is that it's easier to support mutating rules. Are there others?

No files, tests, or entry points are named. Start by reviewing the current frule implementation and its mutating-rule constraints; a complete outcome would determine whether caching can support forward-mode computations without breaking those constraints.