split dual AD and mathematical types

[juliadiff]

split dual AD and mathematical types

[papamarkou]

It's ok @mlubin, there is no actual conflict, at least not on my part. I think we both agree that the mathematically correct definition of dual equality differs from the one that serves the practical purposes of autodiff. We also both agree that the latter is what is used in practice. For this reason, as I have previously mentioned, I don't mind if we change the existing mathematical definition of equality to match that of autodiff. By that I mean that you can simply change the definition without needing to proceed and define a second type to satisfy mathematical equality. If for any reason mathametical equality arises in the future as a programming need, then we can create the second type - for now simply change the definition of == to save yourself from extra unnecessary work.

[andreasnoack]

I also don't think that it is worth spending time on a DualAlg type. The purpose for this package is to allow automatic differentiation so I think it is fine just to use the special definitions here. It would be nice to keep a note about that in the documentation though.