Determine widening conversion of NumPy types

I have a library based on NumPy with a few classes that overload the arithmetic operations. The internals are a little hairy due to a significant amount of error checking, but I've run across a serious problem with how I'm doing it. The idea behind the library is to make it very easy and intuitive to use with minimal syntax and effort by the programmer. As such, I want it to be easy to combine arrays of different data types and simply convert the narrower data type to the wider case. For example, if I have two arrays, one with dtype `float64` and the other with dtype `complex128`, when adding them together I would like to convert the `float64` to `complex128`, but if it's `float64` and `complex192`, I want to convert to that instead. However, if it was a combination of `float64` and `complex64`, there isn't a valid conversion between the two without losing precision of the `float64`, so I would want to convert both to `complex128`. I immediately saw the problem that I would have to look for each combination of types and determine their narrowest common widened type (think least common multiple) if I wanted my library to be fully robust. I don't want to convert everything to the widest type possible, since that becomes memory inefficient very quickly, and I often have very large arrays stored in memory. Is there a good way to determine the narrowest common widened type between two NumPy types?