> I think we might be better with two consistently behaving float classes

That sounds a bit extreme.  Personally, I'd look for a solution involving only one float class and a "with float.nonstopmode: ..." setting under which we get infinities and nans as prescribed by IEEE 754;  the default would be to produce exceptions in places where IEEE 754 would signal a floating-point exception (any of divide-by-zero, overflow or invalid-operation, but not for inexact or underflow).  This is definitely PEP territory, though;  and someone needs to undertake to implement it, which is distinctly nontrivial.

For this issue, I'd avoid making any small changes to current behaviour without having at least a clear vision set out of where we want to end up.