I like the idea of storing the precision by default. Some algorithms already do this in order to keep track of what they're losing in lossy arithmetic operation

I like the idea of storing the precision by default. Some algorithms already do this in order to keep track of what they're losing in lossy arithmetic operations, such as adding small numbers to big ones. It would be nontrivial to implement this efficiently - the simplest implementation would require a second float for each original float that has an uncertainty, doubling the time and memory requirements. A tensorflow-like system which can see the whole flow of numbers might be able to provide precision estimates efficiently only where needed.