I mean, as others pointed out, a number format typically needs much more than just a representation. If there is a bit-size limitation any operation has to round any excess back into the limit. A computational representation based on Turing machines or lambda calculus---again, which is not very different from a constructive real number in usage---do not provide an easy way to do that. That wouldn't have been an issue if there was no bit-size limitation.
Okay, let's ignore arithmetics and just allow comparison. As you've said, a common practice is to normalize it into some standard notation with a well-founded ordering. But there is no mechanical way to convert (or even bound) a computational representation to such notation---the general approach is therefore to compute a difference and check its sign. Not really good when it can continue even after the heat death of universe...
Frankly speaking, I rather expected to see some improvement over Level-Index number systems [1], but it turns out that this post is completely unrelated to number formats. Otherwise it is good, hence my mild frustration here :S
Okay, let's ignore arithmetics and just allow comparison. As you've said, a common practice is to normalize it into some standard notation with a well-founded ordering. But there is no mechanical way to convert (or even bound) a computational representation to such notation---the general approach is therefore to compute a difference and check its sign. Not really good when it can continue even after the heat death of universe...
Frankly speaking, I rather expected to see some improvement over Level-Index number systems [1], but it turns out that this post is completely unrelated to number formats. Otherwise it is good, hence my mild frustration here :S
[1] https://www.mrob.com/pub/math/altnum.html#li_sli