No. The set of "representable" (i.e. computable) numbers is countable (because the set of programs is countable), so in fact almost all real numbers cannot be represented.
Regardless if it is representing number, ranges, or a mix of both, there are still 2^n maximum possible representations for n bits. Perhaps you could create a mix of numbers and ranges equal to (2^n)-2 and then add one range that is from the lowest number represented to negative infinity and add another that does the same for positives. But I can do that system with -, 0, +, NAN. See, I just represented all numbers. It isn't as useful, but boy are there savings on storage, cost, and computation time.
Unums shouldn't be seen as better than ints, longs, floats, or doubles anymore than a double is seen as better than an int. They have different uses and strengths, and there are problems where none are a good choice. Perhaps unums are the best choice for some problems. Perhaps they are clearly and significantly better in some areas. But they should be thought of as an alternative to the existing formats and not a superior replacement.
I would argue that unums are a "superior replacement" for doubles in many cases, though: in the case that you support unums that are "wide" enough, you can represent doubles exactly, plus you have additional values, plus some nice rules about when approximation error occurs/is propagated and not as many bits need to be stored or moved around on buses. It'll be a while before there's an implementation anywhere near as fast as existing FPUs, but Gustafson makes a good argument for his format. Personally, I'm more interested in the correctness benefits than the space/power/time savings - even if unums are never faster than 64-bit floats, they present an interesting way to do real-number arithmetic and my brief exposure to them leaves me much more confident that I could write numerical algorithms correctly than with doubles - I do numerical/statistical algorithms with doubles in my work, and it's really a pain to reason about things that the "uncertain" (open-inverval) values of the unum format would greatly simplify.
They're also an inferior replacement in the case that you want to take advantage of highly-optimized hardware, and that getting a correct answer doesn't really matter. I don't see unums replacing floats for, say, video game graphics. But for numerical computation, it seems like the only real flaw with unums compared to doubles is the nonexistence of a hardware implementation, and the existing popularity of doubles.
Agreed. For neural networks. I would argue the opposite is true, you should just have a 16bit float that casts really large values to infinity silently without throwing errors, with a logistic lookup that maps "inf" to +/- 1... A mathematically incorrect float is operationally superior to the correct one.
For hardware without a FPU, you can do floats in software when accuracy is more important than speed (vs fixed point), one can imagine that if this thing actually works then for hardware without a "UPU" one can just do unums in software when needed.
Rougly speaking It would be 1.414... + plus a bit that indicates the result is between two adjacent exact values. This bit would not be set if the result were exact.
It's not the same as dyadic interval math, it's kind of a hybrid.
And you could set the environmental variables to be as precise as you need. If you need 29 digits of precision, you can quickly show that you know 1.4142135623730950488016887242 and nothing more.
A reasonable metaphor would be that you cannot draw all of the Mandelbrot diagram, but you can render any piece of it to any resolution you choose. Being able to describe it precisely, and to know the limits of your accuracy, is useful.
