[jlgustafson]

Isaac Yonemoto has suggested the use of ±∞ as a NaN, and it seems to work exactly like NaN does other than x/±∞ = 0 whereas you want x/NaN = NaN. I hate hardware exceptions. If you have a code that occasionally hits a NaN, that means the code is still under development. You would never release such a code into the wild, because it clearly is not safe to use and has not done the basic blocking and tackling of guarding against input arguments that will prevent any indeterminate forms, square roots of negative values, etc.

I haven't written them up yet, but _valids_ are what you want if you are want software that can gracefully and mathematically handle the results that make floats generate a NaN. Think of the valid computing environment as the numerical debugging environment for posits. It's slower and ultra-careful and rigorous, but once you get your algorithm to the point where it never tries to color outside the lines, then switch to posits and go FAST.

Leaving a NaN in a number system designed for lean speed is a mixing of computing esthetics. Which do you want? Rigorous and careful and mathematical, or good enough, fast, and cheap? You have to make up your mind, because if you _mix_ the two esthetics in one number system, guess what: You get neither. It won't be fast, because it has to check for exceptions all the time, and it won't be mathematical because it keeps replacing correct answers with answers within its vocabulary (that is, it rounds). IEEE floats are a mixture of the two esthetics, and that is their fatal flaw.