It occurs to me that maybe you are referring to unums? The "unum" format discussed in "End of Error" is variable sized. However they since dropped that design and switched to fixed-sized number formats (16-, 32-, or 64-bit) with adjustable division between exponent and fraction bits, the "posit."

So "posit" numbers get you the ability to tradeoff precision and range, but they're not anymore highly compressed than regular old IEEE floats. Unless, as Gustafson argues, you didn't need a double in the first place and the added features of posits let you switch to a float.

Not really... there is a standard 32-bit and 64-bit representation of posits. They use the same memory as a float or a double, just with better accuracy and other desirable properties. You only reduce storage space if you switch from “double” to “float”, which you can only do if you didn’t need most of the precision of a double in the first place. They’re not THAT much better.