Alternative real number representation

I'm currently doing work on an optimisation algorithm that requires real numbers (Java doubles, to be specific) be converted into bit strings (again, to be specific: an ordered collection of Java booleans) and back again*. The scheme used is that the first bit indicates whether the value lies in 开发者_运维问答the first or second half of the range; the second bit subdivides the halves again, and so on.

The self-made algorithm I'm using to do this works fine when working with ranges that have a specified upper and lower value, however things get problematic when I try to use the full range of the Java double (I think, although I'm not sure, that the problem is related to the fact that Double.MAX_VALUE - (-1.0 * Double.MAX_VALUE) > Double.MAX_VALUE). That, however, isn't my question today. I'm sure I'm not the first one to try to use a number representation like this, so I'm sure the answers to my problems are out there on the Interwebs; I'm just not sure what to search for. What do you call such a representation of real numbers?

* Obviously, the number of bits used places a limit on the precision; I'm aware of and accept this.