Trans-Floating-Point Arithmetic Removes Nine Quadrillion Redundancies From 64-bit IEEE 754 Floating-Point Arithmetic

IEEE 754 floating-point arithmetic is widely used in modern, general-purpose computers. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. Modifying the IEEE arithmetic so that it uses transreal arithmetic has a number of advantages. It removes one redundant binade from IEEE floating-point objects, doubling the numerical precision of the arithmetic. It removes eight redundant, relational, floating-point operations and removes the redundant total order operation. It replaces the non-reflexive, floating-point, equality operator with a reflexive equality operator and it indicates that some of the exceptions may be removed as redundant – subject to issues of backward compatibility and transient future compatibility as programmers migrate to the transreal paradigm.

There are some typos in this version of the paper. In every truth table, r_1 in the heading row should be r_2. In the not-equal table, the last row should be: T, T, T, F. Sorry, folks, I'm dyslexic and those, unfortunately, are the kinds of mistake I am prone to.