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

<p>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.</p> <p> </p> <p>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.