Kulisch U. (2012) An Axiomatic Approach to Computer Arithmetic with an Appendix on Interval Hardware. In: Wyrzykowski R., Dongarra J., Karczewski K., Waśniewski J. (eds) Parallel Processing and Applied Mathematics. PPAM 2011. Lecture Notes in Computer Science, vol 7204. Springer, Berlin, Heidelberg
Different kinds of computer arithmetic follow an abstract mathematical pattern and are just special realizations of it. The basic mathematical concepts are sketched here. The concepts of rounding, of a screen, and of rounded arithmetic operations are defined in an axiomatic manner fully independent of special data formats and encodings. These abstract concepts are then applied and illustrated by the two elementary models of computer arithmetic for real numbers and for real intervals. In the latter case definition of the arithmetic operations as set operations does not suffice. Executable formulas have to be derived. We also demonstrate how this can be achieved. In an appendix we sketch a hardware realization of interval arithmetic and show that most of what is needed for it is already available on current x86-processors.