An Axiomatic Approach to Computer Arithmetic with an Appendix on Interval Hardware
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.
KeywordsArithmetic Operation High Dimensional Space Complete Lattice Interval Arithmetic Great Element
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- 1.American National Standards Institute / Institute of Electrical and Electronics Engineers: A Standard for Binary Floating-Point Arithmetic. ANSI/IEEE Std. 754-1987, New York (1985) (reprinted in SIGPLAN 22(2), 9–25, 1987); Also adopted as IEC Standard 559:1989Google Scholar
- 2.American National Standards Institute / Institute of Electrical and Electronics Engineers: A Standard for Radix-Independent Floating-Point Arithmetic. ANSI/IEEE Std. 854-1987, New York (1987)Google Scholar
- 4.Kulisch, U.: Implementation and Formalization of Floating-Point Arithmetics. IBM T. J. Watson-Research Center, Report Nr. RC 4608, 1–50 (1973); Invited talk at the Caratheodory Symposium, September 1973 in Athens, published in: The Greek Mathematical Society, C. Caratheodory Symposium, 328–369 (1973), and in Computing 14, 323–348 (1975)Google Scholar
- 5.Kulisch, U.: Grundlagen des Numerischen Rechnens - Mathematische Begründung der Rechnerarithmetik, Bibliographisches Institut, Mannheim Wien Zürich (1976)Google Scholar
- 7.Kulisch, U.: Arithmetic Operations for Floating-Point Intervals, as Motion 5 accepted by the IEEE Standards Committee P1788 as definition of the interval operations Google Scholar
- 8.Pryce, J.D. (ed.): P1788, IEEE Standard for Interval Arithmetic, http://grouper.ieee.org/groups/1788/email/pdfOWdtH2mOd9.pdf