Abstract

Floating-point arithmetic was standardized more half a century ago. Now it is currently the main computing tool commonly used in computer systems: almost every language has a floating-point datatype; computers from PCs to supercomputers have floating-point accelerators etc. At the same time, the Floating-point arithmetic continues to be an esoteric subject, which is confirmed by more and more refinements of Std IEEE - 754 and even the emergence of an alternative standard Interval arithmetic.

The article presents an attempt to clarify some of the unlit Std IEEE - 754 features of the Floating-point arithmetic, as well as clarify some aspects of floating-point that have a direct impact on designers of computer systems.

It begins with background on alternative forms of representation standardized floating-point data, continues with a discussion on rounding error of the IEEE floating-point standard with the construction of a priori assessment forms rounding error, and concludes the construction of constructive forms for solving forward and backward analysis problems with examples of how computer system builders can better support floating point.

Categories and Subject Descriptors: (Primary) C.0 [Computer Systems Organization]:

General– instruction set design; D.3.4 [Programming Languages]: Processors —computers, optimization; G. 1.0 [Numerical Analysis]: General—computer arithmetic, error analysis, numerical algorithms

(Secondary) D. 2.1 [Software Engineering]: Requirements/Specifications– languages; D, 3.1 [Programming Languages]: Formal Definitions and Theory —semantics D. 4.1 [Operating Systems]: Process Management—synchronization

General Terms: Algorithms, Design, Languages

Additional Key Words and Phrases: normalized number, exception, floating-point, floating-point standard, representable number unrepresentable number, relative error, rounding error, rounding mode, Ulp, lexicographic order, predicate, method for describing a set, topology of sets of representable numbers.

References:

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