Since most floating-point calculations have rounding error anyway, does it matter if the basic arithmetic operations introduce a little bit more rounding error than necessary? That question is a main theme throughout this section. The section Relative Error and Ulps describes how it is measured. This rounding error is the characteristic feature of floating-point computation. Therefore the result of a floating-point calculation must often be rounded in order to fit back into its finite representation. In contrast, given any fixed number of bits, most calculations with real numbers will produce quantities that cannot be exactly represented using that many bits. Although there are infinitely many integers, in most programs the result of integer computations can be stored in 32 bits. Squeezing infinitely many real numbers into a finite number of bits requires an approximate representation. When a proof is not included, the z appears immediately following the statement of the theorem. The end of each proof is marked with the z symbol. In particular, the proofs of many of the theorems appear in this section. Those explanations that are not central to the main argument have been grouped into a section called 'The Details,' so that they can be skipped if desired. I have tried to avoid making statements about floating-point without also giving reasons why the statements are true, especially since the justifications involve nothing more complicated than elementary calculus. Topics include instruction set design, optimizing compilers and exception handling. The third part discusses the connections between floating-point and the design of various aspects of computer systems. The discussion of the standard draws on the material in the section Rounding Error. Included in the IEEE standard is the rounding method for basic operations. The second part discusses the IEEE floating-point standard, which is becoming rapidly accepted by commercial hardware manufacturers. It also contains background information on the two methods of measuring rounding error, ulps and relative error. The first section, Rounding Error, discusses the implications of using different rounding strategies for the basic operations of addition, subtraction, multiplication and division. It consists of three loosely connected parts. This paper is a tutorial on those aspects of floating-point arithmetic ( floating-point hereafter) that have a direct connection to systems building. One of the few books on the subject, Floating-Point Computation by Pat Sterbenz, is long out of print. There are, however, remarkably few sources of detailed information about it. Introductionīuilders of computer systems often need information about floating-point arithmetic. General Terms: Algorithms, Design, LanguagesĪdditional Key Words and Phrases: Denormalized number, exception, floating-point, floating-point standard, gradual underflow, guard digit, NaN, overflow, relative error, rounding error, rounding mode, ulp, underflow. It begins with background on floating-point representation and rounding error, continues with a discussion of the IEEE floating-point standard, and concludes with numerous examples of how computer builders can better support floating-point.Ĭategories and Subject Descriptors: (Primary) C.0 : General - instruction set design D.3.4 : Processors - compilers, optimization G.1.0 : General - computer arithmetic, error analysis, numerical algorithms (Secondary)ĭ.2.1 : Requirements/Specifications - languages D.3.4 Programming Languages]: Formal Definitions and Theory - semantics D.4.1 Operating Systems]: Process Management - synchronization. This paper presents a tutorial on those aspects of floating-point that have a direct impact on designers of computer systems. Almost every language has a floating-point datatype computers from PCs to supercomputers have floating-point accelerators most compilers will be called upon to compile floating-point algorithms from time to time and virtually every operating system must respond to floating-point exceptions such as overflow. This is rather surprising because floating-point is ubiquitous in computer systems. Copyright 1991, Association for Computing Machinery, Inc., reprinted by permission.įloating-point arithmetic is considered an esoteric subject by many people. Note This appendix is an edited reprint of the paper What Every Computer Scientist Should Know About Floating-Point Arithmetic, by David Goldberg, published in the March, 1991 issue of Computing Surveys. Appendix D What Every Computer Scientist Should Know About Floating-Point Arithmetic

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