What Every Computer Scientist Should Know About Floating-Point Arithmetic

This document presents a tutorial on those aspects of floating point (data types, accelerators, algorithms and exceptions) that have a direct impact on designers of computer systems.

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**Post time**: 06 Jun 2007 11:56:06

What Every Computer Scientist Should Know About Floating-Point Arithmetic

This document presents a tutorial on those aspects of floating point (data types, accelerators, algorithms and exceptions) that have a direct impact on designers of computer systems.

Excerpts from the Introduction:

Floating-point arithmetic is considered an esoteric subject by many people. This is rather surprising because floating-point is ubiquitous in computer systems. Almost every language has a floating-point data type; computers from PC's 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 paper presents a tutorial on those aspects of floating point that have a direct impact on designers of computer systems. 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.

Builders of computer systems often need information about floating-point arithmetic. There are, however, remarkably few sources of detailed information about it. One of the few books on the subject, Floating-Point Computation by Pat Sterbenz, is long out of print. This paper is a tutorial on those aspects of floating-point arithmetic (floating-point hereafter) that have a direct connection to systems building. It consists of three loosely connected parts. The first Section, "Rounding Error," on page 173, discusses the implications of using different rounding strategies for the basic operations of addition, subtraction, multiplication and division. It also contains background information on the two methods of measuring rounding error, ulps and relative error. The second part discusses the IEEE floating-point standard, which is becoming rapidly accepted by commercial hardware manufacturers. included in the IEEE standard is the rounding method for basic operations. The discussion of the standard draws on the material in the Section, "Rounding Error," on page 173. The third part discusses the connections between floating-point and the design of various aspects of computer systems. Topics include instruction set design, optimizing compilers and exception handling.

Floating-point arithmetic is considered an esoteric subject by many people. This is rather surprising because floating-point is ubiquitous in computer systems. Almost every language has a floating-point data type; computers from PC's 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 paper presents a tutorial on those aspects of floating point that have a direct impact on designers of computer systems. 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.

Builders of computer systems often need information about floating-point arithmetic. There are, however, remarkably few sources of detailed information about it. One of the few books on the subject, Floating-Point Computation by Pat Sterbenz, is long out of print. This paper is a tutorial on those aspects of floating-point arithmetic (floating-point hereafter) that have a direct connection to systems building. It consists of three loosely connected parts. The first Section, "Rounding Error," on page 173, discusses the implications of using different rounding strategies for the basic operations of addition, subtraction, multiplication and division. It also contains background information on the two methods of measuring rounding error, ulps and relative error. The second part discusses the IEEE floating-point standard, which is becoming rapidly accepted by commercial hardware manufacturers. included in the IEEE standard is the rounding method for basic operations. The discussion of the standard draws on the material in the Section, "Rounding Error," on page 173. The third part discusses the connections between floating-point and the design of various aspects of computer systems. Topics include instruction set design, optimizing compilers and exception handling.

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