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What Every Scientist Should Know About Floating-Point Arithmetic

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What Every Computer Scientist Should Know About Floating-Point Arithmetic 2550 Garcia Avenue Mountain View, CA 94043 U.S.A. Part No: 800-7895-10 Revision A, June 1992 1994 Sun Microsystems, Inc. 2550 Garcia Avenue, Mountain View, California 94043-1100 U.S.A. All rights reserved. This product and related documentation are protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation. No part of this product or related documentation may be reproduced in any form by any means without prior written authorization of Sun and its licensors, if any. Portions of this product may be derived from the UNIX® and Berkeley 4.3 BSD systems, licensed from UNIX System Laboratories, Inc., a wholly owned subsidiary of Novell, Inc., and the University of California, respectively. Third-party font software in this product is protected by copyright and licensed from Sun’s font suppliers. RESTRICTED RIGHTS LEGEND: Use, duplication, or disclosure by the United States Government is subject to the restrictions set forth in DFARS 252.227-7013 (c)(1)(ii) and FAR 52.227-19. The product described in this manual may be protected by one or more U.S. patents, foreign patents, or pending applications. TRADEMARKS Sun, the Sun logo, Sun Microsystems, Sun Microsystems Computer Corporation, Solaris, are trademarks or registered trademarks of Sun Microsystems, Inc. in the U.S. and certain other countries. UNIX is a registered trademark of Novell, Inc., in the United States and other countries; X/Open Company, Ltd., is the exclusive licensor of such trademark. OPEN LOOK® is a registered trademark of Novell, Inc. PostScript and Display PostScript are trademarks of Adobe Systems, Inc. All other product names mentioned herein are the trademarks of their respective owners. All SPARC trademarks, including the SCD Compliant Logo, are trademarks or registered trademarks of SPARC International, Inc. SPARCstation, SPARCserver, SPARCengine, SPARCstorage, SPARCware, SPARCcenter, SPARCclassic, SPARCcluster, SPARCdesign, SPARC811, SPARCprinter, UltraSPARC, microSPARC, SPARCworks, and SPARCompiler are licensed exclusively to Sun Microsystems, Inc. Products bearing SPARC trademarks are based upon an architecture developed by Sun Microsystems, Inc. The OPEN LOOKand Sun™ Graphical User Interfaces were developed by Sun Microsystems, Inc. for its users and licensees. Sun acknowledges the pioneering efforts of Xerox in researching and developing the concept of visual or graphical user interfaces for the computer industry. Sun holds a non-exclusive license from Xerox to the Xerox Graphical User Interface, which license also covers Sun’s licensees who implement OPEN LOOK GUIs and otherwise comply with Sun’s written license agreements. X Window System is a product of the Massachusetts Institute of Technology. THIS PUBLICATION IS PROVIDED “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, OR NON-INFRINGEMENT. THIS PUBLICATION COULD INCLUDE TECHNICAL INACCURACIES OR TYPOGRAPHICAL ERRORS. CHANGES ARE PERIODICALLY ADDED TO THE INFORMATION HEREIN; THESE CHANGES WILL BE INCORPORATED IN NEW EDITIONS OF THE PUBLICATION. SUN MICROSYSTEMS, INC. MAY MAKE IMPROVEMENTS AND/OR CHANGES IN THE PRODUCT(S) AND/OR THE PROGRAM(S) DESCRIBED IN THIS PUBLICATION AT ANY TIME Please Recycle Contents Abstract . 1 Introduction . 2 Rounding Error . 2 Floating-point Formats . 3 Relative Error and Ulps . 5 Guard Digits. 6 Cancellation . 8 The IEEE Standard. 17 Formats and Operations . 18 Special Quantities . 24 NaNs. 25 Exceptions, Flags and Trap Handlers . 32 Systems Aspects. 37 Instruction Sets. 38 Languages and Compilers . 40 iii Exception Handling. 48 The Details . 50 Rounding Error . 50 Errors In Summation . 61 Summary. 62 Acknowledgments. 63 References . 63 Theorem 14 and Theorem 8 . 65 iv What Every Computer Scientist Should Know About Floating-Point Arithmetic—June 1992 What Every Computer Scientist Should Know About Floating-Point Arithmetic Note – This document is a reprint of the paper What Every Computer Scientist Should Know About Floating-Point Arithmetic, published in the March, 1991 issue of Computing Surveys. Copyright 1991, Association for Computing Machinery, Inc., reprinted by permission. Abstract 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 datatype; 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. Categories and Subject Descriptors: (Primary) C.0 [Computer Systems Organization]: General — instruction set design; D.3.4 [Programming Languages]: Processors — compilers, optimization; G.1.0 [Numerical Analysis]: General — computer arithmetic, error analysis, numerical algorithms (Secondary) 1 D.2.1 [Software Engineering]: Requirements/Specifications — languages; D.3.4 [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: Denormalized number, exception, floating- point, floating-point standard, gradual underflow, guard digit, NaN, overflow, relative error, rounding error, rounding mode, ulp, underflow. Introduction 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 2) 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 Section , “Rounding Error,” on page 2. 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. 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. 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. In particular, the proofs of many of the theorems appear in this section. The end of each proof is marked with the ❚ symbol; when a proof is not included, the ❚ appears immediately following the statement of the theorem. Rounding Error Squeezing infinitely many real numbers into a finite number of bits requires an approximate representation. Although there are infinitely many integers, in most programs the result of integer computations can be stored in 32 bits. In 2 What Every Computer Scientist Should Know About Floating-Point Arithmetic contrast, given any fixed number of bits, most calculations with real numbers will produce quantities that cannot be exactly represented using that many bits. Therefore the result of a floating-point calculation must often be rounded in order to fit back into its finite representation. This rounding error is the characteristic feature of floating-point computation. “Relative Error and Ulps” on page 5 describes how it is measured. 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. “Guard Digits” on page 6 discusses guard digits, a means of reducing the error when subtracting two nearby numbers. Guard digits were considered sufficiently important by IBM that in 1968 it added a guard digit to the double precision format in the System/360 architecture (single precision already had a guard digit), and retrofitted all existing machines in the field. Two examples are given to illustrate the utility of guard digits. The IEEE standard goes further than just requiring the use of a guard digit. It gives an algorithm for addition, subtraction, multiplication, division and square root, and requires that implementations produce the same result as that algorithm. Thus when a program is moved from one machine to another, the results of the basic operations will
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