IEEE 754-2008 en Floating-Point Arithmetic (IEEE Computer Society)《浮点算法》.pdf

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1、IEEE Std 754-2008(Revision of IEEE Std 754-1985)IEEE Standard for Floating-Point ArithmeticIEEE3 Park Avenue New York, NY 10016-5997, USA29 August 2008IEEE Computer SocietySponsored by theMicroprocessor Standards Committee754TMIEEE Std 754-2008(Revision ofIEEE Std 754-1985)IEEE Standard for Floating

2、-PointArithmeticSponsorMicroprocessor Standards Committeeof theIEEE Computer SocietyApproved 12 June 2008IEEE-SA Standards BoardAbstract: This standard specifies interchange and arithmetic formats and methods for binary anddecimal floating-point arithmetic in computer programming environments. This

3、standard specifiesexception conditions and their default handling. An implementation of a floating-point systemconforming to this standard may be realized entirely in software, entirely in hardware, or in anycombination of software and hardware. For operations specified in the normative part of this

4、standard, numerical results and exceptions are uniquely determined by the values of the inputdata, sequence of operations, and destination formats, all under user control.Keywords: arithmetic, binary, computer, decimal, exponent, floating-point, format, interchange,NaN, number, rounding, significand

5、, subnormalThe Institute of Electrical and Electronics Engineers, Inc. 3 Park Avenue, New York, NY 10016-5997, USA Copyright 2008 by the Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Published 29 August 2008. Printed in the United States of America.IEEE is a registered

6、 trademark in the U.S. Patent +1 978 750 8400. Permission to photocopy portions of any individual standard foreducational classroom use can also be obtained through the Copyright Clearance Center.IntroductionThis introduction is not part of IEEE Std 754-2008, IEEE Standard for Floating-Point Arithme

7、tic.This standard is a product of the Floating-Point Working Group of, and sponsored by, the MicroprocessorStandards Committee of the IEEE Computer Society. This standard provides a discipline for performing floating-point computation that yields resultsindependent of whether the processing is done

8、in hardware, software, or a combination of the two. Foroperations specified in the normative part of this standard, numerical results and exceptions are uniquelydetermined by the values of the input data, the operation, and the destination, all under user control.This standard defines a family of co

9、mmercially feasible ways for systems to perform binary and decimalfloating-point arithmetic. Among the desiderata that guided the formulation of this standard were:a) Facilitate movement of existing programs from diverse computers to those that adhere to thisstandard as well as among those that adhe

10、re to this standard.b) Enhance the capabilities and safety available to users and programmers who, although not expertin numerical methods, might well be attempting to produce numerically sophisticated programs. c) Encourage experts to develop and distribute robust and efficient numerical programs t

11、hat areportable, by way of minor editing and recompilation, onto any computer that conforms to thisstandard and possesses adequate capacity. Together with language controls it should be possible towrite programs that produce identical results on all conforming systems.d) Provide direct support for e

12、xecution-time diagnosis of anomalies smoother handling of exceptions interval arithmetic at a reasonable cost.e) Provide for development of standard elementary functions such as exp and cos high precision (multiword) arithmetic coupled numerical and symbolic algebraic computation.f) Enable rather th

13、an preclude further refinements and extensions.In programming environments, this standard is also intended to form the basis for a dialog between thenumerical community and programming language designers. It is hoped that language-defined methods forthe control of expression evaluation and exception

14、s might be defined in coming years, so that it will bepossible to write programs that produce identical results on all conforming systems. However, it isrecognized that utility and safety in languages are sometimes antagonists, as are efficiency and portability.Therefore, it is hoped that language d

15、esigners will look on the full set of operation, precision, and exceptioncontrols described here as a guide to providing the programmer with the ability to portably controlexpressions and exceptions. It is also hoped that designers will be guided by this standard to provideextensions in a completely

16、 portable way.ivCopyright 2008 IEEE. All rights reserved. Notice to usersLaws and regulationsUsers of these documents should consult all applicable laws and regulations. Compliance with theprovisions of this standard does not imply compliance to any applicable regulatory requirements.Implementers of

17、 the standard are responsible for observing or referring to the applicable regulatoryrequirements. IEEE does not, by the publication of its standards, intend to urge action that is not incompliance with applicable laws, and these documents may not be construed as doing so. CopyrightsThis document is

18、 copyrighted by the IEEE. It is made available for a wide variety of both public and privateuses. These include both use, by reference, in laws and regulations, and use in private self-regulation,standardization, and the promotion of engineering practices and methods. By making this documentavailabl

19、e for use and adoption by public authorities and private users, the IEEE does not waive any rights incopyright to this document.Updating of IEEE documentsUsers of IEEE standards should be aware that these documents may be superseded at any time by theissuance of new editions or may be amended from t

20、ime to time through the issuance of amendments,corrigenda, or errata. An official IEEE document at any point in time consists of the current edition of thedocument together with any amendments, corrigenda, or errata then in effect. In order to determine whethera given document is the current edition

21、 and whether it has been amended through the issuance ofamendments, corrigenda, or errata, please visit the IEEE Standards Association Web site athttp:/ieeexplore.ieee.org/xpl/standards.jsp, or contact the IEEE at the address listed previously.For more information about the IEEE Standards Associatio

22、n or the IEEE standards development process,visit the IEEE-SA Web site at http:/standards.ieee.org.ErrataErrata, if any, for this and all other standards can be accessed at the following URL: http:/standards.ieee.org/reading/ieee/updates/errata/index.html. Users are encouraged to check that URL for

23、errata periodically.InterpretationsCurrent interpretations can be accessed at the following URL: http:/standards.ieee.org/reading/ieee/interp/index.html.vCopyright 2008 IEEE. All rights reserved. Patents Attention is called to the possibility that implementation of this standard may require use of s

24、ubject mattercovered by patent rights. By publication of this standard, no position is taken with respect to the existenceor validity of any patent rights in connection therewith. A patent holder or patent applicant has filed astatement of assurance that it will grant licenses under these rights wit

25、hout compensation or underreasonable rates, with reasonable terms and conditions that are demonstrably free of any unfairdiscrimination to applicants desiring to obtain such licenses. Other Essential Patent Claims may exist forwhich a statement of assurance has not been received. The IEEE is not res

26、ponsible for identifying EssentialPatent Claims for which a license may be required, for conducting inquiries into the legal validity or scopeof Patents Claims, or for determining whether any licensing terms or conditions provided in connection withsubmission of a Letter of Assurance, if any, or in

27、any licensing agreements are reasonable or non-discriminatory. Users of this standard are expressly advised that determination of the validity of any patentrights, and the risk of infringement of such rights, is entirely their own responsibility. Further informationmay be obtained from the IEEE Stan

28、dards Association.ParticipantsThe following participants in the Floating-Point Working Group contributed to the development of thisstandard:Dan Zuras, Chair Mike Cowlishaw, Editor Alex AikenMatthew ApplegateDavid BaileySteve BassDileep BhandarkarMahesh BhatDavid BindelSylvie BoldoStephen CanonSteven

29、 R. CarloughMarius CorneaMike Cowlishaw John H. CrawfordJoseph D. DarcyDebjit Das SarmaMarc DaumasBob Davis Mark DavisDick DelpJim DemmelMark A. ErleHossam A. H. FahmyJ.P. FasanoRichard FatemanEric FengWarren E. FergusonAlex Fit-FloreaLaurent FournierChip FreitagIvan GodardRoger A. Golliver David Gu

30、stafsonMichel HackJohn R. HarrisonJohn HauserYozo HidaChris N. HindsGraydon HoareDavid G. Hough Jerry HuckJim HullMichael IngrassiaDavid V. JamesRick JamesWilliam Kahan John KapernickRichard KarpinskiJeff Kidder Plamen KoevRen-Cang LiZhishun Alex LiuRaymond MakPeter Markstein David MatulaGuillaume M

31、elquiondNobuyoshi MoriRicardo MorinNed NedialkovCraig NelsonStuart ObermanJon OkadaIan OllmannMichael ParksTom PittmanEric PostpischilJason RiedyEric M. SchwarzDavid ScottDon SenzigIlya SharapovJim ShearerMichael SiuRon SmithChuck StevensPeter TangPamela J. TaylorJames W. ThomasBrandon ThompsonWendy

32、 ThrashNeil TodaSon Dao TrongLeonard Tsai Charles TsenFred TydemanLiang Kai WangScott WestbrookSteve WinklerAnthony WoodUmit YalcinalpFred ZemkePaul ZimmermannDan Zuras identifies those who also participated in the Ballot Review Committee.viCopyright 2008 IEEE. All rights reserved. The following ind

33、ividual members of the balloting committee voted on this standard. Balloters might havevoted for approval, disapproval, or abstention.Ali Al AwaziTomo BogatajMark BrownSteven R. CarloughJuan C. CarreonDanila ChenetsovSrinivas ChennupatyDanila ChernetsovNaveen CherukuriKeith ChowGlenn Colon-BonetTomm

34、y P. CooperRobert CorbettMarius CorneaMike CowlishawJohn H. CrawfordBob DavisFlorent de DinechinKenneth DockserUlrich DrepperSourav K. DuttaCarol T. EidtBo EinarssonMark A. ErleHossam A. H. FahmyJohn W. FendrichWarren E. FergusonAndrew FieldsendRabiz N. FodaRoger A. GolliverSergiu R. GomaRandall C.

35、GrovesScott A. GudgelMichel HackJohn R. HarrisonPeter L. HarrodBarry E. HedquistRutger HeunksChris N. HindsWerner HoelzlDennis HorwitzDavid G. HoughJerome HuckDavid V. JamesMuzaffer KalPiotr KarockiMark J. KnightTheodore E. KubaskaSusan LandChristoph LauterShawn M. LeardDavid J. LecistonSolomon LeeV

36、incent LefevreVincent J. LipsioZhishun A. LiuWilliam LumpkinsPeter MarksteinEdward M. MccallGeorge J. MiaoGary L. MichelJames MooreJean-Michel MullerBruce MuschlitzMichael S. NewmanCharles K. NgetheGregory D. PetersonUrichl PohlSubburajan PonnuswamyJose PuthenkulamMarko RadmilacGary S. RobinsonRober

37、t A. RobinsonMichael D. RushM. S. SachdevSridhar SamudralaRandy SaundersBartien SayogoThomas SchossigMichael J. SchulteStephen C. SchwarmEric M. SchwarzMathew L. SmithMitchell W. SmithThomas E. StaraiWalter StrupplerPing T. TangPamela J. TaylorJames W. ThomasWendy ThrashLeonard TsaiCharles TsenS. Tu

38、lasidasSrinivasa R. VemuruSteven WallachPaul R. WorkForrest D. WrightOren YuenJanusz ZalewskiAlexandru ZamfirescuDan ZurasviiCopyright 2008 IEEE. All rights reserved. When the IEEE-SA Standards Board approved this standard on 12 June 2008, it had the followingmembership:Robert M. Grow, Chair Thomas

39、Prevost, Vice Chair Steve M. Mills, Past Chair Judith Gorman, Secretary Victor BermanRichard DeBlasioAndy DrozdMark EpsteinAlexander GelmanWilliam GoldbachArnie GreenspanKen HanusJim HughesRichard HulettYoung Kyun KimJoseph L. Koepfinger *John KulickDavid J. LawGlenn ParsonsRon PetersenChuck PowersN

40、arayanan RamachandranJon Walter RosdahlAnne-Marie SahazizianMalcolm ThadenHoward WolfmanDon Wright* Member EmeritusAlso included are the following non-voting IEEE-SA Standards Board liaisons:Satish K. Aggarwal, NRC RepresentativeMichael H. Kelley, NIST RepresentativeLisa PerryIEEE Standards Project

41、EditorMalia ZamanIEEE Standards Program Manager, Technical Program DevelopmentviiiCopyright 2008 IEEE. All rights reserved. Contents1. Overview 11.1 Scope 11.2 Purpose .11.3 Inclusions 11.4 Exclusions .21.5 Programming environment considerations .21.6 Word usage .22. Definitions, abbreviations, and

42、acronyms 32.1 Definitions 32.2 Abbreviations and acronyms 53. Floating-point formats .63.1 Overview 63.2 Specification levels .73.3 Sets of floating-point data .73.4 Binary interchange format encodings .93.5 Decimal interchange format encodings 103.6 Interchange format parameters .133.7 Extended and

43、 extendable precisions .144. Attributes and rounding .154.1 Attribute specification 154.2 Dynamic modes for attributes .154.3 Rounding-direction attributes .165. Operations 175.1 Overview 175.2 Decimal exponent calculation .185.3 Homogeneous general-computational operations .195.4 formatOf general-c

44、omputational operations .215.5 Quiet-computational operations .235.6 Signaling-computational operations .245.7 Non-computational operations .245.8 Details of conversions from floating-point to integer formats 265.9 Details of operations to round a floating-point datum to integral value .275.10 Detai

45、ls of totalOrder predicate .285.11 Details of comparison predicates 295.12 Details of conversion between floating-point data and external character sequences 306. Infinity, NaNs, and sign bit 346.1 Infinity arithmetic .346.2 Operations with NaNs 346.3 The sign bit .357. Default exception handling 36

46、7.1 Overview: exceptions and flags 367.2 Invalid operation .377.3 Division by zero .377.4 Overflow .377.5 Underflow .387.6 Inexact 388. Alternate exception handling attributes .398.1 Overview 398.2 Resuming alternate exception handling attributes 398.3 Immediate and delayed alternate exception handl

47、ing attributes .40ixCopyright 2008 IEEE. All rights reserved. 9. Recommended operations 419.1 Conforming language- and implementation-defined functions 419.2 Recommended correctly rounded functions .429.3 Operations on dynamic modes for attributes 469.4 Reduction operations 4610. Expression evaluati

48、on 4810.1 Expression evaluation rules 4810.2 Assignments, parameters, and function values .4810.3 preferredWidth attributes for expression evaluation 4910.4 Literal meaning and value-changing optimizations 5011. Reproducible floating-point results .51Annex A (informative) Bibliography .53Annex B (in

49、formative) Program debugging support .55Index of operations 57xCopyright 2008 IEEE. All rights reserved. IEEE Standard for Floating-PointArithmetic IMPORTANT NOTICE: This standard is not intended to assure safety, security, health, or environmentalprotection in all circumstances. Implementers of the standard are responsible for determining appropriate safety,security, environmental, and health practices or regulatory requirements. This IEEE document is made available for use subject to important notices and legal disclaimers. These noticesand di