1、IEEE Standard for Interval Arithmetic Sponsored by the IEEE Microprocessor Standards Committee IEEE 3 Park Avenue New York, NY 10016-5997 USA IEEE Computer Society IEEE Std 1788-2015 IEEE Std 1788TM-2015IEEE Standard for Interval ArithmeticSponsorMicroprocessor Standards Committeeof theIEEE Computer
2、 SocietyApproved 11 June 2015IEEE-SA Standards BoardAbstract: This standard speci es basic interval arithmetic (IA) operations selecting and following one of thecommonly used mathematical interval models. This standard supports the IEEE 754TM oating-point formats ofpractical use in interval computat
3、ions. Exception conditions are de ned, and standard handling of these conditionsis speci ed. Consistency with the interval model is tempered with practical considerations based on input fromrepresentatives of vendors, developers and maintainers of existing systems.The standard provides a layer betwe
4、en the hardware and the programming language levels. It does not mandatethat any operations be implemented in hardware. It does not de ne any realization of the basic operations asfunctions in a programming language.Keywords: arithmetic, computing, decoration, enclosure, hull, IEEE 1788TM, interval,
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34、icipantsAt the time this IEEE standard was completed, the Interval Standard Working Group had the followingmembership:Nathalie Revol, ChairR. Baker Kearfott, Vice Chair and Acting ChairWilliam Edmonson, SecretaryJ. Wol von Gudenberg, Web MasterGuillaume Melquiond, ArchivistGeorge Corliss, Voting Tab
35、ulatorJohn Pryce, Senior Technical EditorChristian Keil, Deputy Technical EditorMichel Hack, Vincent Lef evre, Ian McIntosh, Dmitry Nadezhin,Ned Nedialkov and J. Wol von Gudenberg, Assistant Technical EditorsAlexandru AmaricaiAyman BakrAhmed BelhaniGerd BohlenderGilles ChabertRudnei Dias da CunhaHen
36、d DawoodBo EinarssonAlan EliasenHossam A. H. FahmyRichard FatemanScott FersonHaitham GadKathy GerberAlexandre GoldsztejnFrederic GoualardMichael GroszkiewiczMohamed GuerfelRobert HanekBehnam HashemiNathan HayesOliver HeimlichTimothy HickeyWerner HofschusterChenyi HuTrevor Jackson, IIIMalgorzata Jank
37、owskaMichel Kie erWalter Kr amerVladik KreinovichUlrich KulischDorina LanzaDavid LesterDominique LohezWolfram LutherAmin MaherSvetoslav MarkovG unter MayerJean-Pierre MerletJean-Michel MullerHumberto MunozJose Antonio MunozKaori NagatouMitsuhiro NakaoMarkus NeherMarco NehmeierDiep NguyenMichael Noon
38、erShinichi OishiSylvain PionAntony PopovEvgenija PopovaTarek RaissiNacim RamdaniAndreas RauhGaby Dos ReisMichael SchulteKyarash ShahriariStefan SiegelIwona SkalnaMark StadtherrJames StinePipop ThienprapasithWarwick TuckerAlfredo VaccaroMaarten van EmdenErik-Jan van KampenVan SnyderJosep VehiJulio Vi
39、llalba-MorenoG. William WalsterYan WangLee WinterPierre-Alain YvarsSergei ZhilinMohamed ZidanDan ZurasThe working group wishes to record with regret the loss of two members. Antony Popov of So a University,Bulgaria, died suddenly in 2012 at the young age of 49. Walter Kr amer, of the Bergische Unive
40、rsit atWuppertal, Germany, died in October 2014 at the age of 62. Despite illness, Walter had remained an activeparticipant until June 2014.The following members of the individual balloting committee voted on this guide. Balloters may have votedfor approval, disapproval, or abstention.Bakul Banerjee
41、Juan CarreonKeith ChowGeorge CorlissHossam FahmyAndrew FieldsendAlexander GelmanFrederic GoualardRandall GrovesMichel HackPeter HarrodOliver HeimlichWerner HoelzlChenyi HuPiotr KarockiRalph KearfottVladik KreinovichVincent LefevreVincent LipsioWilliam LumpkinsGuillaume MelquiondJames MooreJeanMichel
42、 MullerDmitry NadezhinMarco NehmeierJohn PryceNathalie RevolEugene StoudenmireGerald StueveJ. Wol Von GudenbergForrest WrightOren YuenWhen the IEEE-SA Standards Board approved this standard on 11 June 2015, it had the following member-ship:John Kulick, ChairJon Walter Rosdahl, Vice ChairRichard H. H
43、ulett, Past ChairKonstantinos Karachalios, SecretaryMasayuki AriyoshiTed BurseStephen DukesJean-Philippe FaureJ. Travis Gri thGary Ho manMichael JanezicJoseph L. Koep nger*David J. LawHung LingAndrew MylesT. W. OlsenGlenn ParsonsRonald C. PetersenAnnette D. ReillyStephen J. ShellhammerAdrian P. Step
44、hensYatin TrivediPhillip WinstonDon WrightYu YuanDaidi Zhong*Member EmeritusDon MessinaIEEE-SA Content Production and ManagementJonathan GoldbergIEEE-SA Operational Program ManagementviiCopyright c 2015 IEEE. All rights reserved.IntroductionThis introduction is not part of IEEE Std 1788-2015, IEEE S
45、tandard for Interval Arithmetic.This introduction explains some of the alternative interpretations, and sometimes competing objectives, thatin uenced the design of this standard. Implementers should study it for a fuller understanding of the designchoices made in this standard among these interpreta
46、tions and objectives. For more information on intervalcomputations, including history, applications and software, seea e.g. B1, B14 and the references therein, andalso the interval computations web site B5.Mathematical contextInterval computation is a collaboration between human programmer and machi
47、ne infrastructure which, cor-rectly done, produces mathematically proven numerical results about continuous problems|for instance,rigorous bounds on the global minimum of a function or the solution of a di erential equation. It is partof the discipline of constructive real analysis.“ In the long ter
48、m, the results of such computations mightbecome su ciently trusted to be accepted as contributing to legal decisions. The machine infrastructure actsas a body of theorems on which the correctness of an interval algorithm relies, so it must be made as reliableas is practical. In its logical chain are
49、 many links|hardware, underlying oating-point system, etc.|overwhich this standard has no control. The standard aims to strengthen one speci c link, by de ning intervalobjects and operations that are theoretically well-founded and practical to implement.This document uses the standard notation a;b for the interval between numbers a and b,“ with variousdetailed meanings depending on the underlying theory. The classical“ interval arithmetic (IA) of R.A. MooreB8 uses only bounded, closed, nonempty intervals in the real numbers R|that is, a;b = fx 2 R ja x bg where a;b2R w
