ANSI IEEE 854-1987 Radix-Independent Floating-Point Arithmetic (IEEE Computer Society).pdf

1、iANSI/IEEE Std 854-1987An American National StandardIEEE Standard for Radix-Independent Floating-Point ArithmeticSponsorTechnical Committee on Microprocessors and Microcomputersof theIEEE Computer SocietyApproved March 12, 1987Reaffirmed March 17, 1994IEEE Standards BoardApproved September 10, 1987R

2、eaffirmed August 23, 1994American National Standards Institute Copyright 1987 byThe Institute of Electrical and Electronics Engineers, Inc345 East 47th Street, New York, NY 10017, USANo part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise, without th

3、eprior written permission of the publisher.iiIEEE Standards documents are developed within the Technical Committees of the IEEE Societies and the StandardsCoordinating Committees of the IEEE Standards Board. Members of the committees serve voluntarily and withoutcompensation. They are not necessaril

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7、med, it is reasonable to conclude that its contents, although still ofsome value, do not wholly reflect the present state of the art. Users are cautioned to check to determine that they havethe latest edition of any IEEE Standard.Comments for revision of IEEE Standards are welcome from any intereste

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9、e tospecific applications. When the need for interpretations is brought to the attention of IEEE, the Institute will initiateaction to prepare appropriate responses. Since IEEE Standards represent a consensus of all concerned interests, it isimportant to ensure that any interpretation has also recei

10、ved the concurrence of a balance of interests. For this reasonIEEE and the members of its technical committees are not able to provide an instant response to interpretation requestsexcept in those cases where the matter has previously received formal consideration.Comments on standards and requests

11、for interpretations should be addressed to:Secretary, IEEE Standards Board345 East 47th StreetNew York, NY 10017USAiiiForeword(This Foreword is not a part of ANSI/IEEE Std 854-1987, IEEE Standard for Radix-Independent Floating-Point Arithmetic.)This standard is a product of the Radix-Independent Flo

12、ating-Point Arithmetic Working Group of the MicroprocessorStandards Subcommittee. This work was sponsored by the Technical Committee on Microprocessors andMinicomputers of the IEEE Computer Society. It generalizes ANSI/IEEE Std 754-1985, IEEE Standard for BinaryFloating-Point Arithmetic, to remove d

13、ependencies on radix and wordlength. The committee believes that, except fora possible conflict with the requirements in 5.6 and 7.2 that unrecognizable decimal input strings signal an exception,and in 6.3 that the sign of zero be preserved in certain conversion operations, any implementation confor

14、ming toANSI/IEEE Std 754-1985 will also conform to this standard. In addition, the definition of logb has been enhanced inthe Appendix, and two new functions, conv and nearbyinteger, have been added. Draft 1.0 of this standard waspublished to solicit public comments.1This standard defines a family o

15、f commercially feasible ways for new systems to perform floating-point arithmetic.Issues of retrofitting were not considered. Among the desiderata that guided the formulation of this standard were thefollowing:1) Facilitate movement of existing programs from diverse computers to those that adhere to

16、 this standard.2) Enhance the capabilities and safety available to programmers who, though not expert in numerical methods,may well be attempting to produce numerically sophisticated programs. However, we recognize that utilityand safety are sometimes antagonists.3) Encourage experts to develop and

17、distribute robust and efficient numerical programs that are portable, viaminor editing and recompilation, onto any computer that conforms to this standard and possesses adequatecapacity.4) Provide direct support fora) execution-time diagnosis of anomalies,b) smoother handling of exceptions, andc) in

18、terval arithmetic at a reasonable cost.5) Provide for development ofa) standard elementary functions like exp and cos,b) very high precision (multiword) arithmetic, andc) coupling of numerical and symbolic algebraic computation.6) Enable rather than preclude further refinements and extensions.Member

19、s of the Radix-Independent Floating-Point Arithmetic Working Group who voted on versions of this proposalor participated by correspondence were as follows:W. J. Cody, Chair J. BolstadJ. BoneyL. BreedJ. T. CoonenJ. DemmelA. A. DuBrulleP. J. FaillaceA. FyfeD. M. GayR. GoodmanK. HansonD. HoughR. E. Jam

20、es, IIIW. KahanR. KarpinskiV. KlemaE. LeBlancC. LewisZ. LiuR. MartinR. MateosianW. H. McAllisterE. McDonnellM. MikalajunasK. C. NgK. A. NormanJ. F. PalmerR. PextonF. N. RisJ. RyshpanD. StevensonR. StewartT. SuyehiroH. C. Thacher, JrJ. W. ThomasR. J. Wytmar1IEEE Micro, vol 4, no 4, Aug 1984.ivThe Cha

21、irman of the Microprocessor Standards Committee at the time of approval was James Davis.At the time of approval, the Technical Committee on Microprocessors and Microcomputers had as its ExecutiveCommittee:Martin Freeman, Chair James Flournoy Michael Smolin Robert StewartThe following persons were on

22、 the balloting committee that approved this document for submission to the IEEEStandards Board:A. AllisonP. AshendemG. BaldwinR. BakerL. BarcenasM. BiewerR. BobergP. BorrillC. CampW. J. CodyS. CooperJ. DavisR. DavisS. DiamondW. FischerJ. FlournoyG. ForceM. FreemanD. GustavsonT. HarkawayR. Hochsprung

23、D. HoughD. JamesR. JamesW. KahanL. KaledaR. KarpinskiH. KirrmanD. KraftG. LangdonT. LeonardG. LyonsR. McLellanK. MondalJ. MooneyG. NelsonD. OgdenT. PittmanS. PritalB. ShieldsM. SmolinD. StevensonR. StewartM. TeenerS. TetrickE. WaltzG. WhiteF. WhittingtonWhen the IEEE Standards Board approved this st

24、andard on March 12, 1987, it had the following membership:Donald C. Fleckenstein, Chair Marco W. Migliaro, Vice Chair Sava I. Sherr, Secretary James H. BeallDennis BodsonMarshall L. CainJames M. DalyStephen R. DillonEugene P. FogartyJay ForsterKenneth D. HendrixIrvin N. HowellLeslie R. KerrJack Kinn

25、Irving KolodnyJoseph L. Koepfinger*Edward LohseJohn MayLawrence V. McCallL. Bruce McClungDonald T. Michael*L. John RankineJohn P. RiganatiGary S. RobinsonFrank L. RoseRobert E. RountreeWilliam R. TackaberryWilliam B. WilkensHelen M. Wood*Member emeritusvCLAUSE PAGE1. Scope.11.1 Implementation Object

26、ives 11.2 Inclusions . 11.3 Exclusions 12. Definitions.13. Precisions 23.1 Sets of Values. 23.2 Basic Precisions . 33.3 Extended Precisions . 43.4 Combinations of Precisions . 44. Rounding.54.1 Round to Nearest 54.2 Directed Roundings 54.3 Rounding Precision 55. Operations .55.1 Arithmetic 65.2 Squa

27、re Root 65.3 Floating-Point Precision Conversions 65.4 Conversion Between Floating Point and Integer . 65.5 Round Floating-Point Number to Integral Value. 65.6 Floating-PointDecimal String Conversion. 65.7 Comparison 86. Infinity, NaNs, and Signed Zero .106.1 Infinity Arithmetic . 106.2 Operations w

28、ith NaNs 106.3 The Algebraic Sign 107. Exceptions.117.1 Invalid Operation . 117.2 Division by Zero 117.3 Overflow 117.4 Underflow 127.5 Inexact 128. Traps138.1 Trap Handler 138.2 Precedence . 13Annex (Informative) Recommended Functions and Predicates .141An American National StandardIEEE Standard fo

29、r Radix-Independent Floating-Point Arithmetic1. Scope1.1 Implementation ObjectivesIt is intended that an implementation of a floating-point system conforming to this standard can be realized entirely insoftware, entirely in hardware, or in any combination of software and hardware. It is the environm

30、ent the programmeror user of the system sees that conforms or fails to conform to this standard. Hardware components that requiresoftware support to conform shall not be said to conform apart from such software.1.2 InclusionsThis standard specifies the following:1) Constraints on parameters defining

31、 values of basic and extended floating-point numbers2) Add, subtract, multiply, divide, square root, remainder and compare operations3) Conversions between integers and floating-point numbers4) Conversions between different floating-point precisions5) Conversion between basic precision floating-poin

32、t numbers and decimal strings6) Floating-point exceptions and their handling, including nonnumbers (NaNs)1.3 ExclusionsThis standard does not specify the following:1) Formats for internal storage of floating-point numbers2) Encodings of integers and formats of strings of characters representing deci

33、mal numbers3) Interpretation of the sign and significand fields of NaNs4) Conversion between extended precision (3.2) floating-point numbers and decimal strings2. Definitionsdestination: The location for the result of a binary or unary operation. A destination may be either explicitlydesignated by t

34、he user or implicitly supplied by the system (that is, intermediate results in subexpressions or argumentsfor procedures). Some languages place the results of intermediate calculations in destinations beyond the users2 Copyright 1987 IEEE All Rights ReservedANSI/IEEE Std 854-1987 IEEE STANDARD FORco

35、ntrol. Nonetheless, this standard defines the result of an operation in terms of that destinations precision as well asthe operands values.exponent: The component of a floating-point number that normally signifies the integer power to which the radix israised in determining the value of the represen

36、ted number. Occasionally, the exponent is called the signed or unbiasedexponent.floating-point number: A digit string characterized by three components: a sign, a signed exponent, and asignificand. Its numerical value, if any, is the signed product of its significand and the radix raised to the powe

37、r of itsexponent. In this standard a digit string is not always distinguished from a number it may represent.fraction: The component of the significand that lies to the right of its implied radix point.mode: A variable that a user may set, sense, save, and restore to control the execution of subsequ

38、ent arithmeticoperations. The default mode is the mode that a program can assume to be in effect unless an explicitly contrarystatement is included in either the program or its specification.The following mode shall be implemented:1) Rounding to control the direction of rounding errors2) In certain

39、implementations, rounding precision, to shorten the precision of results3) The implementor may, at his option, implement the following modes: traps disabled or enabled, to handleexceptionsNaN: Not a number; a symbolic entry encoded in a floating-point format. There are two types of NaNs (see 6.2).Si

40、gnaling NaNs signal the invalid operation exception (see 7.1) whenever they appear as operands. Quiet NaNspropagate through almost every arithmetic operation without signaling exceptions.normal number: A nonzero number that is finite and not subnormal.radix: The base for the representation of floati

41、ng-point numbers.result: The digit string (usually representing a number) that is delivered to the destination.significand: The component of a floating-point number that consists of a leading digit to the left of its implied radixpoint and a fraction field to the right.status flag: A variable that m

42、ay take two states, set and clear. A user may clear a flag, copy it, or restore it to a previousstate. When set, a status flag may contain additional system-dependent information, possibly inaccessible to someusers. The operations of this standard may as a side effect set some of the following flags

43、: inexact result, underflow,overflow, divide by zero, and invalid operation.subnormal number: A nonzero floating-point number whose exponent is the precisions minimum and whose leadingsignificant digit is zero.2user: Any person, hardware, or program not itself specified by this standard, having acce

44、ss to and controlling thoseoperations of the programming environment specified in this standard.3. PrecisionsThis standard defines four floating-point precisions in two groups, basic and extended, each having two widths, singleand double. The standard levels of implementation are distinguished by th

45、e combinations of precisions supported.3.1 Sets of ValuesThe standard does not specify how to encode numbers for internal storage. Four integer parameters specify eachprecision:2Subnormal numbers are called denormalized numbers in ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-point Arith

46、metic.Copyright 1987 IEEE All Rights Reserved 3RADIX-INDEPENDENT FLOATING-POINT ARITHMETIC ANSI/IEEE Std 854-1987b = the radixp = the number of base-b digits in the significandEmax= the maximum exponentEmin= the minimum exponentThe parameters are subject to the following constraints:1) b shall be ei

47、ther 2 or 10 and shall be the same for all supported precisions2) (EmaxEmin)/p shall exceed 5 and should exceed 103) bp-1 105The balance between the overflow threshold and the underflow threshold is characterized by theirproduct , which should be the smallest integral power of b that is 4.Each preci

48、sion allows for the representation of just the following entities:1) Numbers of the form (1)sbE(d0.d1d 2dp1), wheres = an algebraic signE = any integer between Eminand Emax, inclusivedi= a base-b digit (0 di b1)2) Two infinities, + and 3) At least one signaling NaN4) At least one quiet NaNThe algebr

49、aic sign provides additional information about any variable that has the value zero. Although all precisionshave distinct representations for +0, 0, + and the signs are significant in some circumstances, such as divisionby zero, and not in others. In this standard, 0 and are written without a sign when the sign does not matter. Animplementation may find it helpful to provide additional information about a variable that is NaN through an algebraicsign, but this standard does not interpret such extensions.The foregoing description enumerates some values redundan