Computer Arithmetic: Algorithms and Hardware Designs

642 Pages·2010·2.97 MB·English

by  Behrooz Parhami

COMPUTER
ARITHMETIC
Algorithms and Hardware Designs
SECOND EDITION
Behrooz Parhami
DepartmentofElectricalandComputerEngineering
UniversityofCalifornia,SantaBarbara
NEWYORK OXFORD
OXFORDUNIVERSITYPRESS
2010
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LibraryofCongressCataloging-in-PublicationData
Parhami,Behrooz.
Computerarithmetic/BehroozParhami.–2nded.
p.cm.
ISBN978-0-19-532848-6
1.Computerarithmetic.2.Computeralgorithms.I.Title.
QA76.9.C62P372009
005.1—dc22
2009034155
Printingnumber:9 8 7 6 5 4 3 2 1
PrintedintheUnitedStatesofAmerica
onacid-freepaper
Tothememoryofmyfather,
SalemParhami(1922–1992),
andtoallothersonwhomIcancount
foraddedinspiration,
multipliedjoy,
anddividedanguish.
PREFACE
to the first edition
THECONTEXTOFCOMPUTERARITHMETIC
Advancesincomputerarchitectureoverthepasttwodecadeshaveallowedtheper-
formanceofdigitalcomputerhardwaretocontinueitsexponentialgrowth,despite
increasingtechnologicaldifficultyinspeedimprovementatthecircuitlevel.Thisphe-
nomenal rate of growth, which is expected to continue in the near future, would
not have been possible without theoretical insights, experimental research, and tool-
buildingeffortsthathavehelpedtransformcomputerarchitecturefromanartintoone
of the most quantitative branches of computer science and engineering. Better under-
standing of the various forms of concurrency and the development of a reasonably
efficientanduser-friendlyprogrammingmodelhavebeenkeyenablersofthissuccess
story.
The downside of exponentially rising processor performance is an unprecedented
increaseinhardwareandsoftwarecomplexity.Thetrendtowardgreatercomplexityis
notonlyatoddswithtestabilityandcertifiabilitybutalsohampersadaptability,perfor-
mancetuning,andevaluationofthevarioustrade-offs,allofwhichcontributetosoaring
developmentcosts.Akeychallengefacingcurrentandfuturecomputerdesignersisto
reversethistrendbyremovinglayerafterlayerofcomplexity,optinginsteadforclean,
robust,andeasilycertifiabledesigns,whilecontinuingtotrytodevisenovelmethodsfor
gainingperformanceandease-of-usebenefitsfromsimplercircuitsthatcanbereadily
adaptedtoapplicationrequirements.
Inthecomputerdesigners’questforuser-friendliness,compactness,simplicity,high
performance, lowcost, andlowpower, computerarithmeticplaysakeyrole. Itisone
ofoldestsubfieldsofcomputerarchitecture.Thebulkofhardwareinearlydigitalcom-
putersresidedinaccumulatorandotherarithmetic/logiccircuits.Thus,first-generation
computer designers were motivated to simplify and share hardware to the extent pos-
sible and to carry out detailed cost–performance analyses before proposing a design.
Manyoftheingeniousdesignmethodsthatweusetodayhavetheirrootsinthebulky,
power-hungrymachinesof30–50yearsago.
In fact computer arithmetic has been so successful that it has, at times, become
transparent.Arithmeticcircuitsarenolongerdominantintermsofcomplexity;registers,
memory and memory management, instruction issue logic, and pipeline control have
xix
xx Preface
become the dominant consumers of chip area in today’s processors. Correctness and
highperformanceofarithmeticcircuitsareroutinelyexpected,andepisodessuchasthe
IntelPentiumdivisionbugofthemid1990sareindeedrare.
Theprecedingcontextischangingforseveralreasons.First,atveryhighclockrates,
theinterfacesbetweenarithmeticcircuitsandtherestoftheprocessorbecomecritical.
Arithmeticunitscannolongerbedesignedandverifiedinisolation.Rather,anintegrated
designoptimizationisrequired,whichmakesthedevelopmentevenmorecomplexand
costly.Second,optimizingarithmeticcircuitstomeetdesigngoalsbytakingadvantageof
thestrengthsofnewtechnologies,andmakingthemtoleranttotheweaknesses,requires
a reexamination of existing design paradigms. Finally, incorporation of higher-level
arithmetic primitives into hardware makes the design, optimization, and verification
effortshighlycomplexandinterrelated.
Thisiswhycomputerarithmeticisaliveandwelltoday.Designersandresearchers
inthisareaproducenovelstructureswithamazingregularity. Carry-lookaheadadders
compriseacaseinpoint.Weusedtothink,inthenotsodistantpast,thatweknewall
therewastoknowaboutcarry-lookaheadfastadders.Yet,newdesigns,improvements,
andoptimizationsarestillappearing.TheIEEE754standardfloating-pointformathas
removed many of the concerns with compatibility and error control in floating-point
computations, thus resulting in new designs and products with mass-market appeal.
Giventhearithmetic-intensivenatureofmanynovelapplicationareas(suchasencryp-
tion, error checking, and multimedia), computer arithmetic will continue to thrive for
yearstocome.
THEGOALSANDSTRUCTUREOFTHISBOOK
Thefieldofcomputerarithmetichasmaturedtothepointthatadozenorsotextsand
referencebookshavebeenpublished. Someofthesebooksthatcovercomputerarith-
metic in general (as opposed to special aspects or advanced/unconventional methods)
arelistedattheendofthepreface.Eachofthesebookshasitsuniquestrengthsandhas
contributedtotheformationandfruitionofthefield.Thecurrenttext,ComputerArith-
metic: Algorithms and Hardware Designs, is an outgrowth of lecture notes the author
developedandrefinedovermanyyears.Herearethemostimportantfeaturesofthistext
incomparisontothelistedbooks:
Divisionofmaterialintolecture-sizechapters.Inmyapproachtoteaching,alecture
isamoreorlessself-containedmodulewithlinkstopastlecturesandpointerstowhat
will transpire in future. Each lecture must have a theme or title and must proceed
frommotivation,todetails,toconclusion.Indesigningthetext,Istrivedtodivide
thematerialintochapters, eachofwhichissuitableforonelecture(1–2hours).A
shortlecturecancoverthefirstfewsubsections,whilealongerlecturecandealwith
variations,peripheralideas,ormoreadvancedmaterialneartheendofthechapter.
Tomakethestructurehierarchical,asopposedtoflatorlinear,lecturesaregrouped
intosevenparts,eachcomposedoffourlecturesandcoveringoneaspectofthefield
(Fig.P.1).
Preface xxi
Emphasisonboththeunderlyingtheoryandactualhardwaredesigns.Theabilityto
copewithcomplexityrequiresbothadeepknowledgeofthetheoreticalunderpin-
nings of computer arithmetic and examples of designs that help us understand the
theory.Suchdesignsalsoprovidebuildingblocksforsynthesisaswellasreference
pointsforcost-performancecomparisons.Thisviewpointisreflectedin,forexample,
thedetailedcoverageofredundantnumberrepresentationsandassociatedarithmetic
algorithms(Chapter3)thatlaterleadtoabetterunderstandingofvariousmultiplier
designsandon-linearithmetic.AnotherexamplecanbefoundinChapter22,where
coordinate rotation digital computer, or CORDIC, algorithms are introduced from
themoreintuitivegeometricviewpoint.
Linkingcomputerarithmetictoothersubfieldsofcomputing. Computerarithmetic
isnourishedby,andinturnnourishes,othersubfieldsofcomputerarchitectureand
technology.Examplesofsuchlinksabound.Thedesignofcarry-lookaheadadders
became much more systematic once it was realized that the carry computation is
a special case of parallel prefix computation that had been extensively studied by
researchersinparallelcomputing.Arithmeticforandbyneuralnetworksisanarea
thatisstillbeingexplored.Theresiduenumbersystemhasprovidedaninvaluable
toolforresearchersinterestedincomplexitytheoryandthelimitsoffastarithmetic,
aswellastothedesignersoffault-tolerantdigitalsystems.
Widecoverageofimportanttopics.Thetextcoversvirtuallyallimportantalgorith-
micandhardwaredesigntopicsincomputerarithmetic, thusprovidingabalanced
and complete view of the field. Coverage of unconventional number representa-
tionmethods(Chapters3and4),arithmeticbytablelookup(Chapter24),whichis
becomingincreasinglyimportant,multiplicationanddivisionbyconstants(Chapters
9and13), errorsandcertifiablearithmetic(Chapters19and20), andthetopicsin
PartVII(Chapters25–28)donotallappearinothertextbooks.
Unifiedandconsistentnotationandterminologythroughoutthetext.Everyeffortis
madetouseconsistentnotationandterminologythroughoutthetext.Forexample,r
alwaysstandsforthenumberrepresentationradixandsfortheremainderindivision
orsquare-rooting.Whileotherauthorshavedonethisinthebasicpartsoftheirtexts,
manytendtocovermoreadvancedresearchtopicsbysimplyborrowingthenotation
and terminology from the reference source. Such an approach has the advantage
ofmakingthetransitionbetweenreadingthetextandtheoriginalreferencesource
easier, but it is utterly confusing to the majority of the students, who rely on the
textanddonotconsulttheoriginalreferencesexcept, perhaps, towritearesearch
paper.
SUMMARYOFTOPICS
The seven parts of this book, each composed of four chapters, were written with the
followinggoals.
Part I sets the stage, gives a taste of what is to come, and provides a detailed per-
spectiveonthevariouswaysofrepresentingfixed-pointnumbers.Includedaredetailed
discussionsofsignednumbers,redundantrepresentations,andresiduenumbersystems.
xxii Preface
PartIIcoversadditionandsubtraction,whichformthemostbasicarithmeticbuilding
blocksandareoftenusedinimplementingotherarithmeticoperations.Includedinthe
discussions are addition of a constant (counting), various methods for designing fast
adders,andmultioperandaddition.
Part III deals exclusively with multiplication, beginning with the basic shift/add
algorithmsandmovingontohigh-radix,tree,array,bit-serial,modular,andavarietyof
othermultipliers.Thespecialcaseofsquaringisalsodiscussed.
PartIVcoversdivisionalgorithmsandtheirhardwareimplementations, beginning
withthebasicshift/subtractalgorithmsandmovingontohigh-radix,prescaled,modular,
array,andconvergencedividers.
PartVdealswithrealnumberarithmetic,includingvariousmethodsforrepresenting
real numbers, floating-point arithmetic, errors in representation and computation, and
methodsforhigh-precisionandcertifiablearithmetic.
Part VI covers function evaluation, beginning with the important special case of
square-rooting and moving on to coordinate rotation digital computer, or CORDIC,
algorithms,followedbygeneralconvergenceandapproximationmethods,includingthe
useoflookuptables.
PartVII deals with broad design and implementation topics, including pipelining,
low-powerarithmetic, andfaulttolerance. Thispartconcludesbyprovidinghistorical
perspectiveandexamplesofarithmeticunitsinrealcomputers.
POINTERSONHOWTOUSETHEBOOK
Forclassroomuse,thetopicsineachchapterofthistextcanbecoveredinalecturelasting
1–2hours.Inhisownteaching,theauthorhasusedthechaptersprimarilyfor1.5-hour
lectures, twice a week, in a 10-week quarter, omitting or combining some chapters to
fitthematerialinto18–20lectures.Butthemodularstructureofthetextlendsitselfto
otherlectureformats,self-study,orreviewofthefieldbypractitioners.Inthelattertwo
cases,readerscanvieweachchapterasastudyunit(foroneweek,say)ratherthanasa
lecture.Ideally,alltopicsineachchaptershouldbecoveredbeforethereadermovesto
thenextchapter.However,iffewerlecturehoursareavailable,someofthesubsections
locatedattheendofchapterscanbeomittedorintroducedonlyintermsofmotivations
andkeyresults.
Problemsofvaryingcomplexities,fromstraightforwardnumericalexamplesorexer-
cisestomoredemandingstudiesorminiprojects, aresuppliedforeachchapter.These
problems form an integral part of the book: they were not added as afterthoughts to
make the book more attractive for use as a text.Atotal of 464 problems are included
(15–18 per chapter). Assuming that two lectures are given per week, either weekly
or biweekly homework can be assigned, with each assignment having the specific
coverage of the respective half-part (two chapters) or full-part (four chapters) as its
“title.”
Aninstructor’ssolutionsmanualisavailable.Theauthor’sdetailedsyllabusforthe
courseECE252BatUCSBisavailableat:
http://www.ece.ucsb.edu/∼parhami/ece_252b.htm.
Preface xxiii
A simulator for numerical experimentation with various arithmetic algorithms is
availableat:
http://www.ecs.umass.edu/ece/koren/arith/simulator/
courtesyofProfessorIsraelKoren.
Referencestoclassicalpapersincomputerarithmetic,keydesignideas,andimpor-
tant state-of-the-art research contributions are listed at the end of each chapter. These
referencesprovidegoodstartingpointsforin-depthstudiesorfortermpapersorprojects.
Alargenumberofclassicalpapersandimportantcontributionsincomputerarithmetic
havebeenreprintedintwovolumes[Swar90].
Newideasinthefieldofcomputerarithmeticappearinpaperspresentedatbiannual
conferences,knownasARITH-n,heldinodd-numberedyears[Arit].Otherconferences
of interest includeAsilomar Conference on Signals, Systems, and Computers [Asil],
InternationalConferenceonCircuitsandSystems[ICCS],MidwestSymposiumonCir-
cuitsandSystems[MSCS],andInternationalConferenceonComputerDesign[ICCD].
RelevantjournalsincludeIEEETransactionsonComputers[TrCo],particularlyitsspe-
cialissuesoncomputerarithmetic,IEEETransactionsonCircuitsandSystems[TrCS],
Computers & Mathematics with Applications [CoMa], IET Circuits, Devices & Sys-
tems[CDS],IETComputers&DigitalTechniques[CDT],IEEETransactionsonVLSI
Systems[TrVL],andJournalofVLSISignalProcessing[JVSP].
ACKNOWLEDGMENTS
Computer Arithmetic: Algorithms and Hardware Designs is an outgrowth of lecture
notestheauthorusedforthegraduatecourse“ECE252B:ComputerArithmetic”atthe
University of California, Santa Barbara, and, in rudimentary forms, at several other
institutionspriorto1988.Thetexthasbenefitedgreatlyfromkeenobservations,curios-
ity, and encouragement of my many students in these courses.Asincere thanks to all
ofthem!
REFERENCESANDFURTHERREADINGS
Note:Referencesappearinupdated2nd-editionform,inordertoavoidtheneedforaseparatelist
forthelatter.
[Arit] InternationalSymposiumonComputerArithmetic,sponsoredbytheIEEEComputer
Society.Thisseriesbeganwithaone-dayworkshopin1969andwassubsequently
heldin1972,1975,1978,andinodd-numberedyearssince1981.The19th
symposiumintheseries,ARITH-19,washeldJune8–10,2009,inPortland,Oregon.
[Asil] AsilomarConferenceonSignalsSystems,andComputers,sponsoredannuallyby
IEEEandheldontheAsilomarConferenceGroundsinPacificGrove,California,
eachfall.The43rdconferenceinthisserieswasheldonNovember1–4,2009.
[Cava84] Cavanagh,J.J.F.,DigitalComputerArithmetic:DesignandImplementation,
McGraw-Hill,1984.
[CDS] IETCircuits,Devices&Systems,journalpublishedbytheInstitutionofEngineering
andTechnology,UnitedKingdom.
xxiv Preface
[CDT] IETComputers&DigitalTechniques,journalpublishedbytheInstitutionof
EngineeringandTechnology,UnitedKingdom.
[CoMa] Computers&MathematicswithApplications,journalpublishedbyPergamonPress.
[Desc06] Deschamps,J.-P.,G.J.A.Bioul,andG.D.Sutter,SynthesisofArithmeticCircuits:
FPGA,ASICandEmbeddedSystems,Wiley-Interscience,2006.
[Erce04] Ercegovac,M.D.,andT.Lang,DigitalArithmetic,MorganKaufmann,2004.
[Flor63] Flores,I.,TheLogicofComputerArithmetic,Prentice-Hall,1963.
[Gosl80] Gosling,J.B.,DesignofArithmeticUnitsforDigitalComputers,Macmillan,1980.
[Hwan79] Hwang,K.,ComputerArithmetic:Principles,Architecture,andDesign,Wiley,
1979.
[ICCD] InternationalConferenceonComputerDesign,sponsoredannuallybytheIEEE
ComputerSociety.ICCD-2009washeldonOctober4–7,inLakeTahoe,California.
[ICCS] InternationalConferenceonCircuitsandSystems,sponsoredannuallybytheIEEE
CircuitsandSystemsSociety.ThelatestinthisserieswasheldonMay24–27,2009,
inTaipei,Taiwan.
[JVSP] J.VLSISignalProcessing,publishedbyKluwerAcademicPublishers.
[Knut97] Knuth,D.E.,TheArtofComputerProgramming,Vol.2:SeminumericalAlgorithms,
3rded.,Addison-Wesley,1997.(Thewidelyusedsecondedition,publishedin1981,
iscitedinPartsVandVI.)
[Kore02] Koren,I.,ComputerArithmeticAlgorithms,2nded.,A.K.Peters,2002.
[Kuli81] Kulisch,U.W.,andW.L.Miranker,ComputerArithmeticinTheoryandPractice,
AcademicPress,1981.
[Lu04] Lu,M.,ArithmeticandLogicinComputerSystems,Wiley,2004.
[MSCS] MidwestSymposiumonCircuitsandSystems,sponsoredannuallybytheIEEE
CircuitsandSystemsSociety.
[Omon94] Omondi,A.R.,ComputerArithmeticSystems:Algorithms,Architectureand
Implementations,Prentice-Hall,1994.
[Rich55] Richards,R.K.,ArithmeticOperationsinDigitalComputers,VanNostrand,1955.
[Scot85] Scott,N.R.,ComputerNumberSystemsandArithmetic,Prentice-Hall,1985.
[Stei71] Stein,M.L.,andW.D.Munro,IntroductiontoMachineArithmetic,
Addison-Wesley,1971.
[Stin04] Stine,J.E.,DigitalComputerArithmeticDatapathDesignUsingVerilogHDL,
Kluwer,2004.
[Swar90] Swartzlander,E.E.,Jr.,ComputerArithmetic,Vols.IandII,IEEEComputerSociety
Press,1990.
[TrCo] IEEETrans.Computers,journalpublishedbytheIEEEComputerSociety.
Occasionallyentirespecialissuesorsectionsaredevotedtocomputerarithmetic
(e.g.,Vol.19,No.8,August1970;Vol.22,No.6,June1973;Vol.26,No.7,July
1977;Vol.32,No.4,April1983;Vol.39,No.8,August1990;Vol.41,No.8,August
1992;Vol.43,No.8,August1994;Vol.47,No.7,July1998;Vol.49,No.7,July
2000;Vol.54,No.3,March2005;Vol.58,No.2,February2009).
[TrCS] IEEETrans.CircuitsandSystems,PartsI&II,journalspublishedbyIEEE.Thetwo
partshavebeendistinguisheddifferentlyovertheyears.Currently,PartIpublishes
“regularpapers,”whilePartIIisdevotedto“expressbriefs.”