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Arithmetic
Account: inchonun
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AN: 511701 ; Kornerup, Peter, Matula, David W..; Finite Precision Number Systems and
Arithmetic
Account: inchonun
r
e
d
n
u
d FINITE PRECISION NUMBER SYSTEMS
e
t
it AND ARITHMETIC
m
r
e
p
s
e
s
u Fundamentalarithmeticoperationssupportvirtuallyalloftheengineering,
r
i
fa scientific,andfinancialcomputationsrequiredforpracticalapplicationsfrom
t
ep cryptography,tofinancialplanning,torocketscience.Thiscomprehensive
c
x
e referenceprovidesresearcherswiththethoroughunderstandingofnumber
,
r
he representationsthatisanecessaryfoundationfordesigningefficientarithmetic
s
i
bl algorithms.
u
p
e Usingtheelementaryfoundationsofradixnumbersystemsasabasisfor
h
t
m arithmetic,theauthorsdevelopandcomparealternativealgorithmsforthe
o
r
f fundamentaloperationsofaddition,multiplication,division,andsquarerootwith
n
o
si preciselydefinedroundings.Variousfiniteprecisionnumbersystemsare
s
i
rm investigated,withthefocusoncomparativeanalysisofpracticallyefficient
e
p
t algorithmsforclosedarithmeticoperationsoverthesesystems.
u
o
th Eachchapterbeginswithanintroductiontoitscontentsandendswith
i
w
m bibliographicnotesandanextensivebibliography.Thebookmayalsobeused
r
o
f forgraduateteaching:problemsandexercisesarescatteredthroughoutthetext
y
n
a andasolutionsmanualisavailableforinstructors.
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Account: inchonun
EncyclopediaofMathematicsanditsApplications
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d
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u AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
d
te UniversityPress.Foracompleteserieslistingvisit
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i
rm http://www.cambridge.org/uk/series/sSeries.asp?code=EOM
e
p
es 80 O.StormarkLie’sStructuralApproachtoPDESystems
us 81 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables
ir 82 J.P.MayberryTheFoundationsofMathematicsintheTheoryofSets
a
f 83 C.Foiasetal.Navier–StokesEquationsandTurbulence
pt 84 B.PolsterandG.F.SteinkeGeometriesonSurfaces
e
xc 85 R.B.ParisandD.KaminskiAsymptoticsandMellin–BarnesIntegrals
e 86 R.McElieceTheTheoryofInformationandCoding,2ndedn
,
er 87 B.A.MagurnAnAlgebraicIntroductiontoK-Theory
h
is 88 T.MoraSolvingPolynomialEquationSystemsI
bl 89 K.BichtelerStochasticIntegrationwithJumps
u
p 90 M.LothaireAlgebraicCombinatoricsonWords
e
h 91 A.A.IvanovandS.V.ShpectorovGeometryofSporadicGroupsII
t
m 92 P.McMullenandE.SchulteAbstractRegularPolytopes
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fr 93 G.Gierzetal.ContinuousLatticesandDomains
n 94 S.R.FinchMathematicalConstants
o
si 95 Y.JabriTheMountainPassTheorem
s
i 96 G.GasperandM.RahmanBasicHypergeometricSeries,2ndedn
m
er 97 M.C.PedicchioandW.Tholen(eds.)CategoricalFoundations
p
t 98 M.E.H.IsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
ou 99 T.MoraSolvingPolynomialEquationSystemsII
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it 100 E.OlivieriandM.Eula´liaVaresLargeDeviationsandMetastability
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101 A.Kushner,V.LychaginandV.RubtsovContactGeometryandNonlinearDifferentialEquations
m
or 102 L.W.BeinekeandR.J.Wilson(eds.)withP.J.CameronTopicsinAlgebraicGraphTheory
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y 103 O.J.StaffansWell-PosedLinearSystems
an 104 J.M.Lewis,S.LakshmivarahanandS.K.DhallDynamicDataAssimilation
in 105 M.LothaireAppliedCombinatoricsonWords
d 106 A.MarkoeAnalyticTomography
e
uc 107 P.A.MartinMultipleScattering
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ro 108 R.A.BrualdiCombinatorialMatrixClasses
ep 109 J.M.BorweinandJ.D.VanderwerffConvexFunctions
r
e 110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
b
111 R.T.CurtisSymmetricGenerationofGroups
t
no 112 H.Salzmannetal.TheClassicalFields
ay 113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLe´vyNoise
M 114 J.BeckCombinatorialGames
.
ed 115 L.BarreiraandY.PesinNonuniformHyperbolicity
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er 116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
es 117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributedParameter
r
s Systems
ht 118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
g
ri 119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
ll 120 T.NishiuraAbsoluteMeasurableSpaces
A 121 M.PrestPurity,SpectraandLocalisation
.
ss 122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
re 123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
P
y 124 F.W.KingHilbertTransformsI
t
si 125 F.W.KingHilbertTransformsII
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ghr 135 V.Berthe´andM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
ri o 136 A.Krista´ly,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,
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op.S andEconomics
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t Kornerup,Peter.
no Finiteprecisionnumbersystemsandarithmetic/PeterKornerup,DavidW.Matula.
ay p. cm.–(Encyclopediaofmathematicsanditsapplications;133)
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. Includesbibliographicalreferencesandindexes.
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ve ISBN978-0-521-76135-2
er 1.Arithmetic–Foundations. I.Matula,DavidW. II.Title.
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ISBN978-0-521-76135-2Hardback
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© 2010. Cambridge Univepplicable copyright law. aCccaiunmratbhcriyisdogpfweuUbeUlbRincsLiaivttseeiosrfsoniisr,t,yaeonxPrdtrewedrsnioslalelhsraoensmrontathoiginrurd,ea-asrpcpaaconruntterysaeitibentihtloeiatrrtynaaefpnotpyrwrctoehopbenrsitpiaetetenersts.iorsentfeesnrucrceehdotro
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AN: 511701 ; Kornerup, Peter, Matula, David W..; Finite Precision Number Systems and Arithmetic
Account: inchonun
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un CONTENTS
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si Preface page xi
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t 1 Radixpolynomialrepresentation 1
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o
th 1.1 Introduction 1
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w
m 1.2 Radixpolynomials 2
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o
f 1.3 Radix-β numbers 7
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a 1.4 Digitsymbolsanddigitstrings 10
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d 1.5 Digitsetsforradixrepresentation 14
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du 1.6 Determiningaradixrepresentation 20
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ep 1.7 Classifyingbase-digitsetcombinations 31
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be 1.8 Finite-precisionandcomplementrepresentations 35
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no 1.8.1 Finite-precisionradix-complementrepresentations 38
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Ma 1.9 Radix-β approximationandroundings 43
.
ed 1.9.1 Bestradix-β approximations 43
v
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se 1.9.2 Roundingintofiniterepresentations 47
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s 1.10 Otherweightedsystems 50
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ig 1.10.1 Mixed-radixsystems 50
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ll 1.10.2 Two-levelradixsystems 52
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s. 1.10.3 Double-radixsystems 52
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Pr 1.11 Notesontheliterature 54
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rs 2 Baseanddigitsetconversion 59
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vi Contents
er 2.4.3 Conversionintoacontiguousdigitset 83
d
un 2.4.4 Conversionintocanonical,non-adjacentform 92
d
te 2.5 Implementingbaseanddigitsetconversions 95
t
i
rm 2.5.1 Implementationinlogic 103
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p
s 2.5.2 On-linedigitsetconversion 108
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u 2.6 Theadditiveinverse 111
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fa 2.7 Notesontheliterature 115
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p
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c
x 3 Addition 119
e
r, 3.1 Introduction 119
e
h
is 3.2 Howfastcanwecompute? 120
l
b
pu 3.3 Digitaddition 125
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th 3.4 Additionwithredundantdigitsets 129
m
ro 3.5 Basiclinear-timeadders 136
f
on 3.5.1 Digitserialandon-lineaddition 144
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is 3.6 Sub-lineartimeadders 147
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pe 3.6.1 Carry-skipadders 148
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ou 3.6.2 Carry-selectadders 149
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t
wi 3.6.3 Carry-look-aheadadders 151
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or 3.7 Constant-timeadders 159
f
ny 3.7.1 Carry-saveaddition 159
a
in 3.7.2 Borrow-saveaddition 163
d
ce 3.8 Additionandoverflowinfiniteprecisionsystems 165
u
d
ro 3.8.1 Additioninredundantdigitsets 165
p
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r 3.8.2 Additioninradix-complementsystems 169
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. 3.9 Subtractionandsign-magnitudeaddition 177
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rv 3.9.1 Sign-magnitudeadditionandsubtraction 180
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ht 3.10 Comparisons 184
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es 3.10.3 Leadingzeroesdetermination 192
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t ap 4.3.3 High-radixmultiplication 217
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Account: inchonun
Contents vii
er 4.4 Sign-magnitudeandradix-complementmultiplication 220
d
un 4.4.1 Mappingintounsignedoperands 221
d
te 4.4.2 2’scomplementoperands 222
t
i
rm 4.4.3 TheBaughandWooleyscheme 222
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s 4.4.4 Usingarecodedmultiplier 224
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u 4.5 Linear-timemultipliers 227
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fa 4.5.1 Theclassicaliterativemultiplier 228
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ep 4.5.2 Arraymultipliers 229
c
x
e 4.5.3 LSB-firstserial/parallelmultipliers 231
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he 4.5.4 Apipelinedserial/parallelmultiplier 237
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i
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b 4.5.5 Least-significantbitfirst(LSB-first)serial/serial
u
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e multipliers 241
h
t
m 4.5.6 On-lineormost-significantbitfirst(MSB-first)
o
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f multipliers 248
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si 4.6 Logarithmic-timemultiplication 252
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rm 4.6.1 Integermultiplierswithoverflowdetection 258
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t 4.7 Squaring 262
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th 4.7.1 Radix-2squaring 263
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m 4.7.2 Recodedradix-4squaring 264
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f 4.7.3 Radix-4squaringbyoperanddualrecoding 266
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a 4.8 Notesontheliterature 269
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ce 5 Division 275
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ro 5.1 Introduction 275
p
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r 5.2 Surveyofdivisionandreciprocalalgorithms 277
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t 5.2.1 Digit-serialalgorithms 279
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. 5.2.3 Resourcerequirements 282
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rv 5.2.4 Reciprocallook-upalgorithms 283
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re 5.3 Quotientsandremainders 284
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ht 5.3.1 Integerquotient,remainderpairs 284
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r 5.3.2 Radix-β quotient,remainderpairs 287
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A 5.3.3 Convertingbetweenradix-β quotient,remainder
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es pairs 289
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rs 5.4.1 Restoringdivision 293
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viii Contents
er 5.5.3 Exploitingsymmetries 321
d
un 5.5.4 Digitselectionbydirectcomparison 324
d
te 5.5.5 Digitselectionbytablelook-up 325
t
i
rm 5.5.6 ArchitecturesforSRTdivision 326
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s 5.6 Multiplicativehigh-radixdivision 329
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fa 5.6.2 Prescaleddivision 334
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ep 5.6.3 Prescaleddivisionwithremainder 338
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he 5.7 Multiplicativeiterativerefinementdivision 344
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bl 5.7.1 Newton–Raphsondivision 346
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e 5.7.2 Convergencedivision 350
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f 5.7.4 Efficiencyofiterativerefinementdivision 359
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si 5.8 Tablelook-upsupportforreciprocals 361
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rm 5.8.1 Directtablelook-up 363
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t 5.8.2 Ulpaccurateandmonotonicreciprocal
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th approximations 370
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m 5.8.3 Bipartitetables 375
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f 5.8.4 Linearandquadraticinterpolation 383
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a 5.9 Notesontheliterature 390
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ce 6 Squareroot 398
u
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ro 6.1 Introduction 398
p
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r 6.2 Rootsandremainders 400
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b
t 6.3 Digit-serialsquareroot 402
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y 6.3.1 Restoringandnon-restoringsquareroot 404
a
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. 6.3.2 SRTsquareroot 407
d
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rv 6.3.3 CombiningSRTsquarerootwithdivision 409
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re 6.4 Multiplicativehigh-radixsquareroot 416
s
ht 6.4.1 Shortreciprocalsquareroot 419
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r 6.4.2 Prescaledsquareroot 422
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A 6.5 Iterativerefinementsquareroot 426
.
s
es 6.5.1 Newton–Raphsonsquareroot 428
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y 6.5.2 Newton–Raphsonroot-reciprocal 432
t
i
rs 6.5.3 Convergencesquareroot 434
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t ap 7.2.1 Floating-pointnumberfactorization 450
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