BinaryArithmetic2IntroductiontoAssemblyLanguage

CS64:ComputerOrganizationandDesignLogicLecture#3Fall2018

ZiadMatni,Ph.D.

Dept.ofComputerScience,UCSB

AdministrativeStuff

•  Theclassisfull–IwillnotbeaddingmorepplL

•  Didyoucheckoutthesyllabus?•  Didyoucheckouttheclasswebsite?•  DidyoucheckoutPiazza(andgetaccesstoit)?•  Didyougotolabyesterday?

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AnyQuestionsFromLastLecture?

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5-MinutePopQuiz!!!

YOUMUSTSHOWYOURWORK!!!1.  Calculateandgiveyouranswerinhexadecimal:

~(0x3E|0xFC)

2.  ConvertfrombinarytodecimalANDtohexadecimal.Useanytechnique(s)youlike:

a)  1001001b)  10010010

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Answers…

1.  Calculateandgiveyouranswerinhexadecimal:~(0x3E|0xFC)

2.  ConvertfrombinarytodecimalANDhexadecimal.Useanytechniqueyoulike:a)  1001001

b)  10010010

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=~(0xFE)=0x01

=01001001=0x49=1+8+64=73=10010010=0x92Iseethatit’s(1001001)x2=146

LectureOutline

•  Two’scomplement•  Additionandsubtractioninbinary•  CarryOutandOverflowbits

•  IntrotoAssembly

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TwosComplementMethod

•  ThisishowTwosComplementfixesthis.•  Let’swriteout-6(10)in2s-Complementbinaryin4bits:

So,–6(10)=1010(2)accordingtothisrule

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011010011010

Firsttaketheunsigned(abs)value(i.e.6)andconverttobinary:

Thennegateit(i.e.doa“NOT”functiononit):Nowadd1:

Let’sdoitBackwards…BydoingitTHESAMEEXACTWAY!

•  2s-ComplementtoDecimalmethodisthesame!

•  Take1010fromourpreviousexample•  Negateitanditbecomes0101•  Nowadd1toit&itbecomes0110,whichis6(10)

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AnotherViewof2sComplement

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NOTE:InTwo’sComplement,ifthenumber’sMSBis“1”,thenthatmeansit’sanegativenumberandifit’s“0”thenthenumberispositive.

AnotherViewof2sComplement

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NOTE:Oppositenumbersshowupassymmetricallyoppositeeachotherinthecircle.

VERYIMPORTANT:Whenwetalkof2scomplement,wemustalsomentionthenumberofbitsinvolved

Ranges

•  Therangerepresentedbynumberofbitsdiffersbetweenpositiveandnegativebinarynumbers

•  GivenNbits,therangerepresentedis:0to

and

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+2N–1forpositivenumbers–2N-1to+2N-1–1for2’sComplementnegativenumbers

Addition•  Wehaveanelementarynotionofaddingsingledigits,along

withanideaofcarryingdigits–  Example:whenadding3to9,weputforward2andcarrythe1

(i.e.tomean12)

•  Wecanbuildonthisnotiontoaddnumberstogetherthataremorethanonedigitlong

•  Example: 123 + 389

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11

215

carrieddigits

AdditioninBinary

•  Samemathematicalprincipalapplies

0011+1101

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1

00

11

00

3+13

161

1

Q:What’sbeingassumedhere???

A:Thatthesearepurelypositivenumbers

Theoretically,Icanaddanybinaryno.withN1digitstoanyotherbinaryno.withN2digits.Practically,aCPUmusthaveadefinedno.ofdigitsthatit’sworkingwith.WHY???

Exercises

Implementingan8-bitadder:•  Whatis(0x52)+(0x4B)?– Ans:0x9D,outputcarrybit=0

•  Whatis(0xCA)+(0x67)?– Ans:0x31,outputcarrybit=1

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BlackBoxPerspectiveofANYN-BitBinaryAdder

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N-bitBINARYADDER

XY

OutputResultsBitsOutputCarryBitCIN

COUT

X+Y+CIN

InputBitsCarry-inbit

N

N

N

ThisisausefulperspectiveforeitherwritinganN-bitadderfunctionincode,

orfordesigningtheactualdigitalcircuitthatdoesthis!

OutputCarryBitSignificance

•  Forunsigned(i.e.positive)numbers,COUT=1meansthattheresultdidnotfitintothenumberofbitsallotted

•  Couldbeusedasanerrorconditionforsoftware–  Forexample,you’vedesigneda16-bitadderandduringsomecalculationofpositivenumbers,yourcarrybit/flaggoesto“1”.Conclusion?

–  Yourresultisoutsidethemaximumrangeallowedby16bits.

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Carryvs.Overflow

•  Thecarrybit/flagworksfor–andislookedat–onlyforunsigned(positive)numbers

•  Asimilarbit/flagworksislookedatforifsigned(two’scomplement)numbersareusedintheaddition: theoverflowbit

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Overflow:forNegativeNumberAddition

•  WhataboutifI’maddingtwonegativenumbers?Like:1001+1011?–  Then,Iget:0100withtheextrabitsetat1–  SanityCheck:

That’sadding(-7)+(-5),soIexpected-12,sowhat’swronghere?–  Theanswerisbeyondthecapabilityof4bitsin2’scomplement!!!

•  Theextrabitinthiscaseiscalledoverflowanditindicatesthattheadditionofnegativenumbershasresultedinanumberthat’s beyondtherangeofthegivenbits.

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HowDoWeDetermineifOverflowHasOccurred?

•  Whenadding2signednumbers: x+y=s

if x,y>0 AND s<0OR if x,y<0 AND s>0---------------------------------------------Then,overflowhasoccurred

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Example1Add:-39and92insigned8-bitbinary-39 92 --- 53

There’sacarry-out(wedon’tcare)Butthereisnooverflow(V)NotethatV=0,whileCout=1andCin_signed_bit=110/9/18 Matni,CS64,Fa18 20

Side-note:Whatistherangeofsignednumbersw/8bits?-27to(27–1),or-128to127

1101100101011100---------100110101

That’s53insigned8-bits!Looksok!

1Cin_signed_bit

Cout

Example2

Add:104and45insigned8-bitbinary

104 45 --- 149

There’snocarry-out(again,wedon’tcare)Butthereisoverflow!

Giventhatthisbinaryresultisnot149,butactually–107!NotethatV=1,whileCout=0andCin_signed_bit=110/9/18 Matni,CS64,Fa18 21

0110100000101101---------10010101

That’sNOT149insigned8-bits!

1Cin_signed_bit

Cout=0

V=Cout+Cin_signed_bit

Side-note(again):Whatistherangeofsignednumbersw/8bits?-27to(27–1),or-128to127

YOURTO-DOs

•  Assignment#2– DueonFriday!!!

•  Nextweek:AssemblyLanguage!– Doyourreadings!!

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