Carnegie Mellon - cs.cmu.edu

Carnegie Mellon

1BryantandO’Hallaron,ComputerSystems:AProgrammer’sPerspective,ThirdEdition

Carnegie Mellon


Bits,Bytes,andIntegers– Part2

15-213:IntroductiontoComputerSystems3rd Lecture,Sept.5,2017

Today’sInstructor:RandyBryant

Carnegie Mellon


AssignmentAnnouncements¢ Lab0availableviacoursewebpageandAutolab.

§ DueThurs.Sept.7,11:59pm§ Nogracedays§ Nolatesubmissions§ Justdoit!

¢ Lab1availableviaTPZandAutolab§ DueThurs,Sept.14,11:55pm§ Readinstructionscarefully:writeup,bits.c,tests.c

§ Quirkysoftwareinfrastructure§ Basedonlectures2,3,and4(CS:APPChapter2)§ Aftertoday’slectureyouwillknoweverythingfortheinteger

problems§ FloatingpointcoveredThursdaySept.7

Carnegie Mellon


SummaryFromLastLecture¢ Representinginformationasbits¢ Bit-levelmanipulations¢ Integers

§ Representation:unsignedandsigned§ Conversion,casting§ Expanding,truncating§ Addition,negation,multiplication,shifting

¢ Representationsinmemory,pointers,strings¢ Summary

Carnegie Mellon


EncodingIntegers

Two’sComplementExamples(w=5)

B2T (X ) = -xw-1 ×2w-1 + xi ×2

i

i=0

w-2

åB2U(X ) = xi ×2i

i=0

w-1

åUnsigned Two’sComplement

SignBit

10 = -16 8 4 2 1

0 1 0 1 0

-10 = -16 8 4 2 1

1 0 1 1 0

8+2 = 10

-16+4+2 = -10

Carnegie Mellon


Unsigned&SignedNumericValues¢ Equivalence

§ Sameencodingsfornonnegativevalues

¢ Uniqueness§ Everybitpatternrepresents

uniqueintegervalue§ Eachrepresentable integerhas

uniquebitencoding

¢ Expressioncontainingsignedandunsignedint:int iscasttounsigned

X B2T(X)B2U(X)0000 00001 10010 20011 30100 40101 50110 60111 7

–88–79–610–511–412–313–214–115

10001001101010111100110111101111

01234567

Carnegie Mellon


SignExtensionandTruncation¢ SignExtension

¢ Truncation

Carnegie Mellon


Today:Bits,Bytes,andIntegers¢ Representinginformationasbits¢ Bit-levelmanipulations¢ Integers



Carnegie Mellon


UnsignedAddition

¢ StandardAdditionFunction§ Ignorescarryoutput

¢ ImplementsModularArithmetics = UAddw(u ,v) = u +v mod2w

• • •• • •

uv+

• • •u + v• • •

TrueSum:w+1bits

Operands:w bits

DiscardCarry:w bits UAddw(u , v)

1110 1001+ 1101 01011 1011 11101011 1110

E9+ D51BEBE

0 0 00001 1 00012 2 00103 3 00114 4 01005 5 01016 6 01107 7 01118 8 10009 9 1001A 10 1010B 11 1011C 12 1100D 13 1101E 14 1110F 15 1111

223+ 213446190

unsigned char

Carnegie Mellon


0 2 4 6 8 10 12 140

2

46

810

1214

0

4

8

12

16

20

24

28

32

Integer Addition

Visualizing(Mathematical)IntegerAddition

¢ IntegerAddition§ 4-bitintegersu,v§ ComputetruesumAdd4(u ,v)

§ Valuesincreaselinearlywithu andv

§ Formsplanarsurface

Add4(u ,v)

u

v

Carnegie Mellon


0 2 4 6 8 10 12 140

2

46

810

1214

0

2

4

6

8

10

12

14

16

VisualizingUnsignedAddition

¢ WrapsAround§ Iftruesum≥2w

§ Atmostonce

0

2w

2w+1

UAdd4(u ,v)

u

v

TrueSum

ModularSum

Overflow

Overflow

Carnegie Mellon


Two’sComplementAddition

¢ TAdd andUAdd haveIdenticalBit-LevelBehavior§ Signedvs.unsignedadditioninC:

int s, t, u, v;s = (int) ((unsigned) u + (unsigned) v);t = u + v

§ Willgive s == t

• • •• • •

uv+

• • •u + v• • •

TrueSum:w+1bits

Operands:w bits

DiscardCarry:w bits TAddw(u , v)

1110 1001+ 1101 01011 1011 11101011 1110

E9+ D51BEBE

-23+ -43446-66

Carnegie Mellon


TAddOverflow

¢ Functionality§ Truesumrequiresw+1

bits§ DropoffMSB§ Treatremainingbitsas

2’scomp.integer

–2w–1

–2w

0

2w–1–1

2w–1

TrueSum

TAdd Result

1 000…0

1 011…1

0 000…0

0 100…0

0 111…1

100…0

000…0

011…1

PosOver

NegOver

Carnegie Mellon


-8 -6 -4 -2 0 2 4 6-8

-6-4

-20

24

6

-8

-6

-4

-2

0

2

4

6

8

Visualizing2’sComplementAddition

¢ Values§ 4-bittwo’scomp.§ Rangefrom-8to+7

¢ WrapsAround§ Ifsum³ 2w–1

§ Becomesnegative§ Atmostonce

§ Ifsum<–2w–1

§ Becomespositive§ Atmostonce

TAdd4(u ,v)

u

vPosOver

NegOver

Carnegie Mellon


CharacterizingTAdd

¢ Functionality§ Truesumrequiresw+1 bits§ DropoffMSB§ Treatremainingbitsas2’s

comp.integer

TAddw (u,v) =u + v + 2w-1 u + v < TMinwu + v TMinw £ u + v £ TMaxwu + v - 2w-1 TMaxw < u + v

ì

í ï

î ï

(NegOver)

(PosOver)

u

v

<0 >0

<0

>0

NegativeOverflow

PositiveOverflow

TAdd(u ,v)

2w

2w

Carnegie Mellon


Multiplication¢ Goal:ComputingProductofw-bitnumbersx,y

§ Eithersignedorunsigned

¢ But,exactresultscanbebiggerthanw bits§ Unsigned:upto2w bits

§ Resultrange:0≤x *y ≤(2w – 1)2 =22w – 2w+1 +1§ Two’scomplementmin(negative):Upto2w-1bits

§ Resultrange:x *y ≥(–2w–1)*(2w–1–1)=–22w–2+2w–1

§ Two’scomplementmax(positive):Upto2w bits,butonlyfor(TMinw)2

§ Resultrange:x *y ≤(–2w–1)2 =22w–2

¢ So,maintainingexactresults…§ wouldneedtokeepexpandingwordsizewitheachproductcomputed§ isdoneinsoftware,ifneeded

§ e.g.,by“arbitraryprecision”arithmeticpackages

Carnegie Mellon


UnsignedMultiplicationinC

¢ StandardMultiplicationFunction§ Ignoreshighorderw bits

¢ ImplementsModularArithmeticUMultw(u ,v)= u ·v mod2w

• • •• • •

uv*

• • •u · v• • •

TrueProduct:2*w bits

Operands:w bits

Discardw bits:w bitsUMultw(u , v)

• • •

1110 1001* 1101 01011100 0001 1101 1101

1101 1101

E9* D5C1DDDD

223* 21347499221

Carnegie Mellon


SignedMultiplicationinC

¢ StandardMultiplicationFunction§ Ignoreshighorderw bits§ Someofwhicharedifferentforsigned

vs.unsignedmultiplication§ Lowerbitsarethesame

• • •• • •

uv*

• • •u · v• • •

TrueProduct:2*w bits

Operands:w bits

Discardw bits:w bitsTMultw(u , v)

• • •

-23* -43

989-35

1110 1001* 1101 01010000 0011 1101 1101

1101 1101

E9* D503DDDD

Carnegie Mellon


Power-of-2MultiplywithShift¢ Operation

§ u << k givesu * 2k

§ Bothsignedandunsigned

¢ Examples§ u << 3 == u * 8§ (u << 5) – (u << 3)== u * 24

§ Mostmachinesshiftandaddfasterthanmultiply§ Compilergeneratesthiscodeautomatically

• • •

0 0 1 0 0 0•••

u2k*

u · 2kTrueProduct:w+k bits

Operands:w bits

Discardk bits:w bits UMultw(u , 2k)

•••

k

• • • 0 0 0•••

TMultw(u , 2k)0 0 0••••••

Carnegie Mellon


UnsignedPower-of-2DividewithShift¢ QuotientofUnsignedbyPowerof2

§ u >> k givesë u / 2kû§ Useslogicalshift

Division Computed Hex Binary x 15213 15213 3B 6D 00111011 01101101 x >> 1 7606.5 7606 1D B6 00011101 10110110 x >> 4 950.8125 950 03 B6 00000011 10110110 x >> 8 59.4257813 59 00 3B 00000000 00111011

0 0 1 0 0 0•••

u2k/

u / 2kDivision:

Operands:•••

k••• •••

•••0 0 0••• •••

ë u / 2k û •••Result:

.

BinaryPoint

0

0 0 0•••0

Carnegie Mellon


SignedPower-of-2DividewithShift¢ QuotientofSignedbyPowerof2

§ x >> k givesë x / 2kû§ Usesarithmeticshift§ Roundswrongdirectionwhenu < 0

0 0 1 0 0 0•••x2k/

x / 2kDivision:

Operands:•••

k••• •••

•••0 ••• •••RoundDown(x / 2k) •••Result:

.

BinaryPoint

0 •••

Division Computed Hex Binary y -15213 -15213 C4 93 11000100 10010011 y >> 1 -7606.5 -7607 E2 49 11100010 01001001 y >> 4 -950.8125 -951 FC 49 11111100 01001001 y >> 8 -59.4257813 -60 FF C4 11111111 11000100

Carnegie Mellon


CorrectPower-of-2Divide¢ QuotientofNegativeNumberbyPowerof2

§ Wanté x / 2kù (RoundToward0)§ Computeasë (x+2k-1)/ 2kû

§ InC:(x + (1<<k)-1) >> k§ Biasesdividendtoward0

Case1:Norounding

Divisor:

Dividend:

0 0 1 0 0 0•••

u

2k/é u / 2k ù

•••

k1 ••• 0 0 0•••

1 •••0 1 1••• .

BinaryPoint

1

0 0 0 1 1 1•••+2k –1 •••

1 1 1•••

1 ••• 1 1 1•••

Biasinghasnoeffect

Carnegie Mellon


CorrectPower-of-2Divide(Cont.)

Divisor:

Dividend:

Case2:Rounding

0 0 1 0 0 0•••

x

2k/é x / 2k ù

•••

k1 ••• •••

1 •••0 1 1••• .

BinaryPoint

1

0 0 0 1 1 1•••+2k –1 •••

1 ••• •••

Biasingadds1tofinalresult

•••

Incrementedby1

Incrementedby1

Carnegie Mellon


Negation:Complement&Increment¢ Negatethroughcomplementandincrease

~x + 1 == -x

¢ Example§ Observation:~x + x == 1111…111 == -1

1 0 0 1 0 11 1x

0 1 1 0 1 00 0~x+

1 1 1 1 1 11 1-1

Decimal Hex Binary x 15213 3B 6D 00111011 01101101 ~x -15214 C4 92 11000100 10010010 ~x+1 -15213 C4 93 11000100 10010011 y -15213 C4 93 11000100 10010011

x=15213

Carnegie Mellon


Complement&IncrementExamples

Decimal Hex Binary x -32768 80 00 10000000 00000000 ~x 32767 7F FF 01111111 11111111 ~x+1 -32768 80 00 10000000 00000000

x=TMin

Decimal Hex Binary 0 0 00 00 00000000 00000000 ~0 -1 FF FF 11111111 11111111 ~0+1 0 00 00 00000000 00000000

x=0

Canonicalcounterexample

Carnegie Mellon



§ Representation:unsignedandsigned§ Conversion,casting§ Expanding,truncating§ Addition,negation,multiplication,shifting§ Summary

¢ Representationsinmemory,pointers,strings

Carnegie Mellon


Arithmetic:BasicRules¢ Addition:

§ Unsigned/signed:Normaladditionfollowedbytruncate,sameoperationonbitlevel

§ Unsigned:additionmod2w

§ Mathematicaladdition+possiblesubtractionof2w

§ Signed:modifiedadditionmod2w(resultinproperrange)§ Mathematicaladdition+possibleadditionorsubtractionof2w

¢ Multiplication:§ Unsigned/signed:Normalmultiplicationfollowedbytruncate,

sameoperationonbitlevel§ Unsigned:multiplicationmod2w

§ Signed:modifiedmultiplicationmod2w(resultinproperrange)

Carnegie Mellon


WhyShouldIUseUnsigned?¢ Don’t usewithoutunderstandingimplications

§ Easytomakemistakesunsigned i;for (i = cnt-2; i >= 0; i--)

a[i] += a[i+1];

§ Canbeverysubtle#define DELTA sizeof(int)int i;for (i = CNT; i-DELTA >= 0; i-= DELTA)

. . .

Carnegie Mellon


CountingDownwithUnsigned¢ Properwaytouseunsignedasloopindex

unsigned i;for (i = cnt-2; i < cnt; i--)

a[i] += a[i+1];

¢ SeeRobertSeacord,SecureCodinginCandC++§ CStandardguaranteesthatunsignedadditionwillbehavelikemodular

arithmetic§ 0– 1à UMax

¢ Evenbettersize_t i;for (i = cnt-2; i < cnt; i--)

a[i] += a[i+1];§ Datatypesize_t definedasunsignedvaluewithlength=wordsize§ Codewillworkevenif cnt =UMax§ Whatifcnt issignedand<0?

Carnegie Mellon


WhyShouldIUseUnsigned?(cont.)¢ Do UseWhenPerformingModularArithmetic

§ Multiprecision arithmetic

¢ Do UseWhenUsingBitstoRepresentSets§ Logicalrightshift,nosignextension

¢ Do UseInSystemProgramming§ Bitmasks,devicecommands,…

Carnegie Mellon



§ Representation:unsignedandsigned§ Conversion,casting§ Expanding,truncating§ Addition,negation,multiplication,shifting§ Summary

¢ Representationsinmemory,pointers,strings

Carnegie Mellon


Byte-OrientedMemoryOrganization

¢ Programs referto databyaddress§ Conceptually,envisionitasaverylargearrayofbytes

§ Inreality,it’snot,butcanthinkofitthatway§ Anaddressislikeanindexintothatarray

§ and,apointervariablestoresanaddress

¢ Note:systemprovides privateaddressspacestoeach“process”§ Thinkofaprocessasaprogrambeingexecuted§ So,aprogramcanclobberitsowndata,butnotthatofothers

• • •

Carnegie Mellon


MachineWords¢ Anygivencomputerhasa“WordSize”§ Nominalsizeofinteger-valueddata

§ andofaddresses

§ Untilrecently,mostmachinesused32bits(4bytes) aswordsize§ Limitsaddressesto4GB(232 bytes)

§ Increasingly,machineshave64-bitwordsize§ Potentially,couldhave18EB(exabytes)ofaddressablememory§ That’s18.4X1018

§ Machinesstillsupportmultipledataformats§ Fractionsormultiplesofwordsize§ Alwaysintegralnumberofbytes

Carnegie Mellon


Word-OrientedMemoryOrganization¢ AddressesSpecifyByte

Locations§ Addressoffirstbyteinword§ Addressesofsuccessivewordsdiffer

by4(32-bit)or8(64-bit)

000000010002000300040005000600070008000900100011

32-bitWords Bytes Addr.

0012001300140015

64-bitWords

Addr =??

Addr =??

Addr =??

Addr =??

Addr =??

Addr =??

0000

0004

0008

0012

0000

0008

Carnegie Mellon


ExampleDataRepresentations

CDataType Typical32-bit Typical64-bit x86-64

char 1 1 1

short 2 2 2

int 4 4 4

long 4 8 8

float 4 4 4

double 8 8 8

pointer 4 8 8

Carnegie Mellon


ByteOrdering¢ So,howarethebyteswithinamulti-byteword orderedin

memory?¢ Conventions§ BigEndian:Sun(OracleSPARC),PPCMac,Internet

§ Leastsignificantbytehashighestaddress§ LittleEndian:x86,ARMprocessorsrunningAndroid,iOS,andLinux

§ Leastsignificantbytehaslowestaddress

Carnegie Mellon


ByteOrderingExample

¢ Example§ Variablex has4-byte valueof0x01234567§ Addressgivenby&x is0x100

0x100 0x101 0x102 0x103

01 23 45 67

0x100 0x101 0x102 0x103

67 45 23 01

Big Endian

Little Endian

01 23 45 67

67 45 23 01

Carnegie Mellon


RepresentingIntegersDecimal: 15213

Binary: 0011 1011 0110 1101

Hex: 3 B 6 D

6D3B0000

IA32, x86-64

3B6D

0000

Sun

int A = 15213;

93C4FFFF

IA32, x86-64

C493

FFFF

Sun

Two’s complement representation

int B = -15213;

long int C = 15213;

00000000

6D3B0000

x86-64

3B6D

0000

Sun6D3B0000

IA32

Increasin

gad

dresses

Carnegie Mellon


ExaminingDataRepresentations¢ CodetoPrintByteRepresentationofData§ Castingpointertounsignedchar* allowstreatmentasabytearray

Printf directives:%p: Printpointer%x: PrintHexadecimal

typedef unsigned char *pointer;

void show_bytes(pointer start, size_t len){size_t i;for (i = 0; i < len; i++)

printf(”%p\t0x%.2x\n",start+i, start[i]);printf("\n");

}

Carnegie Mellon


show_bytes ExecutionExampleint a = 15213;printf("int a = 15213;\n");show_bytes((pointer) &a, sizeof(int));

Result (Linux x86-64):int a = 15213;0x7fffb7f71dbc 6d0x7fffb7f71dbd 3b0x7fffb7f71dbe 000x7fffb7f71dbf 00

Carnegie Mellon


RepresentingPointers

Differentcompilers&machinesassigndifferentlocationstoobjects

Evengetdifferentresultseachtimerunprogram

int B = -15213;int *P = &B;

x86-64Sun IA32EF

FF

FB

2C

AC

28

F5

FF

3C

1B

FE

82

FD

7F

00

00

Carnegie Mellon


char S[6] = "18213";

Representing Strings

¢ StringsinC§ Representedbyarrayofcharacters§ EachcharacterencodedinASCIIformat

§ Standard7-bitencodingofcharacterset§ Character“0”hascode0x30

– Digiti hascode0x30+i§ Stringshouldbenull-terminated

§ Finalcharacter=0

¢ Compatibility§ Byteorderingnotanissue

IA32 Sun31

38

32

31

33

00

31

38

32

31

33

00

Carnegie Mellon


Address Instruction Code Assembly Rendition8048365: 5b pop %ebx8048366: 81 c3 ab 12 00 00 add $0x12ab,%ebx804836c: 83 bb 28 00 00 00 00 cmpl $0x0,0x28(%ebx)

ReadingByte-ReversedListings¢ Disassembly§ Textrepresentationofbinarymachinecode§ Generatedbyprogramthatreadsthemachinecode

¢ ExampleFragment

¢ DecipheringNumbers§ Value: 0x12ab

§ Padto32bits: 0x000012ab

§ Splitintobytes: 00 00 12 ab

§ Reverse: ab 12 00 00

Carnegie Mellon


IntegerCPuzzles

x < 0 Þ ((x*2) < 0)ux >= 0x & 7 == 7 Þ (x<<30) < 0ux > -1x > y Þ -x < -yx * x >= 0x > 0 && y > 0 Þ x + y > 0x >= 0 Þ -x <= 0x <= 0 Þ -x >= 0(x|-x)>>31 == -1ux >> 3 == ux/8x >> 3 == x/8x & (x-1) != 0

int x = foo();

int y = bar();

unsigned ux = x;

unsigned uy = y;

Initialization

Carnegie Mellon


Summary¢ Representinginformationasbits¢ Bit-levelmanipulations¢ Integers



Carnegie Mellon - cs.cmu.edu

Documents

Transcript of Carnegie Mellon - cs.cmu.edu