University*of*Washington*

Transcription

1 Roadmap C: car c = malloc(sizeof(car)); c->miles = 1; c->gals = 17; float mpg = get_mpg(c); free(c); Assembly language: Machine code: Computer system: get_mpg: pushq movq... popq ret %rbp %rsp, %rbp %rbp Java: Car c = new Car(); c.setmiles(1); c.setgals(17); float mpg = c.getmpg(); OS: UniversityofWashington Memory&data Integers&floats Machinecode&C x86assembly Procedures&stacks Arrays&structs Memory&caches Processes Virtualmemory MemoryallocaIon Javavs.C Integers! RepresentaIonofintegers:unsignedandsigned! CasIng! ArithmeIcandshiPing! Signextension Autumn213 Integers&Floats 1 Autumn213 Integers&Floats 2 Butbeforewegettointegers.! Encodeastandarddeckofplayingcards.! 52cardsin4suits! Howdoweencodesuits,facecards?! WhatoperaIonsdowewanttomakeeasytoimplement?! Whichisthehighervaluecard?! Aretheythesamesuit? TwopossiblerepresentaIons! 52cards 52bitswithbitcorrespondingtocardsetto1! One>hot encoding! Drawbacks:! Hardtocomparevaluesandsuits! Largenumberofbitsrequired low-order 52 bits of 64-bit word Autumn213 Integers&Floats 3 Autumn213 Integers&Floats 4

2 TwopossiblerepresentaIons! 52cards 52bitswithbitcorrespondingtocardsetto1! One>hot encoding! Drawbacks:! Hardtocomparevaluesandsuits! Largenumberofbitsrequired low-order 52 bits of 64-bit word Twobe\errepresentaIons! Binaryencodingofall52cards only6bitsneeded low-order 6 bits of a byte! Fitsinonebyte! Smallerthanone>hotencodings.! Howcanwemakevalueandsuitcomparisonseasier?! 4bitsforsuit,13bitsforcardvalue 17bitswithtwosetto1! Pairofone>hotencodedvalues! Easiertocomparesuitsandvalues! SJllanexcessivenumberofbits! Canwedobe\er? Autumn213 Integers&Floats 5 Autumn213 Integers&Floats 6 Twobe\errepresentaIons CompareCardSuits mask:abitvectorthat,whenbitwise ANDedwithanotherbitvectorv,turns allbutthebitsofinterestinvto! Binaryencodingofall52cards only6bitsneeded! Fitsinonebyte! Smallerthanone>hotencodings.! Howcanwemakevalueandsuitcomparisonseasier? low-order 6 bits of a byte! Binaryencodingofsuit(2bits)andvalue(4bits)separately suit value! Alsofitsinonebyte,andeasytodocomparisons Autumn213 Integers&Floats 7 #define SUIT_MASK x3 int samesuitp(char card1, char card2) { return (! (card1 & SUIT_MASK) ^ (card2 & SUIT_MASK)); //return (card1 & SUIT_MASK) == (card2 & SUIT_MASK); } returnsint SUIT_MASK = x3 = 1 1 suit value char hand[5]; // represents a 5-card hand char card1, card2; // two cards to compare card1 = hand[]; card2 = hand[1];... if ( samesuitp(card1, card2) ) {... } equivalent Autumn213 Integers&Floats 8

3 CompareCardValues VALUE_MASK = xf = suit value char hand[5]; // represents a 5-card hand char card1, card2; // two cards to compare card1 = hand[]; card2 = hand[1];... if ( greatervalue(card1, card2) ) {... } mask:abitvectorthat,whenbitwise ANDedwithanotherbitvectorv,turns allbutthebitsofinterestinvto worksevenifvalue #define VALUE_MASK xf isstoredinhighbits int greatervalue(char card1, char card2) { return ((unsigned int)(card1 & VALUE_MASK) > (unsigned int)(card2 & VALUE_MASK)); } Autumn213 Integers&Floats 9 EncodingIntegers! Thehardware(andC)supportstwoflavorsofintegers:! unsigned onlythenon>negajves! signed bothnegajvesandnon>negajves! Thereareonly2 W disinctbitpa\ernsofwbits,so...! Cannotrepresentalltheintegers! Unsignedvalues:...2 W a1! Signedvalues:a2 Wa1...2 Wa1 a1! Reminder:terminologyforbinaryrepresentaIons Most>significant or high>order bit(s) Least>significant or low>order bit(s) Autumn213 Integers&Floats 1 UnsignedIntegers SignedIntegers:SignaandaMagnitude! Unsignedvaluesarejustwhatyouexpect! b 7 b 6 b 5 b 4 b 3 b 2 b 1 b =b b b b b 2! Usefulformula: N>1 =2 N >1! Addandsubtractusingthenormal carry and borrow rules,justinbinary.! Howwouldyoumakesigned)integers? ! Let'sdothenaturalthingfortheposiIves! Theycorrespondtotheunsignedintegersofthesamevalue! Example(8bits):x=,x1=1,,x7F=127! But,weneedtoletabouthalfofthembenegaIve! Usethehighaorderbittoindicatenega,ve:callitthe signbit! Callthisa sign>and>magnitude representajon! Examples(8bits):! x= 2 isnon>negajve,becausethesignbitis! x7f= isnon>negajve! x85=1 2 isnegajve! x8=1 2 isnegajve... Autumn213 Integers&Floats 11 Autumn213 Integers&Floats 12

4 SignedIntegers:SignaandaMagnitude! Howshouldwerepresenta1inbinary?! 11 2 UsetheMSBfor+or>,andtheotherbitstogivemagnitude. MostSignificantBit SignaandaMagnitudeNegaIves! Howshouldwerepresenta1inbinary?! 11 2 UsetheMSBfor+or>,andtheotherbitstogivemagnitude. (Unfortunatesideeffect:therearetworepresentaIonsof!) Autumn213 Integers&Floats Autumn213 Integers&Floats 14 SignaandaMagnitudeNegaIves! Howshouldwerepresenta1inbinary?! 11 2 UsetheMSBfor+or>,andtheotherbitstogivemagnitude. (Unfortunatesideeffect:therearetworepresentaJonsof!)! Anotherproblem:arithmeIciscumbersome.! Example: 4>3!=4+(>3) How do we solve these problems? Autumn213 Integers&Floats Two scomplementnegaives! Howshouldwerepresenta1inbinary? Autumn213 Integers&Floats 16

5 Two scomplementnegaives! Howshouldwerepresenta1inbinary? Ratherthanasignbit,letMSBhavesamevalue,butnega+veweight.! b fori)<)w31:b i =1adds+2 i w31 =1addsa2 w31 tothevalue. tothevalue. b w31 ) b w32 ).).).) b ) Autumn213 Integers&Floats Two scomplementnegaives! Howshouldwerepresenta1inbinary? Ratherthanasignbit,letMSBhavesamevalue,butnega+veweight.! b w31 =1addsa2 w31 tothevalue. fori)<)w31:b i =1adds+2 i tothevalue. e.g.unsigned 2 : b w31 ) b w32 ).).).) b ) =1 1 2 scompl. 2 : > => Autumn213 Integers&Floats Two scomplementnegaives! Howshouldwerepresenta1inbinary? Ratherthanasignbit,letMSBhavesamevalue,butnega+veweight.! b w31 =1addsa2 w31 tothevalue. fori)<)w31:b i =1adds+2 i tothevalue. e.g.unsigned 2 : b w31 ) b w32 ).).).) b ) =1 1 2 scompl. 2 : > => ! >1isrepresentedas =>2 3 +(2 3 1) AllnegaJveintegerssJllhaveMSB=1.! Advantages:singlezero,simplearithmeJc! TogetnegaIverepresentaIonof anyinteger,takebitwisecomplement andthenaddone! ~x + 1 == -x Autumn213 Integers&Floats Autumn213 4abitUnsignedvs.Two scomplement x1+2 2 x+2 1 x1+2 x1 >2 3 x1+2 2 x+2 1 x1+2 x1 Integers&Floats

6 4abitUnsignedvs.Two scomplement 2 3 x1+2 2 x+2 1 x1+2 x1 >2 3 x1+2 2 x+2 1 x1+2 x1 11 >5 (math)difference=16=2 4 4abitUnsignedvs.Two scomplement 2 3 x1+2 2 x+2 1 x1+2 x1 >2 3 x1+2 2 x+2 1 x1+2 x1 11 >5 (math)difference=16= Autumn Integers&Floats Autumn Integers&Floats Two scomplementarithmeic! ThesameaddiIonprocedureworksforbothunsignedand two scomplementintegers! Simplifieshardware:onlyonealgorithmforaddiJon! Algorithm:simpleaddiJon,discardthehighestcarrybit! Examples:! Called modular addijon:resultissummodulo2 W1 Two scomplement! Whydoesitwork?! Putanotherway,forallposiJveintegersx,wewant:! bits(1x1)1+1bits(1 x1)1=(ignoringthecarry>outbit)! Thisturnsouttobethebitwise1complement1plus1one! Whatshouldthe8>bitrepresentaJonof>1be? 1 +???????? (wewantwhicheverbitstringgivestherightresult) +???????? +???????? Autumn213 Integers&Floats 23 Autumn213 Integers&Floats 24

7 Two scomplement! Whydoesitwork?! Putanotherway,forallposiJveintegersx,wewant:! bits(1x1)1+1bits(1 x1)1=(ignoringthecarry>outbit)! Thisturnsouttobethebitwise1complement1plus1one! Whatshouldthe8>bitrepresentaJonof>1be? (wewantwhicheverbitstringgivestherightresult) 1 +???????? +???????? Two scomplement! Whydoesitwork?! Putanotherway,forallposiJveintegersx,wewant:! bits(1x1)1+1bits(1 x1)1=(ignoringthecarry>outbit)! Thisturnsouttobethebitwise1complement1plus1one! Whatshouldthe8>bitrepresentaJonof>1be? (wewantwhicheverbitstringgivestherightresult) Autumn213 Integers&Floats 25 Autumn213 Integers&Floats 26 Unsigned&SignedNumericValues ConversionVisualized bits1 Unsigned Signed ! Signedandunsignedintegershavelimits.! Ifyoucomputeanumberthatistoobig (posijve),itwraps: 6+4=?15U+2U=?! Ifyoucomputeanumberthatistoo small(negajve),itwraps: >7>3=?U>2U=?! Answersareonlycorrectmod2 b! TheCPUmaybecapableof throwingan excepion foroverflowonsignedvalues.! Itwon'tforunsigned.! ButCandJavajustcruisealongsilently whenoverflowoccurs...oops.! Two scomplement Unsigned! OrderingInversion! NegaJve BigPosiJve 2 scomplement Range TMax1 1 2 TMin1 UMax1 UMax 11 TMax11+11 TMax1 Unsigned Range Autumn213 Integers&Floats 27 Autumn213 Integers&Floats 28

8 Overflow/Wrapping:Unsigned addijon:dropthecarrybit Overflow/Wrapping:Two scomplement addijon:dropthecarrybit Autumn Integers&Floats ModularArithmeJc Autumn213 > > Integers&Floats ModularArithmeJc 3 ValuesToRemember! UnsignedValues! UMin =!! UMax = 2 w 1! ValuesforW=32! Two scomplementvalues! TMin = 2 w 1! 1! TMax = 2 w 1 1! 11 1! NegaIveone! 111 1xF...F Decimal Hex Binary UMax 4,294,967,296 FF FF FF FF TMax 2,147,483,647 7F FF FF FF TMin a2,147,483, a1 a1 FF FF FF FF Signedvs.UnsignedinC! Constants! Bydefaultareconsideredtobesignedintegers! Use U suffixtoforceunsigned:! U, U Autumn213 Integers&Floats 31 Autumn213 Integers&Floats 32

9 Signedvs.UnsignedinC! CasIng! int tx, ty;! unsigned ux, uy;! ExplicitcasIngbetweensigned&unsigned:! tx = (int) ux;! uy = (unsigned) ty;! ImplicitcasIngalsooccursviaassignmentsandfuncIoncalls:! tx = ux;! uy = ty;! Thegccflag?Wsign?conversionproduceswarningsforimplicitcasts, but?walldoesnot!! HowdoescasIngbetweensignedandunsignedwork?! Whatvaluesaregoingtobeproduced?!!! Autumn213 Integers&Floats 33 Signedvs.UnsignedinC! CasIng! int tx, ty;! unsigned ux, uy;! ExplicitcasIngbetweensigned&unsigned:! tx = (int) ux;! uy = (unsigned) ty;! ImplicitcasIngalsooccursviaassignmentsandfuncIoncalls:! tx = ux;! uy = ty;! Thegccflag?Wsign?conversionproduceswarningsforimplicitcasts, but?walldoesnot!! HowdoescasIngbetweensignedandunsignedwork?! Whatvaluesaregoingtobeproduced?!!!! Bits)are)unchanged,justinterpreteddifferently! Autumn213 Integers&Floats 34 CasIngSurprises! ExpressionEvaluaIon! Ifyoumixunsignedandsignedinasingleexpression,then signed)values)are)implicitly)cast)to)unsigned.)! IncludingcomparisonoperaJons<,>,==,<=,>=! ExamplesforW=32:TMIN=a2,147,483,648TMAX=2,147,483,647! Constant 1 Constant 2 RelaIon EvaluaIon U U == unsigned >1-1 < signed >1-1 U U > unsigned > > signed U > < unsigned >1-1 >2-2 > signed (unsigned)>1-1 >2-2 > unsigned U < unsigned (int) u > signed!!! Autumn213 Integers&Floats 35 SignExtension! Whathappensifyouconverta32abitsignedintegertoa64a bitsignedinteger? Autumn213 Integers&Floats 36

10 SignExtension! Task:! Givenw>bitsignedintegerx! Convertittow+k>bitintegerwith1same1value1! Rule:! Makekcopiesofsignbit:! X=x w 1,,x w 1,x w 1,x w 2,,x 8abitrepresentaIons 1 11 kcopiesofmsb X # w X # k1 Autumn213 Integers&Floats 37 w1 C:casJngbetweenunsignedandsignedjustreinterpretsthesamebits. Autumn213 Integers&Floats 38 SignExtension SignExtension 4>bit2 4>bit2 8>bit2 8>bit2 4>bit>4 4>bit>4???? 8>bit>4 8>bit12 Autumn213 Integers&Floats 39 Autumn213 Integers&Floats 4

11 SignExtension SignExtension 4>bit2 4>bit2 8>bit2 8>bit2 4>bit>4 4>bit>4 1 8>bit> >bit>4 Autumn213 Integers&Floats 41 Autumn213 Integers&Floats 42 SignExtensionExample ShiPOperaIons! ConverIngfromsmallertolargerintegerdatatype! CautomaIcallyperformssignextension(Javatoo)! LePshiP: x<<y! ShipbitvectorxlepbyyposiJons! Throwawayextrabitsonlep! Fillwithsonright Argumentx <<3 Logical>>2 short int x = 12345; int ix = (int) x; short int y = ; int iy = (int) y; Decimal Hex Binary x ix y CF C iy FF FF CF C ! RightshiP: x>>y! Shipbit>vectorxrightbyyposiJons! Throwawayextrabitsonright! Logicalship(forunsignedvalues)! Fillwithsonlep! ArithmeJcship(forsignedvalues)! Replicatemostsignificantbitonlep! Maintainssignofx ArithmeJc>>2 Argumentx <<3 Logical>>2 ArithmeJc>>2 111 Autumn213 Integers&Floats 43 Thebehaviorof>>inCdependsonthecompiler!Itisarithme,cshiprightinGCC. Java:>>>islogicalshipright;>>isarithmeJcshipright. Autumn213 Integers&Floats 44

12 ShiPOperaIons! LePshiP: x<<y! ShipbitvectorxlepbyyposiJons! Throwawayextrabitsonlep! Fillwithsonright! RightshiP: x>>y! Shipbit>vectorxrightbyyposiJons! Throwawayextrabitsonright! Logicalship(forunsignedvalues)! Fillwithsonlep! ArithmeIcshiP(forsignedvalues)! Replicatemostsignificantbitonlep! Maintainssignofx! Why)is)this)useful?) Argumentx <<3 Logical>>2 ArithmeJc>>2 Argumentx <<3 Logical>>2 ArithmeJc>>2 x >> 9? 111 Thebehaviorof>>inCdependsonthecompiler!Itisarithme,cshiprightinGCC. Java:>>>islogicalshipright;>>isarithmeJcshipright. Autumn213 Integers&Floats 45 Whathappenswhen! x>>n?! x<<m? Autumn213 Integers&Floats 46 Whathappenswhen ShiPingandArithmeIc! x>>n:divideby2 n x=27; 1 x2 n y=x<<2; logicalshiplep: y==18 shipinzerosfromtheright! x<<m:muliplyby2 m fasterthangeneralmuljpleordivideoperajons overflow x/2 n logicalshipright: shipinzerosfromthelep rounding(down) 1 unsigned x=237; y=x>>2; y==59 Autumn213 Integers&Floats 47 Autumn213 Integers&Floats 48

13 ShiPingandArithmeIc UsingShiPsandMasks signed x=>11; y=x<<2; y==18 11 x2 n logicalshiplep: shipinzerosfromtheright 1! Extractthe2ndmostsignificantbyteofaninteger? x 1 overflow x/2 n arithmejcshipright: shipincopiesofmostsignificantbit fromthelep rounding(down) 11 signed x=>19; y=x>>2; y==>5 clarificajonfrommon.:shipsbyn<orn>=wordsizeareundefined Autumn213 Integers&Floats 49 Autumn213 Integers&Floats 5 UsingShiPsandMasks! Extractthe2ndmostsignificantbyteofaninteger:! Firstship,thenmask:(x>>16)&xFF UsingShiPsandMasks! Extractthe2ndmostsignificantbyteofaninteger:! Firstship,thenmask:(x>>16)&xFF x 1 x 1 x>>16 1 x>>16 1 (x>>16)&xff (x>>16)&xff ! Extractthesignbitofasignedinteger?! Extractthesignbitofasignedinteger:! (x>>31)&1>needthe &1 toclearoutallotherbitsexceptlsb! CondiIonalsasBooleanexpressions(assumingxisor1)! if(x)a=yelsea=z;whichisthesameasa=x?y:z;! Canbere>wriwen(assumingarithmeJcrightship)as: a=(((x<<31)>>31)&y) (((!x)<<31)>>31)&z); Autumn213 Integers&Floats 51 Autumn213 Integers&Floats 52

14 MulIplicaIon! WhatdoyougetwhenyoumulIply9x9?! Whatabout2 3 x3?! 2 3 x5?! a2 31 xa2 31? UnsignedMulIplicaIoninC Operands:wbits TrueProduct:2wbits Discardwbits:wbits u v!! StandardMulIplicaIonFuncIon! Ignoreshighorderwbits! ImplementsModularArithmeIc UMult w (u,v)=uyvmod2 w1 u! v! UMult w (u, v)! Autumn213 Integers&Floats 53 Autumn213 Integers&Floats 54 Poweraofa2MulIplywithShiP! OperaIon! u << kgivesu 2 k)! Bothsignedandunsigned Operands:wbits TrueProduct:w+kbits u 2 k! Discardk1bits:wbits UMult w (u, 2 k )! TMult w (u, 2 k )!! Examples! u << 3 == u 8! u << 5 - u << 3 == u 24! MostmachinesshipandaddfasterthanmulJply! CompilergeneratesthiscodeautomaJcally u! 2 k! k! 1 CodeSecurityExample / Kernel memory region holding user-accessible data / #define KSIZE 124 char kbuf[ksize]; / Copy at most maxlen bytes from kernel region to user buffer / int copy_from_kernel(void user_dest, int maxlen) { / Byte count len is minimum of buffer size and maxlen / int len = KSIZE < maxlen? KSIZE : maxlen; memcpy(user_dest, kbuf, len); return len; } #define MSIZE 528 void getstuff() { char mybuf[msize]; copy_from_kernel(mybuf, MSIZE); printf( %s\n, mybuf); } Autumn213 Integers&Floats 55 Autumn213 Integers&Floats 56

15 MaliciousUsage / Declaration of library function memcpy / void memcpy(void dest, void src, size_t n); / Kernel memory region holding user-accessible data / #define KSIZE 124 char kbuf[ksize]; / Copy at most maxlen bytes from kernel region to user buffer / int copy_from_kernel(void user_dest, int maxlen) { / Byte count len is minimum of buffer size and maxlen / int len = KSIZE < maxlen? KSIZE : maxlen; memcpy(user_dest, kbuf, len); return len; } FloaIngpointtopics! Background:fracIonalbinarynumbers! IEEEfloaIngapointstandard! FloaIngapointoperaIonsandrounding! FloaIngapointinC! Therearemanymoredetailsthatwewon tcover! It sa58>pagestandard #define MSIZE 528 void getstuff() { char mybuf[msize]; copy_from_kernel(mybuf, -MSIZE);... } Autumn213 Integers&Floats 57 Autumn213 Integers&Floats 58 FracIonalBinaryNumbers FracIonalBinaryNumbers! 2 i! 2 i 1! 4! 2! 1!.! b i! b i 1!! b 2! b 1! b! b 1! b 2! b 3!! b j! 1/2! 1/4! 1/8! 2 j!!! RepresentaIon! Bitstorightof binarypoint representfracjonalpowersof2! RepresentsraJonalnumber: i b k 2 k k = j! Value RepresentaIon! 5and3/4! 2and7/ ! 47/ ! ObservaIons! Shiplep=mulJplybypowerof2! Shipright=dividebypowerof2! Numbersoftheform arejustbelow1.! LimitaIons:! ExactrepresentaJonpossibleonlyfornumbersoftheformx2 y! OtherraJonalnumbershaverepeaJngbitrepresentaJons! 1/3= =.[1] 2 Autumn213 Integers&Floats 59 Autumn213 Integers&Floats 6

16 FixedPointRepresentaIon! Impliedbinarypoint.Examples: #1:thebinarypointisbetweenbits2and3 b 7 b 6 b 5 b 4 b 3 [.]b 2 b 1 b #2:thebinarypointisbetweenbits4and5 b 7 b 6 b 5 [.]b 4 b 3 b 2 b 1 b! SamehardwareasforintegerarithmeIc. #3:integers!thebinarypointisaperbit b 7 b 6 b 5 b 4 b 3 b 2 b 1 b [.]! Fixedpoint=fixedrangeandfixedprecision)! range:differencebetweenlargestandsmallestnumberspossible! precision:smallestpossibledifferencebetweenanytwonumbers IEEEFloaIngPoint! AnalogoustoscienIficnotaIon! x1 7 C:1.2e7! x1 >6 C:1.2e>6! IEEEStandard754usedbyallmajorCPUstoday! Drivenbynumericalconcerns! Rounding,overflow,underflow! Numericallywell>behaved,buthardtomakefastinhardware Autumn213 Integers&Floats 61 Autumn213 Integers&Floats 62 FloaIngPointRepresentaIon FloaIngPointRepresentaIon! Numericalform: V 1 =( 1) s M2 E! Numericalform: V 1 =( 1) s M2 E! SignbitsdetermineswhethernumberisnegaJveorposiJve! Significand(manJssa)MnormallyafracJonalvalueinrange[1.,2.)! ExponentEweightsvaluebya(possiblynegaJve)poweroftwo! SignbitsdetermineswhethernumberisnegaJveorposiJve! Significand(manJssa)MnormallyafracJonalvalueinrange[1.,2.)! ExponentEweightsvaluebya(possiblynegaJve)poweroftwo! RepresentaIoninmemory:! MSBsissignbits)! expfieldencodese)(butisnot1equal1toe)! fracfieldencodesm)(butisnot1equal1tom) s exp frac Autumn213 Integers&Floats 63 Autumn213 Integers&Floats 64

17 Precisions! Singleprecision:32bits s exp frac 1bit 8bits 23bits! Doubleprecision:64bits s exp frac 1bit 11bits 52bits NormalizaIonandSpecialValues V=( 1) s M2 E s exp frac! Normalized =Mhastheform1.xxxxx! AsinscienJficnotaJon,butinbinary!.11x2 5 and1.1x2 3 representthesamenumber,butthelawermakes beweruseoftheavailablebits! SinceweknowthemanJssastartswitha1,wedon'tbothertostoreit! Howdowerepresent.?Orspecial/undefinedvalueslike 1./.?! FiniterepresentaIonmeansnotallvaluescanberepresented exactly.somewillbeapproximated. Autumn213 Integers&Floats 65 Autumn213 Integers&Floats 66 NormalizaIonandSpecialValues V=( 1) s M2 E s exp frac! Normalized =Mhastheform1.xxxxx! AsinscienJficnotaJon,butinbinary!.11x2 5 and1.1x2 3 representthesamenumber,butthelawermakes beweruseoftheavailablebits! SinceweknowthemanJssastartswitha1,wedon'tbothertostoreit.! Specialvalues:! zero: s== exp==... frac==...! +,a : exp== frac==... 1./.= 1./.=+,1./.= 1./.=! NaN( NotaNumber ): exp==11...1frac!=... ResultsfromoperaJonswithundefinedresult:sqrt(>1),,,etc.! note:exp=11 1andexp= arereserved,limijngexprange Autumn213 Integers&Floats 67 FloaIngPointOperaIons:BasicIdea V=( 1) s M2 E s exp frac! x + f y = Round(x + y)! x f y = Round(x y)! BasicideaforfloaIngpointoperaIons:! First,computetheexactresult! Then,roundtheresulttomakeitfitintodesiredprecision:! Possiblyoverflowifexponenttoolarge! Possiblydropleast>significantbitsofsignificandtofitintofrac Autumn213 Integers&Floats 68

18 FloaIngPointMulIplicaIon FloaIngPointAddiIon!( 1) s1 )M1))2 E1 )))( 1) s2 )M2))2 E2)! ExactResult:( 1) s )M))2 E! Signs: s1^s2! SignificandM: M1M2! ExponentE: E1+E21! Fixing! IfM 2,shipMright,incrementE! IfEoutofrange,overflow! RoundMtofitfracprecision ( 1) s1 M12 E1 +(a1) s2 M22 E2 AssumeE1>E2! ExactResult:( 1) s )M))2 E! Signs,significandM:! Resultofsignedalign&add! ExponentE: E11! Fixing! IfM1 2,shipMright,incrementE! ifm<1,shipmlepkposijons,decrementebyk! OverflowifEoutofrange! RoundMtofitfracprecision + ( 1) s1 1M111 ( 1) s M E1 E21 ( 1) s2 M2 Autumn213 Integers&Floats 69 Autumn213 Integers&Floats 7 Roundingmodes! Possibleroundingmodes(illustratewithdollarrounding): $1.4 $1.6 $1.5 $2.5 $1.5! Round>toward>zero $1 $1 $1 $2 $1! Round>down(> ) $1 $1 $1 $2 $2! Round>up(+ ) $2 $2 $2 $3 $1! Round>to>nearest $1 $2??????! Round>to>even $1 $2 $2 $2 $2! RoundatoaevenavoidsstaIsIcalbiasinrepeatedrounding.! RoundsupabouthalftheJme,downabouthalftheJme.! DefaultroundingmodeforIEEEfloaJng>point MathemaIcalProperIesofFPOperaIons! Exponentoverflowyields+ ora #! Floatswithvalue+,a,andNaNcanbeusedinoperaIons! ResultusuallysJll+,>,orNaN;someJmesintuiJve,someJmesnot! FloaIngpointoperaIonsarenotalwaysassociaIveor distribuive,duetorounding!! (3.14+1e1)>1e1!=3.14+(1e1>1e1)! 1e2(1e2>1e2)!=(1e21e2)>(1e21e2) Autumn213 Integers&Floats 71 Autumn213 Integers&Floats 72

19 FloaIngPointinC! Cofferstwolevelsofprecision float singleprecision(32>bit) double doubleprecision(64>bit)!!!! #include <math.h>togetinfinityandnanconstants! Equality(==)comparisonsbetweenfloaIngpointnumbersare tricky,andopenreturnunexpectedresults! Justavoidthem! FloaIngPointinC! Conversionsbetweendatatypes:! CasJngbetweenint,float,anddoublechangesthebit representajon.! int float! Mayberounded;overflownotpossible! int doubleorfloat double! Exactconversion(32>bitints;52>bitfrac+1>bitsign)! long int double! Roundedorexact,dependingonwordsize! doubleorfloat int! TruncatesfracJonalpart(roundedtowardzero)! NotdefinedwhenoutofrangeorNaN:generallysetstoTmin!!! Autumn213 Integers&Floats 73 Autumn213 Integers&Floats 74 NumberRepresentaIonReallyMa\ers! 1991:PatriotmissiletargeIngerror! clockskewduetoconversionfromintegertofloajngpoint!!!! 1996:Ariane5rocketexploded($1billion)! overflowconverjng64>bitfloajngpointto16>bitinteger! 2:Y2Kproblem! limited(decimal)representajon:overflow,wrap>around! 238:Unixepochrollover! Unixepoch=secondssince12am,January1,197! signed32>bitintegerrepresentajonrollsovertotminin238! otherrelatedbugs! 1994:IntelPenJumFDIV(floaJngpointdivision)HWbug($475million)! 1997:USSYorktown smart warshipstranded:dividebyzero! 1998:MarsClimateOrbitercrashed:unitmismatch($193million) Autumn213 Integers&Floats 75 FloaIngPointandtheProgrammer #include <stdio.h> int main(int argc, char argv[]) { float f1 = 1.; float f2 =.; int i; for ( i=; i<1; i++ ) { f2 += 1./1.; } printf(x%8x x%8x\n, (int)&f1, (int)&f2); printf(f1 = %1.8f\n, f1); printf(f2 = %1.8f\n\n, f2); f1 = 1E3; f2 = 1E-3; float f3 = f1 + f2; printf (f1 == f3? %s\n, f1 == f3? yes : no ); return ; } $./a.out x3f8 x3f81 f1 = 1. f2 = 1.9 f1 == f3? yes Autumn213 Integers&Floats 76

20 MemoryReferencingBug double fun(int i) { volatile double d[1] = {3.14}; volatile long int a[2]; a[i] = ; / Possibly out of bounds / return d[]; } fun() > 3.14 fun(1) > 3.14 fun(2) > fun(3) > fun(4) > 3.14, then segmentation fault RepresenIng3.14asaDoubleFPNumber! =! 3.14=11.111! ( 1) s )M))2 E)! S=encodedas! M=1.111.(leading1lepout)! E=1encodedas124(withbias) s exp (11) frac (first 2 bits) 11 Autumn213 ExplanaJon: Saved State 4 d7 d4 3 d3 d 2 a[1] 1 a[] Integers&Floats LocaJonaccessedby fun(i) 77 Autumn213 frac (the other 32 bits) 1 Integers&Floats 78 MemoryReferencingBug(Revisited) double fun(int i) { volatile double d[1] = {3.14}; volatile long int a[2]; a[i] = ; / Possibly out of bounds / return d[]; } fun() > 3.14 fun(1) > 3.14 fun(2) > fun(3) > fun(4) > 3.14, then segmentation fault MemoryReferencingBug(Revisited) double fun(int i) { volatile double d[1] = {3.14}; volatile long int a[2]; a[i] = ; / Possibly out of bounds / return d[]; } fun() > 3.14 fun(1) > 3.14 fun(2) > fun(3) > fun(4) > 3.14, then segmentation fault Autumn213 Saved State d7 d4 d3 d a[1] a[] Integers&Floats LocaJon accessed byfun(i) 79 Autumn213 Saved State d7 d4 d3 d a[1] a[] Integers&Floats LocaJon accessed byfun(i) 8

21 MemoryReferencingBug(Revisited) Summary double fun(int i) { volatile double d[1] = {3.14}; volatile long int a[2]; a[i] = ; / Possibly out of bounds / return d[]; } fun() > 3.14 fun(1) > 3.14 fun(2) > fun(3) > fun(4) > 3.14, then segmentation fault Saved State d7 d4 d3 d a[1] a[] LocaJon accessed byfun(i)! Aswithintegers,floatssufferfromthefixednumberofbits availabletorepresentthem! Cangetoverflow/underflow,justlikeints! Some simplefracjons havenoexactrepresentajon(e.g.,.2)! Canalsoloseprecision,unlikeints! EveryoperaJongetsaslightlywrongresult! MathemaIcallyequivalentwaysofwriInganexpressionmay computedifferentresults! ViolatesassociaJvity/distribuJvity! NevertestfloaIngpointvaluesforequality!! CarefulwhenconverIngbetweenintsandfloats! Autumn213 Integers&Floats 81 Autumn213 Integers&Floats 82 Manymoredetailsforthecurious...! Exponentbias! Denormalizedvalues togetfinerprecisionnearzero! DistribuIonofrepresentablevalues! FloaIngpointmulIplicaIon&addiIonalgorithms! Roundingstrategies! Wewon tbeusingortesingyouonanyoftheseextrasin 351. Autumn213 Integers&Floats 83 Autumn213 Integers&Floats 84

22 NormalizedValues V=( 1) s M2 E s exp frac k n! CondiIon:exp andexp 111 1! Exponentcodedasbiasedvalue:E)=))exp)3)Bias)! expisanunsignedvaluerangingfrom1to2 k >2(k==#bitsinexp)! Bias=2 k>1 >1! Singleprecision:127(soexp:1 254,E:> )! Doubleprecision:123(soexp:1 246,E:> )! TheseenablenegaJvevaluesforE,forrepresenJngverysmallvalues! Significandcodedwithimpliedleading1:M))= 1.xxx x 2! xxx x:thenbitsoffrac! Minimumwhen (M=1.)! Maximumwhen111 1 (M=2. ε)! Getextraleadingbitfor free Autumn213 Integers&Floats 85 NormalizedEncodingExample V=( 1) s M2 E s exp frac k n! Value:float f = ;! = 2 =1.1 2 x2 13 (normalizedform)! Significand: M = frac= 1 2! Exponent:E=exp>Bias,soexp=E+Bias E 1 = 13 Bias = 127 exp = 14 = 1 2! Result: 1 1 s exp frac Autumn213 Integers&Floats 86 DenormalizedValues! CondiIon:exp=! Exponentvalue:E)=exp Bias+1(insteadofE=exp Bias)! Significandcodedwithimpliedleading:M)=).xxx x 2! xxx x:bitsoffrac! Cases! exp=,frac=! Representsvalue! NotedisJnctvalues:+and (why?)! exp=,frac! Numbersverycloseto.! Loseprecisionasgetsmaller! Equispaced Autumn213 Integers&Floats 87 SpecialValues! CondiIon:exp=111 1! Case:exp=111 1,frac=! Representsvalue (infinity)! OperaJonthatoverflows! BothposiJveandnegaJve! E.g.,1./.= 1./.=+,1./.= 1./.=! Case: exp=111 1,frac! Not>a>Number(NaN)! Representscasewhennonumericvaluecanbedetermined! E.g.,sqrt( 1),, # Autumn213 Integers&Floats 88

23 VisualizaIon:FloaIngPointEncodings TinyFloaIngPointExample s exp frac NaN > # >Normalized >Denorm > + +Denorm +Normalized + # NaN! 8abitFloaIngPointRepresentaIon! thesignbitisinthemostsignificantbit.! thenextfourbitsaretheexponent,withabiasof7.! thelastthreebitsarethefrac! SamegeneralformasIEEEFormat! normalized,denormalized! representajonof,nan,infinity Autumn213 Integers&Floats 89 Autumn213 Integers&Floats 9 DynamicRange(PosiIveOnly) Denormalized numbers Normalized numbers s exp frac E Value /81/64 = 1/ /81/64 = 2/ /81/64 = 6/ /81/64 = 7/ /81/64 = 8/ /81/64 = 9/ /81/2 = 14/ /81/2 = 15/ /81 = /81 = 9/ /81 = 1/ /8128 = /8128 = n/a inf closesttozero largestdenorm smallestnorm closestto1below closestto1above largestnorm DistribuIonofValues! 6abitIEEEalikeformat! e=3exponentbits! f=2fracjonbits! Biasis2 3>1 >1=3 s exp frac 1 3 2! NoIcehowthedistribuIongetsdensertowardzero Denormalized Normalized Infinity Autumn213 Integers&Floats 91 Autumn213 Integers&Floats 92

24 DistribuIonofValues(closeaupview)! 6abitIEEEalikeformat! e=3exponentbits! f=2fracjonbits! Biasis3 s exp Denormalized Normalized Infinity frac InteresIngNumbers {single,double} DescripIon exp frac NumericValue! Zero.! SmallestPos.Denorm. 1 2 {23,52} 2 {126,122}! Single ! Double ! LargestDenormalized (1. ε)2 {126,122}! Single ! Double ! SmallestPos.Norm {126,122}! Justlargerthanlargestdenormalized! One ! LargestNormalized (2. ε)2 {127,123}! Single ! Double Autumn213 Integers&Floats 93 Autumn213 Integers&Floats 94 SpecialProperIesofEncoding! FloaIngpointzero( + )exactlythesamebitsasintegerzero! Allbits=! Can(Almost)UseUnsignedIntegerComparison! Mustfirstcomparesignbits! Mustconsider a = + =! NaNsproblemaJc! Willbegreaterthananyothervalues! Whatshouldcomparisonyield?! OtherwiseOK! Denormvs.normalized! Normalizedvs.infinity FloaIngPointMulIplicaIon ( 1) s1 M12 E1 ( 1) s2 M22 E2! ExactResult:( 1) s M2 E! Signs: s1^s2//xorofs1ands2! SignificandM: M1M2! ExponentE: E1+E2! Fixing! IfM 2,shipMright,incrementE! IfEoutofrange,overflow! RoundMtofitfracprecision Autumn213 Integers&Floats 95 Autumn213 Integers&Floats 96

25 FloaIngPointAddiIon ( 1) s1 M12 E1 +( 1) s2 M22 E2 Assume1E11>1E21! ExactResult:( 1) s M2 E! Signs,significandM:! Resultofsignedalign&add! ExponentE:E1 ( 1) s1 1M111! Fixing! IfM 2,shipMright,incrementE! ifm<1,shipmlepkposijons,decrementebyk! OverflowifEoutofrange! RoundMtofitfracprecision + ( 1) s M E1 E21 ( 1) s2 M2 Autumn213 Integers&Floats 97 CloserLookatRoundaToaEven! DefaultRoundingMode! Hardtogetanyotherkindwithoutdroppingintoassembly! AllothersarestaJsJcallybiased! SumofsetofposiJvenumberswillconsistentlybeover>orunder> esjmated! ApplyingtoOtherDecimalPlaces/BitPosiIons! Whenexactlyhalfwaybetweentwopossiblevalues! Roundsothatleastsignificantdigitiseven! E.g.,roundtonearesthundredth (Lessthanhalfway) (Greaterthanhalfway) (Halfway roundup) (Halfway rounddown) Autumn213 Integers&Floats 98 RoundingBinaryNumbers! BinaryFracIonalNumbers! Halfway whenbitstorightofroundingposijon=1 2! Examples! Roundtonearest1/4(2bitsrightofbinarypoint) Value Binary Rounded AcJon RoundedValue 23/ (<1/2 down) 2 23/ (>1/2 up) 21/4 27/ (1/2 up) 3 25/ (1/2 down) 21/2 Autumn213 Integers&Floats 99