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1 FloatingpointvericationinHOLLight:theexponentialfunction 1 Floatingpointverication theexponentialfunction inhollight: UniversityofCambridge JohnHarrison Introduction Floatingpointcorrectness Ourimplementationlanguage Thealgorithm OutlineoftheHOLproof Generalconclusions JohnHarrison UniversityofCambridge,December1997

2 FloatingpointvericationinHOLLight:theexponentialfunction 2 Introduction Floatingpointalgorithmsarefairlysmall,but Therehavebeenerrorsincommercial oftencomplicatedmathematically. Inthecaseoftranscendentalfunctionsit's systems,e.g.thepentiumfdivbugin1994. Vericationusingmodelcheckersisdicult diculteventosaywhatcorrectnessmeans. apparatus. becauseoftheneedformathematical Itcanevenbedicultusingtheoremprovers sincenotmanyofthemhavegoodtheoriesof realnumbersetc. JohnHarrison UniversityofCambridge,December1997

3 FloatingpointvericationinHOLLight:theexponentialfunction 3 Floatingpointcorrectness Wewanttospecifythecorrectnessaccordingto thefollowingdiagram: v(a) exp 6 -v(exp(a)) exp(v(a)) 6 v v a EXP -EXP(a) andexp(v(a))in`unitsinthelastplace'of Wemeasurethedierencebetweenv(EXP(a)) EXP(a). JohnHarrison UniversityofCambridge,December1997

4 FloatingpointvericationinHOLLight:theexponentialfunction 4 Ourimplementationlanguage Thisincludesthefollowingconstructs: command=variable:= command; expression if expressionthen command else command command while if expressionthen expressiondo do commandwhile expression command skip WedeneasimplerelationalsemanticsinHOL, fexpressiong correctnessrules.wethenprovetotalcorrectness andderiveweakestpreconditionsandtotal Theideaisthatthislanguagecanbeformally viavcgeneration. linkedtoc,verilog,handel,...

5 FloatingpointvericationinHOLLight:theexponentialfunction 5 Thealgorithm TanginACMTransactionsonMathematical Thealgorithmweverifyistakenfromapaperby Similartechniquesarewidelyusedforoating Software,1989. pointlibraries,and,probably,forhardware implementations. constants.tang'spapergivesactualvaluesashex Thealgorithmreliesonatableofprecomputed Thealgorithmworksinthreephases: representationsofieeenumbers. Performrangereduction Usepolynomialapproximation Reconstructanswerusingtables Thecorrectnessproofreectsthis.

6 FloatingpointvericationinHOLLight:theexponentialfunction 6 Codeforthealgorithm ifisnan(x)thene:=x elseifabs(x)>threshold_1then elseifx==plus_infinitythene:=plus_infinity elseifx==minus_infinitythene:=plus_zero elseifabs(x)<threshold_2thene:=plus_one+x elsee:=plus_zero ifx>plus_zerothene:=plus_infinity (N:=INTRND(X*Inv_L); N2:=N%Int_32; N1:=N-N2; else ifabs(n)>=int_2e9then R1:=X-Tofloat(N)*L1; R1:=(X-Tofloat(N1)*L1)-Tofloat(N2)*L1 M:=N1/Int_32; J:=N2; R2:=Tofloat(--N)*L2; R:=R1+R2; Q:=R*R*(A1+R*A2); E1:=S_Lead(J)+(S_Trail(J)+S*P); S:=S_Lead(J)+S_Trail(J); P:=R1+(R2+Q); )E:=Scalb(E1,M)

7 FloatingpointvericationinHOLLight:theexponentialfunction 7 StructureoftheHOLproof / Realnumbers / \ Programming / \ language / \ IEEEspec / Realanalysis \ / // Squarefreedecomp& // Sturm'stheorem Algorithm / /// \\\ /// \\\ FPlemmas /// \\\ /// Verification

8 FloatingpointvericationinHOLLight:theexponentialfunction 8 Floatingpointlemmas(1) roundingarealnumberxtoaoatingpoint Wedenetheerrorerror(x)resultingfrom value. Becauseoftheregularwayinwhichthe operationsaredened,alltheoperationsthen counterpartsaccordingtothesamepattern: relatetotheirabstractmathematical -Finite(a)^Finite(b)^ =)Finite(a+b)^ abs(val(a)+val(b))<threshold(float_format) error(val(a)+val(b))) (Val(a+b)=(Val(a)+Val(b))+ Thecomparisonsareevenmorestraightforward: -Finite(a)^Finite(b) =)(a<b=val(a)<val(b)) JohnHarrison UniversityofCambridge,December1997

9 FloatingpointvericationinHOLLight:theexponentialfunction 9 Floatingpointlemmas(2) Wehaveseverallemmasquantifyingtheerror,e.g. -abs(x)<threshold(float_format)^ abs(x)<(&2powj/&2pow125) Therearemanyimportantsituations,however, =)abs(error(x))<=&2powj/&2pow150 isexactlyrepresentable,e.g.subtractionof wheretheoperationsareexact,becausetheresult nearbyvalueswiththesamesign: -Finite(a)^Finite(b)^ =)Finite(a-b)^ &2*abs(Val(a)-Val(b))<=abs(Val(a)) Thisisaclassicresultinoatingpointerror (Val(a-b)=Val(a)-Val(b)) analysis. JohnHarrison UniversityofCambridge,December1997

10 FloatingpointvericationinHOLLight:theexponentialfunction 10 Informalerroranalysis HOL.Oneneedsto: Tang'serroranalysistranslatesquitedirectlyinto 1.Provethatcleverimplementationtricks ensurecertainremaindertermsarecalculated exactly.thisreliesoncancellation,andthe zeroes. factthatpre-storedconstantshavetrailing 2.Provethatthepolynomialapproximation 3.Provethattheroundingerrorswhen obeystheappropriateerrorbounds. large. reconstructingthenalanswerdonotgettoo afewlines,while3isgivenalonganddetailed InTang'spaper,1isquitebrief,2isdismissedin proof.

11 FloatingpointvericationinHOLLight:theexponentialfunction 11 HOLerroranalysis reversed! IntheHOLversion,thisorderofdicultyis 1.Therstpartisnotfundamentallydicult, butquitetrickybecauseitinvolvesalotof 2.Thesecondpartinvolvesnumerical specialcasesandlow-levelproofs. translateintoaformalproof(e.g.taylor approximation,whichneedsalotofworkto makesasmallmistakehere,thoughitdoesn't series,sturm'stheorem...).infacttang 3.Thelastpartisquiteroutine,andwecan aectthenalresult. automatically.actually,wederivebetter programholtocomposetheroundingerrors boundsthantangdoessinceweavoid thework. makingsimplifyingassumptionstocutdown

12 FloatingpointvericationinHOLLight:theexponentialfunction 12 Thenalresult conrmtang'sbottom-lineresult: Underthevarious`denitional'assumptions,we (X==Plus_infinity_ (Isnan(X)=)Isnan(E))^ Finite(X)^ (X==Minus_infinity=)E==Plus_zero)^ exp(valx)>=threshold(float_format) (Finite(X)^exp(ValX)<threshold(float_format) =)E==Plus_infinity)^ =)Isnormal(E)^ _(Isdenormal(E)_Iszero(E))^ abs(val(e)-exp(valx))<(&54/&100)*ulp(e) Infact,thisspecicationisabitmoreprecise abs(val(e)-exp(valx))<(&77/&100)*ulp(e)) overowthreshold. thantang's,e.g.weareexplicitaboutthe JohnHarrison UniversityofCambridge,December1997

13 FloatingpointvericationinHOLLight:theexponentialfunction 13 Conclusions Weconrm(andstrengthen)themainresults anduncoversubtleissues.thisclassofproofs ofthehandproof.butwedetectafewslips Theproofwasverylong(over3monthsof isagoodtargetforverication. work),butmostofthiswasdevotedto It'samistaketobelievethatonly`trivial' generalresultsthatcouldbere-used. mathematicsisneededforverication theoriesareessential. applications.hollight'smathematical Automationoflineararithmeticispractically indispensable.bettertoolsfornonlinear Theproofruntimesareverylongowingtothe reasoningareneeded. extensiveuseofarithmeticdonebyinference.

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