Data-Types Optimization for Floating-Point Formats by Program Transformation

Transcription

1 Dt-Types Optimiztion for Floting-Point Formts by Progrm Trnsformtion Nsrine Dmouche University of Perpignn LAbortory of Mthemtics nd PhysicS 52 Avenue Pul Alduy Perpignn, Frnce Emil: Mtthieu Mrtel University of Perpignn LAbortory of Mthemtics nd PhysicS 52 Avenue Pul Alduy Perpignn, Frnce Emil: Alexndre Chpoutot U2IS, ENSTA PrisTech Pris-Scly University 828 bd des Mréchux Pliseu, Frnce Abstrct In floting-point rithmetic, desirble property of computtions is to be ccurte, since in mny industril context smll or lrge perturbtions due to round-off errors my cuse considerble dmges. To cope with this mtter of fct, we hve developed tool which corrects these errors by utomticlly trnsforming progrms in source to source mnner. Our trnsformtion, relying on sttic nlysis by bstrct bstrction, concerns pieces of code with ssignments, conditionls nd loops. By trnsforming progrms, we cn significntly optimize the numericl ccurcy of computtions by minimizing the error reltively to the exct result. An interesting side-effect of our technique is tht more ccurte computtions my mke it possible to use smller dt-types. In this rticle, we show tht our trnsformed progrms, executed in single precision, my compete with not trnsformed codes executed in double precision. I. INTRODUCTION The floting-point numbers described by IEEE754 Stndrd [1], [16] re more nd more used in mny industril pplictions, including criticl embedded softwre. In these computtions, recurrent problem ppers becuse of roundoff errors tht reduce the ccurcy of the results. This perturbtion becomes prticulrly crucil when ccumulted errors cuse dmges whose grvity vries depending on the context of the ppliction. Obviously, embedded softwre is not the only pplictive domin concerned by these ccurcy problems which re ubiquitous in ll numericl scientific computtions. In the light of this problem, we correct these errors by utomticlly trnsforming progrms in source to source mnner. Bsiclly, we trnsform not only rithmetic expressions but lso pieces of code contining ssignments, conditionls, loops nd sequences of commnds. Our trnsformtion significntly improves the numericl ccurcy of the progrm considered. To optimize progrms, we generte lrge rithmetic expressions corresponding to the computtions of the originl progrm nd further, we consider mny expressions mthemticlly equivlent to the originl ones in order to, finlly, choose more ccurte mong them in polynomil time. There exists severl methods for vlidting [3], [7], [8], [10], [20] nd improving [12], [17] the ccurcy of numericl This work ws supported by the ANR Project ANR-12-INSE-0007 CAFEIN /16/$31.00 c 2016 IEEE codes in order to void filures. Here, s in our previous work, we rely on sttic nlysis by bstrct interprettion [4] to compute vrible rnges nd error bounds. We use set of trnsformtion rules for rithmetic expressions nd commnds [6]. These rules, which re pplied in deterministic order, llow one to obtin more ccurte code mong ll these which re considered. We hve shown tht the numericl ccurcy of our cse study progrms is significntly improved. In most cses it is bout of 20%. In this pper, we propose new experiment, tht compres the optimized progrms obtined by our tool nd executed in single precision to the initil progrms executed in both single nd double precision. Our min contribution is to show tht the trnsformed progrms in single precision re close to the originl progrms in double precision. This offers to the progrmmer the possibility to degrde to the single precision dt-type without loosing much informtion. To ensure tht our tool is useful in prctice, we write the source nd trnsformed progrm in C nd compile them by using GCC Compiler. This rticle is orgnized s follows. We first review in Section II the bsics of our trnsformtion method bsed on floting-point rithmetic. We explin briefly how to trnsform rithmetic expressions. We then introduce in Section III the different trnsformtion rules tht llow us to utomticlly trnsform progrms. We present in Section IV our min contribution which demonstrtes the interest of the comprison between single, double precision nd optimized codes. We illustrte this with some experiments concerning the computtion of the integrl of polynomil using Simpson s Rule [2]. Relted work is discussed in Section V. Finlly, we give some concluding remrks nd perspectives in Section VI. II. ANALYSIS AND TRANSFORMATION OF EXPRESSIONS In this section we introduce the bckground needed to understnd our pproch to improve the numericl ccurcy of progrms. We recll the key notions on our wy of computing the rounding errors on floting-point rithmetic expressions. Next, we briefly explin how to trnsform rithmetic expressions using n intermediry representtion clled APEGs. A. Sttic Anlysis of Arithmetic Expressions Floting-point numbers re used to represent rel numbers. Becuse of their finite representtion, round-off errors rise

2 during the computtions which my cuse dmges in criticl contexts. IEEE754 Stndrd formlizes binry floting-point number s triplet of sign s {0, 1}, significnd nd exponent. We consider tht number x is written: x = s d 0.d 1...d p 1 b e = s m b e p+1, 1 where, s is the sign { 1, 1}, m is the mntiss, m = d 0.d 1...d p 1 with digits 0 d i < b, 0 i p 1, p is the precision, nd e is the exponent [e min,e mx ]. A floting-point number x is normlized whenever d 0 0. Normliztion voids multiple representtions of the sme number. IEEE754 Stndrd specifies some prticulr vlues for p, e min nd e mx which re summrized in Figure 1 s well s denormlized numbers which re floting-point numbers with d 0 = d 1 =... = d k = 0, k < p 1 nd e = e min. Denormlized numbers mke underflow grdul [9]. Finlly, the following specil vlues lso re defined: Fig. 1. NN Not Number result of n invlid opertion, the vlues ± corresponding to overflows, the vlues +0 nd 0 signed zeros. Formt Nme p e bits e min e mx Binry16 Hlf precision Binry32 Single precision Binry64 Double precision Binry128 Qudrtic precision Bsic IEEE754 formts. IEEE754 Stndrd defines four rounding modes for elementry opertions over floting-point numbers. These modes re towrds, towrds +, towrds zero nd to the nerest respectively denoted by +,, 0 nd. The semntics of the elementry opertions specified by IEEE754 Stndrd is given by Eqution 2. x r y = r x y, with r : R F 2 where floting-point opertion, denoted by r {+,,, }, is computed using the rounding mode r nd denotes n exct opertion. Obviously, the results of the computtions re not exct becuse of the round-off errors. This is why, we use lso the function r : R R tht returns the round-off errors. We hve r x = x r x. 3 In order to compute the errors during the evlution of rithmetic expressions [14], we use vlues which re pirs x,µ FR = E where x denotes the floting point number used by the mchine nd µ denotes the exct error ttched to F, i.e., the exct difference between the rel nd flotingpoint numbers s defined in Eqution 3. For exmple, the rel number 1 3 is represented by the vluev = 1 1 3, 3 = , The semntics of the elementry opertions on E is defined in [14]. Our tool uses n bstrct semntics [4] bsed on E. The bstrct vlues re represented by pir of intervls. The first intervl contins the rnge of the floting-point vlues of the progrm nd the second one contins the rnge of the errors obtined by subtrcting the floting-point vlues from the exct ones. In the bstrct vlue denoted by x, µ E, we hve x the intervl corresponding to the rnge of the vlues nd µ the intervl of errors on x. This vlue bstrcts set of concrete vlues {x,µ : x x nd µ µ } by intervls in component-wise wy. We now introduce the semntics of rithmetic expressions on E. We pproximte n intervl x with rel bounds by n intervl bsed on floting-point bounds, denoted by x. Here bounds re rounded to the nerest, see Eqution 4. [x,x] = [ x, x]. 4 We denote by the function tht bstrcts the concrete function. It over-pproximtes the set of exct vlues of the error x = x x. Every error ssocited to x [x,x] is included in [x,x]. We lso hve for rounding mode to the nerest [x,x] = [ y,y] with y = 1 2 ulp mx x, x. 5 Formlly, the unit in the lst plce, denoted by ulpx, consists of the weight of the lest significnt digit of the flotingpoint number x. Equtions 6 to 7 give the semntics of the ddition nd multipliction over E, for other opertions see [14]. If we sum two numbers, we must dd errors on the opernds to the error produced by the round-off of the result. When multiplying two numbers, the semntics is given by the development of x 1 +µ 1 x 2 +µ 2. x 1,µ 1 + x 2,µ 2 = x 1 + x 2,µ 1 + µ 2 + x 1 + x 2, 6 x 1,µ 1 x 2,µ 2 = x 1 x 2,x 2 µ 1 + x 1 µ 2 + µ 1 µ 2 + x 1 x 2. B. Accurcy Improvement The work in [12] concerns the definition of new intermediry representtion clled APEG for Abstrct Progrm Expression Grph. This pproch represents in polynomil size n exponentil number of equivlent rithmetic expressions. Becuse APEGs hold in bstrction boxes mny equivlent expressions up to ssocitivity nd commuttivity, hence, it prevents the combintoril problem. A box contining n opernds cn represent up to n 3 possible formuls. In order to build lrge APEGs, two lgorithms re used propgtion nd expnsion lgorithm. The first one serches recursively in the APEG where symmetric binry opertor is repeted nd introduces bstrction boxes. Then, the second lgorithm finds homogeneous prt nd inserts polynomil number of boxes. In order to dd new shpes of expressions in n APEG, one propgtes recursively subtrctions nd divisions into the concerned opernds, propgte products, nd fctorizing common fctors. Finlly, n ccurte formul is serched mong ll the equivlent formuls represented in n APEG using the bstrct semntics of Section II-A. The APEGs re n extension of the Equivlence Progrm Expression Grphs EPEGs introduced by R. Tte et l. [21]. An APEG is defined inductively s follows: 1 A constnt cst or n identifier id is n APEG, 2 An expression p 1 p 2 is n APEG, where p 1 nd p 2 re APEGs nd is binry opertor mong {+,,, }, 7

3 3 A box p 1,...,p n is n APEG, where {+,} is commuttive nd ssocitive opertor nd the p i,1 i n, re APEGs, 4 A non-empty set {p 1,...,p n } of APEGs consists in n APEG where p i,1 i n, is not set of APEGs itself. We cll the set {p 1,...,p n } the equivlence clss. An exmple of APEG is given in Figure 2, it represents ll the following expressions: + 2 Fig. 2. b +,,b c c b APEG for the expression e = ++b c. Ap = + + b c, + b + c, b + + c, 2 + b c, c + + b, c + b +, c b + +, c 2 + b, + c + b c, 2 c + b c, b c + + c, b c + 2 c c. 8 R. Tte et l. in their rticle [21], use rewriting rules to extend the structure up to sturtion. In our context, such rules would consist of performing some pttern mtching in n existing APEG p nd then dding new nodes in p, once pttern hs been recognized. III. TRANSFORMATION OF COMMANDS In this section, we introduce the trnsformtion rules, implemented in our tool nd used to improve the numericl ccurcy of progrms. The syntx of commnds is given by: Com c ::= id = e c 1 ;c 2 ifφ e thenc 1 elsec 2 while Φ e do c nop, 9 where e is n expression in Expr mde of rithmetic opertions nd comprison. The principle of the trnsformtion of commnds relies on set of hypotheses: Progrms re defined by tuple c,δ,c,β ν c,δ,c,β : c is the progrm t optimizing, δ the forml environment tht mps vribles to expressions δ: V Expr, C is the context, i.e., the progrm enclosing the commnd to trnsform, β is the blck list tht contins vribles tht must not be removed from the progrm, ν is the reference vrible t optimizing, Progrms re written in SSA form single sttic ssignments, Trnsformtion rules re pplied deterministiclly, The best progrm, the most ccurte, is obtined by compring the reference vrible of the originl nd trnsformed progrm. Here, we survey briefly the different kinds of trnsformtion rules. We refer the interested reder to [6] to see the detils of these trnsformtion rules bsed on the syntx nd semntics. We strt with the ssignments. We hve two rules, the first consists in removing the ssignment from the progrm nd sving it in the memory δ if the conditions below re verified: The vribles of the expression e does not pper in δ, b The vrible v does not belong to the blck list β, c The vrible v is different of the reference vrible ν. Otherwise, we build lrge expression by substituting in it the forml expressions memorized previously by the first rule in δ. Remrk tht by inlining expressions in vribles when trnsforming progrms, we crete lrge formuls. In our implementtion, in order to fcilitte their mnipultion, we slice these formuls, t defined level of the syntctic tree, in severl sub-expressions, nd we ssign them to intermediry vribles. Finlly, we inject these new ssignments into the min progrm. Exmple 3.1: To explin the use of the trnsformtion rules of ssignments, let us consider Eqution 10 in which three vribles x, y nd z re ssigned. In this exmple, ν consists of the vrible z tht we im t optimizing, nd = 0.1, b = 0.01, c = nd d = re constnts. x = +b; y = c +d; z = x +y, δ, [ ], {z} = ν nop; y = c +d; z = x +y, δ = δ[x +b], [ ], {z} = ν y = c +d; z = x +y, δ = δ[x +b], [ ], {z} = ν nop; z = x +y, δ = δ [y c +d], [ ], {z} = ν z = x +y, δ = δ [y c +d], [ ], {z} = ν z = d +c +b +,δ, [ ],{z} In Eqution 10, initilly, the environment δ is empty, the blck list contins z nd the reference vrible ν t optimizing is z. If we pply the first rule of ssignment, we my remove the vrible x nd memorize it in δ. So, the line corresponding to the vrible discrded is replced by nop nd the new environment is δ = [x + b]. We then repet the sme process, we remove the vrible y nd sved it in δ. Next, we pply rule for sequences which discrds the nop sttement. For the lst step, we my not discrd z becuse the condition is not stisfied z = ν. Then, we substitute x nd y by their vlue in δ nd we trnsform the expression. Another kind of trnsformtion rules is for the sequences of commnds. If one member of the sequence is nop, then we trnsform only the other member, else, we trnsform both of them. Our implementtion trnsforms lso conditionls. If the condition is stticlly known, then we keep just the evluted brnch nd we trnsform it, else, we trnsform both brnches of the conditionl. In some cses, we del with undefined vribles becuse they hve been discrded from the progrm nd sved in the environmentδ s indicted in the first trnsformtion rule for ssignments. We then re-inject them into the progrm nd we do the necessry trnsformtions. The next trnsformtion rules concern the while loop. We trnsform the body of the loop ensuring tht the vribles of the condition do not belong to the environment δ. Otherwise, we hve to re-insert the vribles memorized in the environment into the progrm s doing for the lst rule of conditionls. 10

4 At the end of this section, we del with complexity considertions. As underlined previously, only one rule my be selected t ech step of the trnsformtion of progrm p. Consequently, the trnsformtion would be liner in the size n, i.e., the number of lines, of p if we would not reinject ssignments. However, given ssignment cnnot be removed twice, so the trnsformtion is qudrtic. Finlly, the entire trnsformtion of progrm p is repeted until nothing chnges, tht is t most n times. Hence, the globl complexity for progrm trnsformtion of size n is On 3. IV. EXPERIMENTS We hve implemented tool bsed on the rules of Section III to improve the numericl ccurcy of the flotingpoint computtions. This tool finds more ccurte progrm mong ll those equivlent. In this section, we emphsize the efficiency of our implementtion in terms of improving the dt-types used by the progrms. More precisely, we show tht by using our tool, we pproximte the results to be ever ccurte nd close to the results obtined under the double precision while using single precision. In the light of these ides, let us confirm our clims by mens of smll exmples such s Simpson s method. We strt by briefly describing wht our progrm computes, nd then we give their listing before nd fter being optimized with our tool. Their ccurcy is then discussed. Note tht our progrms re written in SSA form [5] to void ny problem deling with the reding nd writing vribles. A. Problem Description nd Simpson s Method Simpson s method consists in technique for numericl integrtion tht pproximtes the computtion of b fxdx. It uses second order pproximtion of the function f by qudrtic polynomil P tht tkes three bsciss points, b nd m with m = +b/2. When integrting the polynomil, we pproximte the integrl of f on the intervl [x,x + h] h R smll with the integrl of P on the sme intervl. Formlly, the smller the intervl is, the better the integrl pproximtion is. Consequently, we divide the intervl [, b] into subintervls [, + h], [ + h, + 2h],... nd then we sum the obtined vlues for ech intervl. We write: n b fxdx h 1 [ 2 fx fx 2j j=1 where n 2 j=1 ] fx 2j 1 + fx n n is the number of subintervls of [,b] with n is even, h = b /n is the length of the subintervls, x i = +ih for i = 0,1,...,n 1,n. In our cse, we hve chosen the polynomil given in Eqution 12. It is well-known by the specilists of flotingpoint rithmetic tht the developed form of the polynomil evlutes very poorly close to multiple root. This motivtes our choice of the function f below for our experiments. f = x 2. 7 = x x x x x x x The listing corresponding of the implementtion of the Simpson s method is described on Figure 3. int min { = 1.9; b = 2.1; n = 100.0; i = 1.0; x = ; h = b - /n; f = x*x*x*x*x*x*x * x*x*x*x*x*x * x*x*x * x*x * x*x*x*x * x*x*x * x * x * x ; x = b ; g = x*x*x*x*x*x*x * x*x*x*x*x*x * x*x*x * x*x * x*x*x*x * x*x*x * x * x * x ; s = f + g; while i < n { x = + i * h; f = x*x*x*x*x*x*x * x*x*x*x*x*x * x*x * x*x*x * x*x*x*x * x*x*x * x*x * x ; s = s * f; i = i + 1.0; }; i = 2.0 ; while i < n-1 { x = + i * h ; f = x*x*x*x*x*x*x * x*x*x*x*x*x * x*x * x*x*x * x*x*x*x * x*x*x * x*x * x ; s = s * f; i = i + 1.0; }; s = s * h / 3.0; } Fig. 3. Listing of the initil Simpson s method. int min { TMP_3 = ; TMP_1 = TMP_3 * 1.9 * 1.9 * 1.9 * * TMP_3 * 1.9 * 1.9 * * * 1.9 * 1.9; TMP_2 = 1.9 * 280. * TMP_3; TMP_15 = ; TMP_13 = TMP_15 * 2.1 * 2.1 * 2.1 * * TMP_15 * 2.1 * 2.1 * * * 2.1 * 2.1; TMP_14 = 2.1 * 280. * TMP_15; TMP_27 = 3.61; TMP_25 = TMP_27 * 1.9 * 1.9 * 1.9 * 1.9 * * TMP_27 * 1.9 * 1.9 * 1.9 * 1.9; TMP_26 = 1.9 * * TMP_3; TMP_32 = TMP_3 * 1.9; i = 1.; s = TMP_1 - TMP_ * TMP_ TMP_13 - TMP_ * TMP_ ; f = TMP_25 + TMP_ * TMP_ * TMP_27 * * TMP_ ; x = 2.1; while i < 100. { x = i * 0.002; TMP_37 = x * x; TMP_35 = TMP_37 * x * x * x * x * x * TMP_37 * x * x * x * x; TMP_36 = 84. * TMP_37 * x * x * x; TMP_42 = x * TMP_37 * x; f = TMP_35 + TMP_ * TMP_ * TMP_37 * x-672. * TMP_ * x ; s = s + 4. * f; i = i + 1.; } ; [... ] s = * s ; } Fig. 4. Listing of the trnsformed Simpson method. When given the initil progrm described in Figure 3 to our tool, it improves its numericl ccurcy by up to 99% depending on the entries. The trnsformed progrm is given in Figure 4. As detiled in Section III, our tool hs:

5 b c d Fig. 5. Simultion results of the Simpson s method with single, double precision nd optimized progrm using our tool. The vlues of x nd y xes correspond respectively to the vlue of n nd s in Eqution 13. creted lrge expressions, trnsformed them into more ccurte expressions, performed prtil evlution of the expressions, split the trnsformed expressions, ssigned the trnsformed expressions to T M P vribles. This process hs been pplied inside nd outside the loop. B. Experimentl results The experimentl results described herefter compre the numericl ccurcy of progrms using single nd double precision with progrm trnsformed with our tool nd running single precision only. Note tht the codes presented in this rticle re written in C, compiled with the GCC compiler version nd executed by n Intel Core i7 processor under Ubuntu In ddition, progrms re compiled with the optimiztion level o0 to void ny optimiztion done by the compiler nd dditionlly, we enforce the copy in the memory t ech execution step by declring ll the vribles s voltile this voids tht vlues re kept in registers using more thn 64 bits. The results observed on Figure 5 when executing the initil progrm in single nd double precision nd the trnsformed progrm in single precision, demonstrte tht our pproch succeeds well to improve the ccurcy. If we interest in the result of computtions round the multiple root 2.0, we cn see tht the behvior of the optimized code is fr closer to the originl progrm executed with double precision thn the single precision originl progrm. This shows tht single precision my suffices in mny contexts. In Figure 5, one cn see the difference, for different vlues of the step h > 0, in the computtion of +nh s = fxdx, 0 n < b h 13

6 between the originl progrm with both single nd double precision nd the trnsformed progrm in single precision in terms of numericl ccurcy of computtions. Obviously, the ccurcy of the computtions of the polynomil fx depends on the vlues of x. The more the vlue of x is close to the multiple root, the worst the result is. For this reson, we mke the intervl [,b] vry by choosing [,b] {[1.9, 2.1],[1.95, 2.05],[1.95, 2.05]} nd by choosing to split them in n = 100, n = 100 nd n = 250 slices respectively for the ppliction of Simpson s rule. Concerning the results obtined, our tool sttes tht the percentge of the optimiztion computed by the bstrct semntics of Section II-A is up to 99.39%. This mens tht the bound obtined by the technique of Section II-A on the numericl error of the computed vlues of the polynomil t ny itertion is reduced by 99.39%. Curve d of Figure 5 displys the function fx t points n = i,100 i 200. Next, if we tke for exmple Curve b of Figure 5, we observe tht our implementtion is s ccurte s the initil progrm in double precision for mny vlues of n since it gives results very close to the double precision while working in single precision. Note tht, for the x xis of Figure 5, we hve chosen to observe only the interesting intervl of n which is round the multiple root 2.0. V. RELATED WORK Other reserch work tries to optimize the numericl dttypes or to increse the precision of computtions. Durlov nd Kunck use n SMT solver to determine the miniml dttype needed to rech certin ccurcy [7]. For the fixed-point rithmetic [18], n lterntive to the floting-point rithmetic, mny pproches hve been proposed to optimize the formt of the numbers [13], [15]. Another kind of work ims t incresing the ccurcy of computtions by mens of dditionl clcultions which significntly slow down the pplictions. Double-double floting-point numbers emulte by softwre numbers with twice the precision of the double hrdwre formt [11]. Finlly, compenstion techniques use error-free trnsformtions to cpture the exct errors of floting-point computtions in order to re-inject the ccumulted error in the result [19], [22]. VI. CONCLUSION In this rticle, we hve shown the usefulness of our implementtion to improve the ccurcy of progrms. This llows one to work in lower precision nd obtin results close to the higher precision when trnsforming progrms using our tool. Our exmples compre the result of trnsformed progrm with the initil progrms executed in single nd double precision. We believe tht this pproch is very promising ccording to the different experiments results obtined. Another reserch direction consists in the interprocedurl progrms trnsformtion, i.e., we im t generlizing our techniques to cover other kinds of progrmming lnguge ptterns like pointers, rrys nd, specilly functions. We hope lso in future work to optimize severl reference vribles simultneously. One difficulty is tht the optimiztion of one vrible my decrese the ccurcy of other vribles. Compromises hve to be done. Another perspective consists t studying the impct of the ccurcy optimiztion on the convergence time of distributed systems nd lso hndle with the importnt issue tht concerns the reproducibility of the results: different runs of the sme ppliction yield different results due to the vritions in the order of evlution of the mthemticl expression. It will be very interesting to study how our technique could improve reproducibility. REFERENCES [1] ANSI/IEEE. IEEE Stndrd for Binry Floting-point Arithmetic, std edition, [2] K.-E. Atkinson. An Introduction to Numericl Anlysis. Second Edition, [3] J. Bertrne, P. Cousot, R. Cousot, J. Feret, L. Muborgne, A. Miné, nd X. Rivl. Sttic nlysis by bstrct interprettion of embedded criticl softwre. ACM SIGSOFT Softwre Engineering Notes, 361:1 8, [4] P. 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