In general, the correctness of numerical computations of programs based on floating-point arithmetic

is not intuitive and developers hope to compute an accurate result without guaranty. To solve this problem, we procceed by automatic source to source transformation of programs to improve their numerical accuracy

Chapoutot, Alexandre

Damouche, Nasrine

Martel, Matthieu

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DraftNumerial Auray Improvement of Programs: Priniples and ExperimentsNasrine Damouhe1, Matthieu Martel1 & Alexandre Chapoutot 21University of Perpignan (Laboratoire LAMPS), 52 av Paul Alduy, 66860 PERPIGNAN Cedex 92Paris-Salay University (U2IS, ENSTA ParisTeh,), 828 bd des Maréhaux, 91762 Palaiseau edex Franenasrine.damouheuniv-perp.fr, matthieu.marteluniv-perp.fr, alexandre.hapoutotensta-paristeh.frProblemIn general, the orretness of numerial ompu-tations of programs [4℄ based on oating-pointarithmeti is not intuitive [1℄ and developershope to ompute an aurate result withoutguaranty. To solve this problem, we proeed by++a bc ≠++ab cautomati soure to souretransformation of programsto improve their numerialauray.Transforming ExpressionsFor automati transformation of single arithmetiexpressions, several tehniques have already beenproposed [5, 6℄. Among them, the APEG (Ab-strat Program Expression Graph) introdue anew intermediary representation that manipulatesin a single data struture a large set of equiva-lent arithmeti expressions [5℄. Basially, APEGdo not dupliate the ommon parts of the syn-tati trees of mathematially equivalent expres-sions. The APEG orresponding to the expressione = (a + a) + b ∗ c is dis-played hereafter. An expres-sion is built from an APEGby seleting one operator ineah dotted box and one ar-bitrary parsing for eah box(e.g. +(a,a,b) ).2 a×+b□×+cb c×a a+×× ++(a,a,b)xcTransforming CommandsThe transformation uses an environment δwhih maps identiers to formal expressions, ablak list β of identiers and a target variable µto be optimized. Programs are assumed to begiven in SSA form and Φ-nodes are assoiatedto onditionals and loops.Assignments c ≡ id = e• If the following onditions are satised ⇒remove the assignment from the programand memorize e[id 7→ e] in δ. the variables of e do not exist in δ, v 6∈ β and v 6= ν,• Otherwise, we inline the variables savedin δ in the onerned expression e and wetransform the resulting expression.Sequenes c ≡ c1; c2• If c1 or c2 is nop ⇒ rewrite the other mem-ber of the sequene,• Otherwise, rewrite both members of thesequene.Conditionals c ≡ ifΦ e then c1 else c2• If e is true ⇒ rewrite c1,• If e is false ⇒ rewrite c2,• If e is statially unknown ⇒ rewrite bothbranhes of the onditionals,• If the variables of e have been removedfrom the program ⇒ re-insert them intothe soure ode and then rewrite it (thevariables of e are inserted in the blaklist β).While Loops c ≡ whileΦ e do c• Rewrite the body of the loop,• re-injet the variables disarded into theprogram and then rewrite it (again theonerned variables are added to β).Experiments and ResultsTo illustrate how we transform programs, we have taken an example from robotis. It omputesthe position of a two wheeled robot by odometry [3℄. Note that sl and sr are the instantaneousrotation speeds of the left and right wheels, c is the irumferene of the wheels and l the length ofits axle. The left Figure represents the initial program of the odometry while the right one gives thetransformed ode.while (t < 100.0 ) do { delta_dl = (c * sl); delta_dr = (c * sr); delta_d = ((delta_dl + delta_dr) * 0.5); delta_theta = ((delta_dr - delta_dl) * inv_l); arg = (theta + (delta_theta * 0.5)); cos = (1.0 - ((arg * arg) * 0.5)) + ((((arg * arg)* arg)* arg) / 24.0); x = (x + (delta_d * cos)); sin = (arg - (((arg * arg)* arg)/6.0)) + (((((arg * arg)* arg)* arg)* arg)/120.0); y = (y + (delta_d * sin)); theta = (theta + delta_theta); t = (t + 0.1);}while (t < 100.0) do { TMP_6 = (0.1 * (0.5 * (9.691813336318980 - (12.34 * sl)))); TMP_23 = ((theta + (((9.691813336318980 - (sl * 12.34)) * 0.1) * 0.5)) * (theta + ((( 9. 691813336318980- (sl * 12.34)) * 0.1) * 0.5))); TMP_25 = ((theta + TMP_6) * (theta + TMP_6)) * (theta + (((9.691813336318980 - (sl * 12.34)) * 0.1) * 0.5)); TMP_26 = (theta + TMP_6) ; x = ((0.5 * (((1.0 - (TMP_23 * 0.5)) + ((TMP_25 * TMP_26) / 24.0)) * ((12.34 * sl) + 9.691813336318980))) + x); TMP_27 = ((TMP_26 * TMP_26) * (theta + (((9.691813336318980 - (sl * 12.34)) * 0.1) * 0.5))); TMP_29 = (((TMP_26 * TMP_26) * TMP_26) * (theta + (((9.691813336318980 - (sl * 12.34)) * 0.1) * 0.5))); y = (((9.691813336318980 + (12.34 * sl)) * (((TMP_26 - (TMP_27 / 6.0)) + ((TMP_29 * TMP_26) / 120.0)) * 0.5)) + y); theta = (theta + (0.1 * (9.691813336318980 - (12.34 * sl)))); t = t + 0.1;}Listing of the initial Odometry program. sl = [0.52,0.53]; sr = 0.785398163397; c = 12.34; theta = t = x = y = 0.0 ;sl = [0.52,0.53]; theta = t = x = y = 0.0 ; inv_l =0.1 ; Listing of the transformed Odometry program.For the demonstration of its eieny, we evaluate through our tool a set of programs in the domainsof robotis, avionis, mathematis, et. The experimental results show the eieny of our approahto optimize the numerial auray. We ompare the initial and the new error of eah of programs(see Figure (a)). We have also demonstrated [2℄ that improving the numerial auray aeleratesthe onvergene time of numerial iterative algorithms (see Figure (b)). Finally, we have omparedinitial odes working in double preision with optimized odes in simple preision (see gure ()).(a) (b) ()Conlusion and Perspetives• Improve numerial auray of programs by up to 20%,• Aelerate onvergene of numerial iterative methods up to 15%,• Transformed programs in single preision are lose to soure pro-grams with double preision.• Extend this work to deal with other or more ompliated stru-tures of programs suh that funtions and arrays.• Validate that programs transformed by our tool are equivalentto the original programs using the Coq proof assistant.• Optimize the numerial auray of parallel programs.NumericalAccuracyNumber ofIterationsPrecision21 %15%Initial Odometry ProgramOptimized Odometry ProgramInitial Jacobi's MethodOptimized Jacobi's MethodOptimized Single Precision Simpson's MethodInitial Double Precision Simpson's MethodReferenes[1℄ ANSI/IEEE. IEEE Standard for Binary Floating-point Arithmeti, std 754-2008 edition, 2008.[2℄ N. Damouhe, M. Martel, and A. Chapoutot. Impat of auray optimization on the onvergene of numerialiterative methods. In Moreno Falashi, editor, LOPSTR'15, volume LNCS 9527 of LNCS. Springer, 2015.[3℄ N. Damouhe, M. Martel, and A. Chapoutot. Intra-proedural optimization of the numerial auray of programs.In M.Núñez and M. Güdemann, editors, FMICS'15, volume 9128 of LNCS, pages 3146. Springer, 2015.[4℄ E. Darulova and V. Kunak. Sound ompilation of reals. In POPL, 2014.[5℄ A. Ioualalen and M.Martel. A new abstrat domain for the representation of mathematially equivalent expressions.In SAS'12, volume 7460 of LNCS, pages 7593. Springer, 2012.[6℄ J. R. Wilox P. Panhekha, A. Sanhez-Stern and Z. Tatlok. Automatially improving auray for oating pointexpressions. In PLDI, 2015.1

Numerical Accuracy Improvement of Programs: Principles and Experiments

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Numerical Accuracy Improvement of Programs: Principles and Experiments

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