Termination analysis of floating-point programs using parameterizable rational approximations

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1 Terminatin analysis f flating-pint prgrams using parameterizable ratinal apprximatins Fnenantsa Maurica Université de La Réunin LIM France fnenantsa.maurica@ gmail.cm Frédéric Mesnard Université de La Réunin LIM France frederic.mesnard@univreunin.fr Étienne Payet Université de La Réunin LIM France etienne.payet@univreunin.fr ABSTRACT Analysis f flating-pint prgrams is a tpic that received an increasing attentin the past few years. Hwever, nly very few wrks have been dne regarding their terminatin analysis. We address that prblem in this paper. We present a technique that takes advantage f the already existing wrks n terminatin analysis f ratinal prgrams. Our apprach cnsists in translating the flating-pint prgrams int ratinal nes by means f sund apprximatins. We apprximate the flating-pint expressins using piecewise linear functins. Our apprximatin differs frm the already existing nes in the sense that it can be as precise as needed. CCS Cncepts Sftware its engineering Frmal methds; Sftware verificatin; Keywrds Terminatin analysis, Flating-pint arithmetic, Linearizatin 1. INTRODUCTION Terminatin analysis f prgrams is a ht research tpic that has already prduced many techniques. Sme f the mst innvative nes are the synthesis f linear ranking functins fr simple lps [15], the generalizatin t eventual linear functins [1], the plyranking [6], Chen s algrithm [7]. Unfrtunately, nne f these techniques can be used fr the terminatin analysis f flating-pint prgrams since flating-pint cmputatins are highly nn-linear due t the runding errrs. T ur knwledge, there is nly very limited wrk n that tpic. [16] extends the adrnments-based ACM apprach. [8] reduces the terminatin prblem t a secndrder satisfiability prblem. This paper develps a new technique that helps addressing that lack f wrk. Its main advantage is that, thrugh the use f an innvative ratinal apprximatin, it transpses the terminatin analysis f flating-pint lps int terminatin analysis f ratinal lps, which is already very well cvered in the literature. The rest f the paper is rganized as fllws. First, we quickly intrduce the basics f flating-pint arithmetic in Sectin 2. Then, we give the details f ur ratinal apprximatin in Sectin 3. We end by presenting its applicatin t terminatin analysis in Sectin FLOATING-POINT ARITHMETIC A real number x R is apprximated in machine by a flating-pint number ˆx F. The IEEE-754 stard defines the flating-pint arithmetic. A flating-pint number is represented by the triplet (s, m, e) such that: ˆx = ( 1) s β e m where s {0, 1 is the sign, β {2, 10 is the radix, e [e min, e max] is the expnent, the fractinal number m = m 0.m 1m 2...m p 1 is the signific is written in p digits where p is called precisin. We call machine epsiln ɛ the distance between 1 the flating-pint number fllwing 1. We call unit in the last place, r ulp, f ˆx the value the least significant digit represents: ulp(ˆx) = ɛ β e. If ˆx is such that ˆx β e min, ˆx is called a nrmal number. Otherwise, ˆx is called a subnrmal number. We call s min the smallest psitive subnrmal, n min the smallest psitive nrmal n max the biggest psitive nrmal. IEEE-754 requires flating-pint arithmetic peratins t be crrectly runded. That is t say, giving a runding mde, a real arithmetic peratin, its flating-pint equivalent 2 flating-pint numbers ˆx 1, ˆx 2, the fllwing hlds: ˆx 1 ˆx 2 = (ˆx 1 ˆx 2). Apart frm the frmat f the representatin the runding mdes, IEEE-754 als defines special numbers exceptin hling. Hwever, fr simplicity, we will nly cnsider the runding t nearest ties t even. Mrever, we will

2 cnsider that there is n special value in ur representatin that n exceptin will ccur in ur cmputatins, ntably n verflw. As illustrative example, cnsider the extremely simple type myflat presented in Figure 1 where β = 10, p = 2, e min = 0, e max = 1. s min step : 0.1 subnrmals ɛ n min step : nrmals step : 1 n max Figure 1: A persnalized flating-pint type, myflat. Symmetry t the rigin fr the negatives. A mre cmplete intrductin t flating-pint cmputatins can be fund in [10]. 3. RATIONAL APPROXIMATION The cmmn methd fr apprximating ˆx is t use the abslute errr the relative errr. We recall the definitin f the abslute errr A (x) = x ˆx the relative errr R(x) = x ˆx, mre details can be fund in [11, Sectin x 2.1]. The linear expressins f these errrs vary accrding t the range f the flating-pint numbers that are studied, resulting in piecewise linear apprximatins. T ur knwledge, literature mainly apprximates by distinguishing the range f the nrmals frm the range f the subnrmals. Such is the case fr example f [4, 5] where the abslute errr is used fr the subnrmals the relative errr fr the nrmals. Smetimes a mix f the tw errrs is used, resulting in a single apprximatin fr bth the subnrmals the nrmals, like in [2, 14]. Thugh these apprximatins are crrect, they may be t imprecise t be f use. Thus, mre precise apprximatins are needed. Hwever, the ptimal bund fr the abslute errr has lng been knwn the ptimal bund fr the relative errr has very recently been prved in [12]. Hence, the traditinal apprximatins cited abve cannt be refined anymre. The nvelty f ur apprximatin is threefld. Firstly, it can be as precise as needed, until the equivalence between the flating-pint expressin its apprximatin is reached. Secndly, it unifies the different types f current linear apprximatins (apprximatin by abslute errr, by relative errr, by cmbinatin f bth) under a single framewrk in which they are classified hierarchically. Thirdly, it gives anther prf fr the ptimality f the errr bund f flatingpint rundings. 3.1 Apprximatin f ˆx with parameterizable precisin 99 Fr easily understing ur idea, let us study the apprximatin f a real x by ˆx = (x) fr the particular case f the type myflat when the runding t nearest ties t even is used. The runding functin is exactly defined as fllws: : R F myflat < x < x < x < 97.5 x (x) = ˆx = x < x < 98.5 The key pint is t ntice that is already a piecewise linear functin, a piecewise cnstant functin actually. As such, is nt different frm the apprximatins by abslute/relative errr which are als piecewise linear functins. Thus, in the same way we can apprximate flating-pint arithmetic peratins using the abslute/relative errr apprximatins, we can als apprximate using the exact definitin f the runding functin, which will lead t the equivalence between the initial expressin its apprximatin. Hwever, the number f pieces used in the exact definitin equals the number N f the flating-pint pssible values, N = 2 β p 1 (e max e min + 2) 1 fr ur case, which is a ridiculusly big number. Hence, even if it may be f interest theretically, the exact definitin is hardly useful practically. Here cmes the idea f linear apprximatin with parameterizable precisin. It can be easily understd that n piecewise linear functin with a number f pieces fewer than N can exactly define ˆx. Apprximatins are used instead. Reducing the number f pieces will strictly reduce the precisin f the apprximatin while increasing the number f pieces will strictly increase it. The prblem f apprximating ptimally ˆx using linear functins defined in k pieces frmally cnsists f finding the upper apprximatin functin µ k the lwer apprximatin functin ν k such that: minimize( X (µ k(x) ˆx)dx) µ k (x) = {a i x + b i, x X i k 1 i k ˆx µ k (x) minimize( X (ˆx ν k(x))dx) ν k (x) = {c i x + d i, x X i k 1 i k ν k (x) ˆx where X = [ n max, n max] {X k a partitin f k elements f X that is t be fund. Optimal means that if we measure the precisin f an apprximatin with the surface between the upper lwer apprximatin functins, then n apprximatin with greater precisin can be fund fr the given number f pieces. Failing at finding a general slutin fr (1) (2), we cnsider a simplified versin f the prblem in which the partitining is already knwn. We will nw give the slutins f that simplified versin. Thugh the ptimality f ur partitining has yet t be prved, the prperty f increasing (1) (2)

3 It is interesting t ntice that the prf fr the ptimal bund fr the relative errr recently given in [12] can be retrieved by slving the simplified versin f (1): minimize( n max n min (µ(x) ˆx)dx) µ(x) = a x + b µ(x) ˆx x n min (a) 1 piece (c) 3 pieces (b) 2 pieces Figure 2: Cmmn piecewise linear apprximatins f ˆx precisin until the equivalence is preserved. Fr brevity, we nly present the slutins fr the simplified versin f (1), that is t say the value f the upper apprximatin functin µ k. We hpe that the simplified versin is clear enugh s that the reader is able t slve it by himself/herself. Fr k = 1, 2, 3 we respectively use the partitining: {X 1 = {[ n max, n max] {X 2 = {[ n max, 0[, [0, n max] {X 3 = {[ n max, n min[, [ n min, n min], ]n min, n max] We respectively find the upper linear apprximatin functin: µ 2(x) = µ 1(x) = x + A, x [ n max, n max] { x (1 + R n ) + As x [0, n max] x (1 Rn) + As x [ n max, 0[ x (1 + Rn) µ 3(x) = x + As x (1 Rn) x ]n min, n max] x [ n min, n min] x [ n max, n min[ where A = ɛ βemax, A 2 s = ɛ βe min, R 2 n = ɛ. 2+ɛ These are the cmmn apprximatins, as used in [2, 4, 5, 14] as illustrated in Figure 2(a), 2(b) 2(c). k = 1 is the cmmn apprximatin by abslute errr. k = 2 is the cmmn mixed use f the abslute errr the relative errr. k = 3 is the cmmn use f the relative errr fr the nrmals the abslute errr fr the subnrmals. which will give µ(x) = x (1 + δ), δ =, after which the 2+ɛ identificatin f δ as Rn easily fllws. This result shws hw far reaching ur apprach wuld be. Using ur frmalizatin, we culd prve the ptimality f the cmmn linear apprximatins classified them in rder f increasing precisin. We can as well call them the 1-piece, 2-pieces 3-pieces linear apprximatins. Nw we will present apprximatins with higher precisin. When k 4, we start frm the apprximatin fr k = 3, at each increase f k, we cnsecutively apprximate the flating-pint numbers with same ulp in decreasing rder f ulp. In ther wrds, we grup by ulp frm the edge t the center. Flating-pint numbers ˆx I = [ˆx m, ˆx M ] with same ulp have a unique ptimal linear upper apprximatin which is x + A I where A I = ulp(ˆx) 2. Fr example, fr the apprximatin with 4 pieces, we start frm the apprximatin with 3 pieces. Then fr the new set, we have the chice between a grup f psitive flating-pint numbers a grup f negative flating-pint numbers. We decide t take the psitive ne. This is illustrated in Figure 3(a). Fr the apprximatin with 5 pieces, we start frm the apprximatin with 4 pieces. Then fr the new set, we have nly a single chice, which is the previus grup f negative flating-pint numbers. This is illustrated in Figure 3(b). We repeat the same prcess fr k 6 until k is such that all flating-pint numbers with same ulp are ptimally apprximated, as illustrated in Figure 3(c). Starting frm there, fr the apprximatins with higher number f pieces, at each increase f k, we cnsecutively define each flating-pint number exactly in decreasing rder f ulp, in ther wrds frm the edge t the center again. The equivalence between ˆx its apprximatin is reached when k = N. It is wrth mentining that apprximating by gruping the the flating-pint numbers with same ulp in decreasing rder f ulp is just ne pssible strategy. Depending n the applicatin n the cncrete flating-pint cnstants used in the prgram, it culd be beneficial fr example t tighten the apprximatin with increasing rder f ulp r in particular arund the ccurring cnstants. We just presented piecewise linear apprximatins which are mre precise than the cmmn piecewise linear apprximatins by increasing the number f pieces. We will nw see hw t use these apprximatins f ˆx t linearize arithmetic expressins. 3.2 Linearizatin f arithmetic flating-pint expressins ɛ

4 (a) Frm 3 t 4 pieces. The new set is in dashed line. Same as fr 3 pieces fr the negatives. (b) Frm 4 t 5 pieces. The new set is in dashed line. Same as fr 4 pieces fr the psitives. Generally, nce such apprximatin by ratinal expressins is btained, the linearizatin is cnsidered dne even if, strictly speaking, it is nt since nn-linear terms, like prduct f tw variables, may still appear in the ratinal expressin. In thse cases, the nn-linear terms have t be linearized by techniques like [13]. Fr example, giving tw ratinal variables x [x m, x M ] y [y m, y M ], their prduct z = x y will be apprximated by the linear cnstraints: z x my y mx + x my m 0 z + x my + y M x x my M 0 z + x M y + y mx x M y m 0 z x M y y M x + x my M 0 We will nw present a cncrete applicatin f ur technique thrugh the terminatin analysis f lps using flatingpint numbers as variables. (c) All gruped by ulp. Symmetry t the rigin fr the negatives. Figure 3: Piecewise linear apprximatins f ˆx with higher precisin Flating-pint arithmetic peratins are nn-linear. Linearizing a flating-pint arithmetic expressin cnsists f translating the expressin int a ratinal linear expressin by means f sund apprximatins. Knwing an apprximatin f ˆx such that ν k (x) ˆx µ k (x) using the prperty f crrectly runded peratins, the apprximatin f a flating-pint arithmetic peratin is straightfrward: fr a given arithmetic peratin, we have ν k (y z) y z µ k (y z). When linearizing an arithmetic expressin with multiple peratins, we use intermediate variables t decmpse the expressin int elementary arithmetic peratins with respect t their precedence. Fr example, the fllwing expressin r = x+ y z x is decmpsed int: r = t 2 x t 2 = x+ t 1 t 1 = y z Then, it nly remains t apprximate each elementary peratin using the previusly presented piecewise linear apprximatins, with the desired precisin. The abve example will give us: ν k (t 2 x) r µ k (t 2 x) ν k (x + t 1) t 2 µ k (x + t 1) ν k (y z) t 1 µ k (y z) 4. APPLICATION TO TERMINATION AN- ALYSIS OF FLOATING-POINT LOOPS Our apprach fr analyzing terminatin f lps invlving flating-pint arithmetic cnsists in translating them int ratinal lps, then applying the already existing terminatin analysis techniques t the btained translatin. As we suppse that n verflw ccurs, the result f the analysis is nly valid fr the initial ranges f the variables that d nt lead t verflw. Abstract interpretatin-based tls, like Fluctuat [9], r frmal methd-based tls, like Gappa [3], already exist fr the determinatin f thse ranges. The nn-ccurrence f verflw is an infrmatin that we can use t enrich the cnstraints f ur apprximatins since fr every arithmetic flating-pint peratin : n max x y n max. Hwever, fr the sake f readability, we will mit these additinal cnstraints. Each flating-pint expressin is then apprximated by ratinal nes using the technique presented in the previus sectin. Expressins in guard cnditins are hled using intermediate variables. We will nw cncretely see hw t prceed thrugh the analysis f the lp alp presented in Figure 4, in which the variables x y are f type myflat. In ur ntatin, the primed variables refer t the value f their unprimed equivalent at the end f ne iteratin. while(x * y >= 0 & y >= -6) { x := x - 0.6; y := y ; Figure 4: A flating-pint lp, alp alp terminates fr every initial value f x y. Indeed, fr any initial value x 0, y 0 f x, y: (i) x is always strictly decreasing, (ii) y is strictly increasing at each iteratin until reaching 10 if 6 y 0 < 10 r y stays unchanged if y 0 10.

5 (i) (ii) imply that if the lp runs lng enugh, x will becme negative while y will becme psitive in which case the cnditin x y 0 will be unsatisfied. Ntice that y 1 = y is different frm y 2 = y due t the runding errrs. Fr example, fr y = 20, y 1 = 20 while y 2 =. We will nw shw hw t analyze alp using ur apprach. We start by decmpsing all the arithmetic expressins int a successin f elementary peratins. If cmplex expressins are encuntered in the guard cnditin, they are put inside intermediate variables s that they can be decmpsed nrmally. This is illustrated in Figure 5. c0 := x * y; while(c0 >= 0 & y >= -6) { x := x - 0.6; y := y ; c0 := x * y ; (a) Treatment f the guard cnditin c0 := x * y; while(c0 >= 0 & y >= -6) { x := x - 0.6; t0 := y + 0.3; y := t ; c0 := x * y ; (b) Decmpsitin int a successin f elementary peratins Figure 5: Preliminary peratins befre translatin Then, each peratin is apprximated by piecewise linear functins with the desired precisin as presented in the previus sectin. We will nw shw that the cmmn apprximatins the 1-piece, the 2-pieces the 3-pieces linear apprximatins in ur terminlgy are nt precise enugh t prve the terminatin f alp while the 4-pieces ne is. Using the 1-piece linear apprximatin, r = y z is apprximated by ν 1(y z) r µ 1(y z) such that µ 1(x) = x + 0.5, 99 x 99 ν 1(x) = x 0.5, 99 x 99. Applying that apprximatin n the transfrmed prgram in Figure 5(b), we will get the ratinal apprximatin f alp in Figure 6 which uses the nn-deterministic assignments <=: :<=. x*y <=: c0 :<= x*y + 0.5; while(c0 >= 0 & y >= -6) { x <=: x :=< x - 0.1; y <=: t0 :<= y + 0.8; t0-0.2 <=: y :<= t ; x *y <=: c0 :<= x *y + 0.5; Figure 6: Ratinal apprximatin f alp using the 1-piece linear apprximatin The 1-piece linear apprximatin is precise enugh t capture the decreasing f x as the btained upper bund f x is strictly less than x. Hwever, it is t lse t capture the increasing t a psitive number f y. Indeed, the btained lwer bund f t0 is less than r equal t y that f y is less than r equal t t0. Thus, with this apprximatin, y may decrease r may stay cnstant during all executins f the lp. If the initial values f x y are negative numbers y stays cnstant, the lp will run infinitely. As there are cases where the translated lp may terminate r may nt, we cannt decide the terminatin f alp. Using the 2-pieces linear apprximatin, r = y z is apprximated by ν 2(y z) r µ 2(y z) such that: µ 2(x) = ν 2(x) = { { x x [0, 99] x x [ 99, 0[ x 0.05 x [0, 99] x 0.05 x [ 99, 0[ Thus, the instructin x := x - 0.6; is translated as in Figure 7. if (x >=0) { 20*(x - 0.6)/ <=: x <=: 22*(x - 0.6)/ ; else { 22*(x - 0.6)/ <=: x <=: 20*(x - 0.6)/ ; Figure 7: Ratinal apprximatin f x := x - 0.6; using the 2-pieces linear apprximatin The ther instructins are translated in a similar way. We easily verify that the 2-pieces linear apprximatin is precise enugh t capture the increasing t a psitive number f y as the btained lwer bund f y is strictly greater than y fr 6 y 0. Hwever, it is t lse t capture the decreasing f x fr any initial value x 0 f x. Indeed, fr x 0 14, the btained upper bund f x is greater than r equal t x. Thus, with this apprximatin, x may increase r may stay cnstant during all executins f the lp. If x 0 14 y 0 0, the lp will run infinitely. As there are cases where the translated lp may terminate r may nt, we cannt decide the terminatin f alp. The same reasning applies fr the 3-pieces linear apprximatin which is als precise enugh t capture the increasing t a psitive number f y but nt the decreasing f x. Using the 4-pieces linear apprximatin, r = y z is apprximated by ν 4(y z) r µ 4(y z) such that: x x [10, 99] 22 x x ]1, 10[ µ 4(x) = x x [ 1, 1] x x [ 99, 1[ 20 x 0.5 x [10, 99] 20 x x ]1, 10[ ν 4(x) = x 0.05 x [ 1, 1] x x [ 99, 1[ 22

6 Thus, the instructin x := x - 0.6; is translated as in Figure 8. if (x >= 10) { x <=: x :=< x - 0.1; else if (1 <= x & x <= 10) { 20*(x - 0.6)/ <=: x <=: 22*(x - 0.6)/; else if (-1 <= x & x <= 1) { x <=: x :<= x ; else { 22*(x - 0.6)/ <=: x <=: 20*(x - 0.6)/; Figure 8: Ratinal apprximatin f x := x - 0.6; using the 4-pieces linear apprximatin The ther instructins are translated in a similar way. Nw, the apprximatin is precise enugh t capture bth the decreasing f x the increasing t a psitive number f y. Indeed, the btained upper bund f x is strictly less than x the btained lwer bund f y is strictly greater than y fr any 6 y 0. As the translated lp terminates fr any initial value f x y, s des alp. 5. CONCLUSION We have presented a new technique fr analyzing terminatin f flating-pint lps by translating them int ratinal nes using sund apprximatins. T achieve that, we use piecewise linear functins. The nvelty lies in the quality f the apprximatins which are ptimal fr the chsen partitining can be as precise as needed by increasing r decreasing the number f pieces. That is the ntin f what we call piecewise linear apprximatin with parameterizable precisin. Unlike the few ther existing techniques fr terminatin analysis f flating-pint lps, urs benefits frm the very well cvered terminatin analysis f ratinal linear lps. T sme extent, the presented technique can be als applied t prgrams that perate n bth flating-pint variables n ther kind f data such as integers, arrays mre generally heap-based data-structures. T ur knwledge, such a cmbined analysis has nt yet been prpsed fr prgram terminatin. Our wrk can be extended in many interesting ways. We can guide the partitining by the ranges f the variables in the expressin t apprximate. Thse ranges can be retrieved by abstract interpretatin-based techniques fr example. We can als use apprximatins with different precisin fr each expressin t apprximate instead f using a unique apprximatin fr all the expressins. The ntin f apprximatin with parameterizable precisin can be easily used in ther tpics invlving flating-pint arithmetic. We ntably think abut the tpic f cnstraint slving ver flating-pint numbers which als uses ratinal apprximatins. Indeed, the mre precise the apprximatins are, the mre efficient the filtering prcess will be. 6. REFERENCES [1] R. Bagnara F. Mesnard. Eventual linear ranking functins. In Principles Practice f Declarative Prgramming, [2] M. S. Belaid, C. Michel, M. Rueher. Bsting lcal cnsistency algrithms ver flating-pint numbers. In Principles Practice f Cnstraint Prgramming [3] S. Bld, J.-C. Filliâtre, G. Melquind. Cmbining Cq Gappa fr certifying flating-pint prgrams. In Intelligent Cmputer Mathematics [4] S. Bld T. M. T. Nguyen. Hardware-independent prfs f numerical prgrams. In NASA Frmal Methds, [5] S. Bld T. M. T. Nguyen. Prfs f numerical prgrams when the cmpiler ptimizes. Innvatins in Systems Sftware Engineering, [6] A. R. Bradley, Z. Manna, H. B. Sipma. The plyranking principle. In Autmata, Languages Prgramming [7] H. Y. Chen, S. Flur, S. Mukhpadhyay. Terminatin prfs fr linear simple lps. In Static Analysis [8] C. David, D. Krening, M. Lewis. Unrestricted terminatin nn-terminatin arguments fr bit-vectr prgrams. In Eurpean Sympsium n Prgramming [9] D. Delmas, E. Gubault, S. Putt, J. Suyris, K. Tekkal, F. Védrine. Twards an industrial use f Fluctuat n safety-critical avinics sftware. In Frmal Methds fr Industrial Critical Systems [10] D. Gldberg. What every cmputer scientist shuld knw abut flating-pint arithmetic. ACM Cmputing Surveys, [11] N. J. Higham. Accuracy Stability f Numerical Algrithms. Sciety fr Industrial Applied Mathematics, [12] C.-P. Jeannerd S. M. Rump. On relative errrs f flating-pint peratins: ptimal bunds applicatins. Preprint, [13] G. P. McCrmick. Cmputability f glbal slutins t factrable nncnvex prgrams: Part I Cnvex underestimating prblems. Mathematical Prgramming, [14] A. Miné. Relatinal abstract dmains fr the detectin f flating-pint run-time errrs. In Eurpean Sympsium n Prgramming [15] A. Pdelski A. Rybalchenk. A cmplete methd fr the synthesis f linear ranking functins. In Verificatin, mdel checking, abstract interpretatin, [16] A. Serebrenik D. De Schreye. Terminatin f flating-pint cmputatins. Jurnal f Autmated Reasning, 2005.