A Formal Model and Efficient Traversal Algorithm for Generating Testbenches for Verification of IEEE Standard Floating Point Division

Transcription

1 A Formal Moel an Efficient Traversal Algorithm for Generating Testbenches for Verification of IEEE Stanar Floating Point Division Davi W. Matula, Lee D. McFearin Department of Computer Science an Engineering Southern Methoist University Dallas, TX {matula,mcfearin}@engr.smu.eu Abstract We utilize a formal moel of ivision for etermining a testbench of pbit (ivien, ivisor) pairs whose output 2pbit quotients have properties characterizing these instances as the most challenging for verifying any ivision algorithm esign an implementation. Specifically, our test suites yiel 2pbit quotients where the leaing pbits traverse all or a pseuoranom sample of leaing bit combinations, an the next pbits comprise a roun bit followe by (p1) ientical bits. These values are pro ven to be closest to the pbit quotient rouning bounaries an shown to possess other esirable coverage properties. We introuce an efficient metho of generating these testbenches. We also escribe applications of these testbenches at the esign simulation stage an the prouct evaluation stage. 1. Introuction The IEEE floating point stanar [IEEE] requires all conforming implementations of ivision an square root to correctly roun the result as if the infinitely precise result ha been calculate an then roune to the target precision. As processors become larger an these implementations become more complex, the chances for expensive flaws increase [SB94]. Single precision unary functions such as square root an square root reciprocal have only 16 million istinct inputs an may be exhaustively teste. However, ivision has two arguments generating 64 trillion istinct normalize single precision input cases, an almost a quintillion times as many istinct ouble precision input cases. Since ivision can not be exhaustively teste in a reasonable amount of time, selective, statistical test suites are typically use for verification. One such type of test suite focuses on those instances which yiel infinitely precise quotients extremely close to the rouning bounaries[ka87,pa99, MM02, MM03]. These instances are the most sensitive to calculation errors as a very small approximation error may cause the result to cross a rouning bounary an roun incorrectly. Note that the problem of etermining extremal rouning bounary cases for transcenentals has no known conveniently scalable solution for higher precisions an has been terme the "table maker s ilemma" [Mu97,LM98,LM04,Zi91,DE05]. While a boy of excellent research on search techniques has evolve for fining extremal "haresttoroun" cases for ouble precision an some ouble extene precision transcenental functions, it appears to be very har to obtain a general characterization. In contrast, the problem of etermining extremal rouning bounary cases for ivision has a firm number theoretic founation that is conveniently scalable to higher precisions. Our solution herein improves the efficiency of generating benchmark sets for this problem an is instructive in that it comprises funamental elementary results from three istinct number types as follows. i. Raix Arithmetic: the problem of etermining extremal rouning bounary cases for floating point ivision is initially cast as a problem with binary raix floating point inputs an outputs. ii. Rational Arithmetic: The extremal rouning bounary cases are recharacterize employing rational arithmetic where Farey fractions an continue fractions conveniently ientify the esire extremal cases. iii. Resiue Arithmetic: The core computational steps are simplifie by utilizing properties of the multiplicative inverse moulo 2 k. This is the principle aition herein to our previously publishe methoology /DATE EDAA

2 2. Backgroun The size of a floating point ivision operation is characterize by the size of the significan of the arguments. A positive p bit number is a binary rational of the form 2 e i, where 1 i 2 p 1 an i is o. Thus, the precision p 1 provies a measure of the size of the pbit number s significan bit string in the normalize floating point factore format z = 2 e (1. b 1 b 2...b p 1 ). A p p bit fraction enotes a fraction n where the numerator an enominator are positive pbit inte gers[mm00,mm03]. The rational value q = n is terme the infinitely precise p p bit quotient. For our purposes herein a normalize p p bit fraction has the enominator in the range 1 2 p 1, an the numerator in the range n <2, so then the normalize p p bit quotient is in the stanar binae 1 q = n = 1. b 1b 2...< 2. Note that if this normalization yiels a numerator greater than 2 p, it must be even to be a pbit number. Following the literature on continue fractions an Farey series, two fractions are ajacent if their cross prouct is unity [HW79,MK85]. With this backgroun, the extremal rouning bounary instances of p p bit fractions n for rountonearest floating point ivision have been characterize by three istinct, an provably equivalent efinitions[mm00, MM02, MM03]: (i) [Distance form] the istance of n from a closest pbit i mipoint satisfies n 2 p i < 1. 2 p 2 2p 1 (ii) [Binary expansion form] n has the stanar binary expansion n = 1. b 1b 2...b p 1 b p... where the tail "rouning critical" b p b p+1...b 2p is either or containing the maximum length run of p 1 like value bits opposite to the roun bit b p. (iii) [Best rational approximation form] n is the best rational approximation of a pbit mipoint, an is given by the closest continue fraction convergent to a pbit mipoint i. 2 p The set of p p bit fractions equivalently efine by these characterizations for a given p is enote by RN p. For example, the set RN 5 of extremal mipoint rouning bounary 5 5 bit fractions has ten members an can be evelope employing the multiplicative inv erse as illustrate in Table 1. For any o integer a with 1 a 31, let a 1 be the unique o integer with 1 a 1 31 an aa 1 1 (mo ). Traversing the rows for a = 1, 3, 5,..., 31 in Table 1, the fractions +a traverse the mipoints of the five bit intervals [ i 16, i+1 ) for 16 i Let n an be etermine by ( + a)a 1 = n + 1 an ( + a)( a 1 )= 1. This assures that +a = 1 an +a a 1 a 1 a 1 ( a 1 ). Then n a 1 = 1 RN 5 when 1 a 1 15 an is a 5 bit number (i.e. either < or is even). Similarly n RN a 1 5 when a 1 31 an n is a 5 bit number. a n +a a a 1 Extremal Value Table 1: 5 5 bit Extremal Mipoint Rouning Bounary Instances 3. Enumeration of Extremal Rouning Bounary Instances For any precision p 2, the set of extremal mipoint rouning bounary p p bit fractions can be etermine as follows. Observation 1 Let a be a pbit o integer an a 1 its p bit multiplicative inverse. Then (i) if 2 p a 1 2 p 1 let n = (2 p +a)a p. If n is even

3 or n <2 p, then n RN a 1 p, (ii) if 1 a 1 2 p 1 1 let = (2 p +a)(2 p a 1 )+1. If is 2 p even or <2 p, then RN 2 p a 1 p. Note that the multiplicative inv erse pairs a, a 1 are istinct o value pairs except for the four cases a = 1, 2 p 1 1, 2 p 1 + 1, an 2 p 1, where a is its own multiplicative inv erse. Once a multiplicative inv erse pair a,a 1 is ientifie, n or may be calculate via multiplication giving an extremal instance in most cases. On average 11 total extremal rouning bounary instances for rountonearest an irecte rouning may be calculate from n (or ) using aition via the symmetric properties iscusse in a 1 2 p a 1 [MM01] an emonstrate in Example 1. Example 1 Given a multiplicative inverse pair moulo (a k, a 1 k ) = (3, 11), the multiplication step escribe earlier, prouces =23. From Observation 1, n = 23 a 1 RN 40 p. Furthermore, from [MM01],, 22 19, 38 27, RN p, an 25, 38, 25 19, 27, 19 RD p. After the multiplicative inv erse pair a, a 1 an n (or n 1 ) has been etermine, the computational cost for each group of extremal rouning bounary instances as foun in Example 1 is 22 aitions. Past approaches have generate these multiplicative inv erse pairs by using the Eucliean GCD Algorithm for each pair. The key to the creation of an efficient algorithm base on Observation 1 lies in the systematic moular generation of the 2 p multiplicative inv erse pairs. Observation 2 For p 3, let a k 3 k (mo 2 p ) an a 1 k 3 2p 2 k (mo 2 p ) be stanar resiues moulo 2 p for k = 0, 1, 2,..., 2 p 3. Then the 2 p multiplicative inverse pairs (a k, a k 1) an (2 p a k, 2 p a k 1) for 0 k 2 p 3 contain all 2 p 1 o integers in [1, 2 p 1]. Tables 2 an 3 illustrate the use of exponential resiues 3 k (mo2 p ) in generating the multiplicative inverse pairs a, a 1 for p = 5 employing Observation 2. Exponential Resiues k 3 k (mo) 3 8 k (mo) Table 2: The 10 multiplicative inv erse resiues moulo. Multiplicative Inv erse Pairs k (a k, a 1 k ) ( a k, a k ) 0 (1, 1) (31, 31) 1 (3, 11) (29, ) 2 (9, 25) (23, 7) 3 (27, 19) (5, 13) 4 (, ) (15, 15) Table 3: The 10 multiplicative inv erse pairs moulo. This metho requires only a multiplication with two shiftana operations for etermining a multiplicative inverse pair along with an n or. Using this computational cost along with the cost of fining the symmetries in [MM01] an average cost can be etermine. Lemma 1 The average cost of 11 extremal rouning bounary instances is 1 multiplication an 24 aitions proviing a cost of slightly over 2 aitions per extremal rouning bounary instance. Using Observation 2, we may etermine all the extremal rouning bounary instances for p p bit binary floating point ivision. The following steps outline the process. Create a storage array containing 2 p 3 elements for a 3 k (mo2 p ) as ARRAY; an a similar array for a p 3 k (mo2 p ) as INV_ARRAY. Compute ARRAY[1] as 3 k (mo 2 p ), where k is the starting value, possibly 1. Compute ARRAY[n + 1] = 3 ARRAY[n] (mo 2 p ). Note this can be compute as ARRAY[n + 1] = (AR RAY[n] + (ARRAY[n] << 1)) (mo 2 p ) where << is a binary shift operation. The last element of ARRAY is the last element of INV_ARRAY.

4 Compute INV_ARRAY[n 1] = 3 INV_ARRAY[n] (mo 2 p ). Once ARRAY an INV_ARRAY hav e been compute, each ARRAY[m] an INV_ARRAY[m] represent an a, a 1 pair. Aitionally 2 p a, 2 p a 1 is a secon pair. Note that the full set of 2 p multiplictive inv erse pairs can be generate employing only 2 p 2 shiftana operations an 2 p complement operations. The complement operation is a 1 s complement since the values are o integers. With at least one of these complementary pairs, extremal rouning bounary instances can be calculate with a multiplication as mentione in Observation 1. Using all k as iscusse in Observation 2 generates the entire set RN p. The benchmark set RN 24 for testing IEEE stanar single precision ivision implementations is enumerate an available at the site [MM04]. Using this proceure the number of extremal mipoint rouning bounary cases RN p was compute for 3 p 28 an is shown in Table 4. p RN p p RN p Table 4: Number of Extremal Mipoint Rouning Bounary Cases for Precisions 3 to 28 To compute a psueo ranom subset of RN p with this metho, a subsequence of the enumeration may be selecte at ranom. Specifically, a first element may be picke at ranom for ARRAY an the "turning point" last element of INV_ARRAY may be obtaine with the extene Eucliean algorithm from the "turning point" last element of ARRAY. It is possible to avoi the nee for the Eucliean algorithm by restricting the turning points to elements a for which a 1 is computable by a single aition. Observation 3 provies a set of such multiplicative inv erse pairs that are uniformly space throughout the sequence a k, a 1 k for a 3 k (mo2 p ) as k traverses the range k = 0, 1,..., 2 p 2. Observation 3 Let j 3 an n be o satisfying 1 n 2 j+3 1. Then with a = n2 j + 1, a 1 = (2 j n)2 j j+3, we obtain that a, a 1 is a multiplicative inv erse pair moulo 2 2 j+3. IEEE stanar ouble precision has p = 53, which woul lea to multiplicative inv erse pairs in the full enumeration as given for p = 5 in Table 2. Employing Observation 3 the full sequence can be partitione into subsequences of size 2 24 pairs selectable by a 27bit ranom integer see q. Thus our enumeration proceure is initiate at a = q , an the turning point occurs after 2 23 shiftanas when a = (q ) 2 53 = q with q o. This provies a reasonable size psueo ranom sample of about 16 million multiplicative inv erse pairs using only the aition operation, which woul provie a psueo ranom subset of RN 53 of size about 22 million. 4. Applications The sets RN 24 an RD 24 along with a C++ program for generating RN p for 3 p 28 are provie by the link [MM04]. The sets RN 24 an RD 24 were evelope for an use in the esign verification stage to test the floating point ivie implementation on the one watt x86 compatible Geoe processor evelope by National Semiconuctor in 2001 an now available from AMD. The set was also employe in an afterthefact prouct evaluation stage to search for ivision implementation errors in the 1993 Pentium Processor having the well known fiv flaw [SB94]. The test suites foun the erroneous fiv quotients with a frequency much higher than that of ranom testing, surprisingly some 10,000 times (4 orers of magnitue) more frequent for single precision. For evience that fining these single precision erroneous quotients was not a fortuitous "accient", further testing showe that these istinct sets for the 15 precisions 22 p 36, uncovere the flaw at each precision at a much higher frequency than ranom testing[mm02].

5 5. Conclusions The extremal rouning bounary instances provie a natural testbench for esign verification of any implementation of IEEE stanar floating point ivision. They provie a large number of test cases, which can be calculate with relative ease. As the size of the ivision problems grow, a simple moular enumeration test suite can be carefully controlle statistically an as a sample still maintain a general coverage over the quotient, ivisor, an ivien ranges. As improve formal moels reveal more structure approaches to testing ivision, fewer flaws are likely to appear in future esigns. 6. Acknowlegments We thank Jason Moore for his assistance in the evelopment of this paper. This work was supporte in part by the Semiconuctor Research Corporation uner contract RID References [LM04] [DE05] F. e Dinechin, A. V. Ershov, an N. Gast, "Tow ars the postultimate libm," Proc. th IEEE Symposium on Computer Arithmetic. IEEE, [HW79] C. H. Hary an E. M. Wright, An Introuction to the Theory of Numbers, 5th e. Lonon, Englan: Oxfor University Press, [IEEE] IEEE Stanar 754 for Binary Floating Point Arithmetic, ANSI/IEEE Stanar No. 754, American National Stanars Institute, Washington DC, [Ka87] W. Kahan, Checking Whether Floating Point Division is Correctly Roune, monograph, April 11, 1987, (see home page [LM98] V. Lefevre, JeanMichel Muller, an A. Tisseran, "The Table Maker s Dilemma," IEEE Transactions on Computers V. Lefevre, an JeanMichel Muller "Worst cases for correct rouning of the elementary functions in ouble precision. " [MK85] D. W. Matula, P. Kornerup "Finite Precision Rational Arithmetic: Slash Number Systems." IEEE Transactions on Computers, Vol. C34 No. 1, January [MM00] D. W. Matula, L. D. McFearin "Number Theoretic Founations of Binary Floating Point Division with Rouning" Proceeings: Fourth Real Numbers an Computers, April 2000, pp [MM01] L. D. McFearin an D. W. Matula, "Generation an Analysis of Har to Roun Cases for Binary Floating Point Division." Proc. 15th IEEE Symposium on Computer Arithmetic. IEEE, pp [MM01a] L. D. McFearin an D. W. Matula, Selecting Test Suites for IEEE Stanar Floating Point Division." Proc. 19 th IEEE International Conference on Computer Design. IEEE, [MM02] L. D. McFearin, "A pbit Moel of Binary Floating Point Division an Square Root with Emphasis on Extremal Rouning Bounaries.", Ph. D. Dissertation, Southern Methoist University, Dallas, Texas, May [MM03] D. W. Matula, L. D. McFearin "A pxp Bit Fraction Moel of Binary Floating Point Division an Extremal Rouning Cases." Journal of Theoretical Computer Science, 291:159182, [MM04] D. W. Matula, See home page [Mu97] J.M. Muller, Elementary Functions, Algorithms an Implementations. Birkhauser, Boston, [Pa99] M. Parks, "Numbertheoretic Test Generation for Directe Rouning," Proc. 14th IEEE Symposium on Computer Arithmetic. IEEE, [SB94] H.P. Sharangpani, M. L. Barton, "Statistical Analysis of Floating Point Flaw in the Pentium Processor", Intel Corporation, [Zi91] A. Ziv. Fast evaluation of elementary mathematical functions with correctly roune last bit. ACM Transactions on Mathematical Software, (3):410423, Sept