+ E. Bit-Alignment for Retargetable Code Generators * 1 Introduction A D T A T A A T D A A. Keen Schoofs Gert Goossens Hugo De Mant

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1 Bit-lignment for Retargetable Coe Generators * Keen Schoofs Gert Goossens Hugo e Mant IMEC, Kapelreef 75, B-3001 Leuven, Belgium bstract When builing a bit-true retargetable compiler, every signal type must be implemente eactly as specifie, even when the wor-length of the signal oes not match the length of the available harware. Etra operations must be introuce in the algorithmic escription in orer to ensure that the remaining bits o not in$uence the atabits an to assure that signal types are correctly converte from one type to anothe~ n algorithm will be presente which generates coe to assure bit-trueness, optimise for the available harware. 1 Introuction In SP-algorithms every signal has a certain signal ~pe, inicating the number of bits in the signal, the number of bits behin the binary point, an the way of encoing (signe, unsigne, two s complement, etc..). In most algorithms, signals with a large variety of types are present. he wor length of these signals (the number of bits they contain) can be ifferent from the size of the functional unit they are mappe upon. If the wor length is bigger, multiple precision arithmetic must be u, if they are smaller, it must be ecie to what value the remaining bits (calle non-ata bits) must be set. We have to ecie how many non-ata bits we allow at the most significant bit (msb) an least significant bit (lsb) sie of the ata wor, an to what values these bits can be set in orer to avoi that they corrupt the ata bits uring arithmetic operations (the operations must remain bit-true). he allowe values of these bits will epen strongly on what kin of operations will be one with these signals. (see figure 1) he presence of ifferent types normally implies that uring computation, certain signals must be converte from one type to another. his is specifie by means of cast operations in the algorithmic speeitication. cast operation may imply that certain bits must be remove from the original signal, that etra copies of the sign-bit must be ae at the msb-sie of the ata wor, or that etra zeroes must be ae at the lsb-sie of the ata wor. El + E + B / El /~ Non-ata bits B Figure 1: two possible alignments for the aition of two 4-bit ata wors on an 8-bit aer to a an remove bits uring type changes, since they generate coe for preefine processor cores. he bit-true character is an important requirement for retargetable coe generation, because for certain applications (like filters) the actuat type of the signals is important to obtain the esire result. More-over programmable SP-cores are often use to o rapi prototyping when esigning custom application specific architectures (SICS). In orer to evaluate what the influence of the actual harware bit-withs in the SIC will be on the esign, we nee to have a bit-true coe generator. In this paper we assume an architecture moel in which comple programmable ata paths, consisting out of multiple functional units (FUS), are interconnecte by busses. n eample with two ata paths, a multiplier/accumulator MPY/CC) with a ownshifter bank an an LU with an upshifter, is shown in figure 2, 10 v : In this paper, we present a technique calle software alignment, which takes care of the bit-trueness of a esign an correctly implements type changes by aing operations to the esign. his is neee because retargetable coe generators can not simply a eicate harware *his research was sponsoreby ESPRI project 2260 SPRIE the E.U. t p~ofe~~orat the Katbolieke Univemitelt ~Uven of Figure 2: Eample architecture of a programmable processor core. O / IEEE 76

2 he research escribe in this paper is part of the CHESS retargetable coe generation project. 2 Literature survey In literature not much attention has been pai to the subject escribe above. raitional, software compilers are only concerne with type changes between integer an floating point, an between integers an oubles [4], an o not really support conversions between integers with a ifferent number of bits. Normally the only kins of integers supporte have a length which is a multiple of the register size of the target processor. For retargetable SP coe generators the importance of signal types an the problem of bit-trueness in simulation an implementation was recognise in [5], but no clear solution was presente. Some relate work, concerning harware alignment, can be foun in [1]. Harware alignment solves the same problem as software alignment but for customizable SIC processors instea of preefine programmable processors. In [1] it is assume that the interconnection network between the ata paths is not fully efine so that the compiler can generate etra wiring to change the alignment an types of the signats. n atgorithm is presente which minimises the amount of etra wiring neee for this task. he resulting architecture eactly implements the signal types as given in the original specification. In [3] an algorithm is presente which moijies the signal types from the specification, without egraing the precision of the calculations, also in orer to minimise the etra wiring neee to implement the ifferent remaining type changes. his oes not necessarily mean that the number of ifferent types is reuce, but only that the number of ifferent wor lengths in the ifferent types is reuce. he algorithm is quite useful as a preprocessing step before solving the software alignment problem iscusse in this paper. It can not, however, take the place of this algorithm, because it only optimises the types, an oes not generate a set of operations which actually implement the type changes. 3 efinitions 3.1 Signal type Every signal in a SP algorithm has a signat type which inicates how the ata bits must be interprete. If we assume that the signal is always in two s complement representation, the signal type can be enote: < w1, fp > where W1 is the number of ata bits of the signal, an ~p inicates the position of the binary point, counting from the least significant ata bit, as illustrate in figure 3. nb enotes the number of bits of the wire or register carrying the signal. 3.2 Signal alignment he alignment of a signal is efine if we know the number of non-ata bits at the msb an lsb sie of a ata wor an if we know the value of these non-ata bits. he alignment-attribute of a signal therefore consists OE he offset: this integer inicates the number of nonata bits either at the msb or at the lsb sie of the ata wor (the size of the lsb or msb etension). If one size is known, the other one can be erive since we know the length of the ata wor an the size of the harware carrying the ata. off F MSB sie El MSB 1 sie LSB sie LSB sie MSB etension LSB-etension Figure 3: signal with type <6,4> mappe on a carrier with nb=12. o the right is a more schematic representation in which the ata bits are represente by a thick line. he alignment sie: his inicates whether the offset is specifie for the msb or lsb etension. he msb etension: his inicates the value of the non-ata bits at the msb sie of the ata wor. he lsb etension: his inicates the value of the nonata bits at the lsb sie of the ata wor. For both etensions, many ifferent bit patterns are possible. In practice, only the following bit-patterns are useful: Zero-etension: all non-ata bits are set to zero. One-etension: all non-ata bits are set to one. Sign-etension: all non-ata bits are set equal to the sign bit of the ata wor. For algorithmic reasons we also efine the following: on t care etension (-et): o be use when the number of non-ata bits is zero, or when the contents of the non-ata bits are irrelevant for the correct eecution of the operation. Unefine etension (u-et): o be use if the contents-of the non-ata bits can not be classifie in any other category. (for eample, because not every bit in the etension is set to the same value) or can nor be etermine at compile time (because of, for instance, carry-rippling). his oes not mean that these bits are irrelevant to the operation that uses them. his etension can be generate for instance at the msb sie of the result of an aition. s an eample, the following alignment attribute aii = (rnsb 1 s O) means that the signal has 1 non-ata bit at the msb sie of the ata wor, of which the value is equal to the sign bit of the ata wor, while all non-ata bits at the lsb sie of the ata wor are set to zero. 77

3 3.3 lignment propagation Most operations o not allow every possible alignment for their input operans. lso for every type of operation there eists a relationship between the input an output offsets an between the input an output etensions. hese relationships are parmneterise epressions, which moel the freeom available in the selection of the actual alignment of the signals. his information is store once an for all in the library of the compiler, in which all supporte operations are eclare. In the case of the CHESS library, over 100 operations are supporte in this way. lignment propagation then means etermining the allowe alignments for every signal in the SP-algorithm base on the allowe alignments of the operans of the signal. More etaile information about alignment propagation is moele can be foun in [1] Offset propagation he relation between input an output offsets can be epresse in mathematical equations. In all practical eamples these equations are linear. Each operation contributes a number of equations equal to or less than the number of inputs of the operation, of the form: #outputs i=l UiX~ + #inputs j=l izjxj = CSt with ai an a integer an in most practical cases equal to one or zero. C!stis also atways integer Etension propagation When we want to epress the relation between the input an output etensions for a certain operation, we can not use a mathematical formulation of the same simplicity as for offsets. Instea we use lookup tables. For each operation possible on each functional unit, we require 2 tables per output (one for the msb an one for the lsb sie). In each table the value of the output etension is given for each allowe combination of input etension values. s an eanmle the Isb etension table for the the aition (table 1) is ~iven. EPOR B s s.- -- f able 1: lsb etension table for the aition he alignment tables of several operations happening one after the other cam be combine into larger alignment tables as is eplaine in [1]. s s i 3.4 lignment conflicts We have an alignment conjlict if the alignment of one operation is unacceptable as input for the net operation. he software alignment algorithm will try to fin alignments for each operation which minimise the number of alignment conflicts. However, sometimes alignment conflicts can not be avoie. In the rest of the paper techniques are presente to solve alignment conflicts. For eample, etra operations can be inserte in the SP-algotithm, calle software alignment operations, which can resolve the alignment conflicts. 3.5 Software alignment operations Some FUS can eecute operations, which can mlfy the alignment of a signal without affecting the values of its ata bits. For instance, a shift operation can be use to change the offset, while logical-or an -N operations can change the etensions. In a library these sofhvare alignment operations for the most common functional units (a superset of the harware available for the particular processor we are generating coe for) are store. Our library currently supports the following software alignment operation~ out = in + zero; out = in zero; out = in + in; out = in one; out = in V zero; out = in 63 zero; Out = in << n : Ci~; Out = in >> n : Ci~; out = in. 2m; where, V, Q << n : Ci. an >> n: ci. represent logical N, OR, EXOR, an up- an ownshift respectively. zero an one are constants with ata bits equal to O an 1 respectively, an with etensions that can be chosen such to set the esire etensions of the result. n is the shift value, which can be chosen to set the result s offse~ ci~ is the value of the bits shifte in. It can be chosen to set the result s etension. On certain functional units (like an aer-subtracter) a number of ifferent software alignment operations are possible. We can inicate this by combining the etension tables, of the possible software alignment operations into a multiple output table. his is an alignment table with for each combination of input values at most n output-values, each corresponing to a ifferent moe of the functional unit. When the most interesting output etension is finally ecie upon, the functional unit is set in the corresponing moe. In the rest of the paper we will ignore this option, in orer to reuce the compleity of the eamples, without any loss of generality. 4 Software alignment algorithm Software alignment has to assure the bit-true character of a esign. It oes this by etermining the correct alignment for every signal in the SP-algorithm, by means of alignment propagation. he propagation minimises the number of alignment conflicts, but in most practical cases, still some conflicts will remain. Remaining conflicts can be solve in two ways by the software alignment algorithm. It can try solve them by replacing eisting pass operations by software alignment operations (@ware alignment without introuction of etra cycles). If this is not sufficient it can 78

4 introuce etra operations (software alignment with introuction of etra cycles). Finally software alignment must also implement type changes, where a signal type changes from one type to another. hese three aspects of software alignment will be eplaine separately, an will be illustrate by means of a small eample, mappe upon the architecture presente in figure Software alignment without the introuction of etra cycles If we want to implement the algorithm z = (a* b) + c on the harware presente in figure 2, we can first eecute tmp == a * bon the multiplier an store tmp in regc. Net, we can compute z = tmp + c. In orer for a signat to get to its estination, it must travel through a number of FUs which are in pass moe. If we eplicitly write the pass moes (this is one alreay by the instruction selection tool of the retargetable coe generator [6]) the algorithm becomes :,z. ((a* b)@@+cq (1) pass pass pass If we now replace the pass moes by software alignment operations, we can at the same time implement alignment changes require to solve possible alignment conflicts an transport the signal from one operation to the net. z=(ab+zero>>n:cin +c*2m (2) ~~ align his equation must now be solve in the offset an etensions of the ifferent signals an constants. If it can not be solve, etra variables can be introuce, by aing etra software alignment operations, which causes an overhea of etra cycles to be eecute by the SP-atgorithm. his can happen when the ata path oes not have any FUs in pass-moe, or if these FUs in paw-moes are unusable to solve the alignment conflict. How often this happens, epens upon the harware the SP-algorithm is mappe upon. 4.2 Software alignment with the introuction of etra cycles If even with the techniques escribe in the previous section some alignment conflicts remain, etra operations must be ae in between to convert the alignment to something that is acceptable. If the conflict occurs between two operations that eecute on ifferent ata paths, the signal is simply reroute through a number of other ata paths, uring which the alignment of the signat is moifie. o etermine an efficient solution, the compiler has to look up the possible software alignment operations on these aitional ata paths. For each possible path between an ata path input an output, these operations are represente in a separate term, calle ata path term (P term). P terms are generate by the compiler in a preprocessing step. For eample, for the MPY-CC ata path in figure 2, a possible P term woul be align out = (in * (2n) + zero) >> 7n : c to inicate that an incoming signal can be multiplie with a power of 2 (which moifies the offset), then ae with zero an then ownshifte over m positions, with the shiftin bits set to c. he contents of these P terms can be prune to take into account encoing restrictions impose by the processor s controller, if two ifferent functional units can not be in a certain moe at the same time. From the list of terms, a term is chosen which can transform the conflicting alignment into an acceptable form. If no such term eists a concatenation of these terms must be use, inicating that the signal must pass through several ata paths in orer to get the correct alignment. If it can be proven that no combination of P terms will yiel a vali solution, we have an error conition. his is the case when the aition of any P term to all chains of P terms alreay compute, oes not yiel anew term which makes new combinations of input an output alignments possible. If the conflict oes not happen at ata path borers, the conflicting signal must first be eporte out of the ata path (using pass-moes, which can be replace by software alignment operations). hen the terms escribe above must be use, an finally the signal must be importe again to the position where the original conflict occurre, again using pass-moes which can be replace by software alignment operations. ssume, in the eample of equation 1 that a conflict occurs after the multiplication a* b, an that the conflict can be solve with the upshift software alignment operation: out = in << n : c~n We use the following P term on the LU-Shifter ata path: out = (in + zero) << n : c~n We woul then get the following equation: z = ((((a* b) + zero)> n, : f+ eport +zero) << ns : c~n3 \ align via P term an align *2m1 + zero) >> nz : Cj~2 + C * 2m2. \ / import an align he more software alignment operations we a, the more egrees of freeom we have to set our etensions an offset correctly. If more than one solution is possible, clearly the solution which requires the least numkr of etra cycles must be chosen. 4.3 Cast operations implemente with software alignment Figure4 shows the cast of a signal of type <4,3> with alignment =(msb 4 ) to a type <5,2> with alignment =(msb101) We can split up each cast operation in a number of elementary transformations using the following proceure (see figure 4): Step First we ientify which of the bits in the original type will still be present in the final type. 79

5 MS B a I LSB Ii!kl E! s Xcs c s / seo 1 E 1 Figure 4: cast-operation, split up in a number of elementary transformation, Steps B, C an E are to be implemente with software alignment. Shown here is the cast of a<4,3>ml=(msb 4)toa <5,2 >ali=(msbl Ol) on a 8 bit architecture. inicates a atabit. Step B: hese bits are then shifte to the position in which they must appear in the final result. Step C: hen the sign-etension at the msb sie an zero-etension at the lsb-sie are generate, if necessary. Step : he bits neee in the signal of the new type are now reay. We still have to ientify which bits are now present in the new type. level of the tree all available P terms are applie to the alignments resulting from the previous level of the tree. Branches which o not contain any new combinations of offset an etension are prune. he tree has a finite epth because there are only a limite number of combinations of etensions an offsets. Inee, the number of etensions is limite, an the offset is an integer number which has as an upper boun the size of the harware. If finally none of the branches of the tree match the initial alignment, we have an alignment conflict which can not be solve on the available harware. It is not require to calculate the entire tree. he tree is built level by level. s soon as one of the branches of the tree at a certain level contains the initial-alignment, no further levels nee to be compute. his is because the cost function which is use to evaluate the quality of ifferent solutions is simply the number of cycles they nee when implemente in the SP-algorithm. his number of cycles correspons to the epth of the tree at which the solution was foun. If more than one branch contains the targete initiakignment, that branch is selecte which contains the operations that generates the least number of resource conflicts uring scheuling. ll this is illustrate in figure 5. Step E: Finally the correct alignment etensions are generate if neee Steps an o not require any special operations, they are just internal bookkeeping. Step B is an offset change, an steps C an E are equivalent to the set(ing of alignment etensions. Steps B, C, an E can be implemente using the software alignment techniques presente in section Outline of the software alignment proceure Base on information about the available processor harware, an ehaustive list of all possible P terms is compile, Where possible, the pass operations in a esign are replace by software alignment operations. lignment attributes are checke throughout the esign for conflicts. fter this step the alignment of every signal in the esign is known. he metho for oing this is ientical to the alignment propagation technique presente in [1]. ll remaining alignment conflicts are solve using the techniques escribe in section 4.2. he cast operations are imemente using the techniques es&ibe in section-4.3. uring the latter two steps, a combination of software alignment operations has to be foun which allow us to go from one preefine alignment to another. his is one with a branch an boun metho. For each alignment conflict which requires the introuction of etra cycles an for the steps B, C an E of each cast operation, a tree is built starting from the consumption alignment (ali-cons) an builing towars the prouction alignment (-pro). t each ali con Sa3 (ali4, ali7).%3 {ali2. (ali-pro) (alill) {) {ali12}.ali7) Sa3 {ali13..ali16) Figure 5: software alignment tree from ali-cons to alipro, for the case where three software alignment operation Sal, Saz an Saq are available. cross inicates a ea-en branch, a circle inicates we have reache the estination alignment. 5 Eperiments In this section a small eperiment is presente to show how a software alignment tree is generate. From the cast eample in section 4.3 we will generate the operations necessary for step C. We assume only the following P terms are availabk out = in + zero out = in >> 1 : sign-bit out = in<<l:o an each P term can be use after itself an after each other P term. We also o not concern ourselves with importing an eporting the signal to an from ata paths. We start with a signal with alignment ali=(msb 3 u), an we want to generate a signal with ali=(msb 3 s u). 80

6 MSB:S LSB, U MSB : S of f:5 LSB:U out. in. <1 of f:4 out. in<<l MSB : s Off : ERROR MSB:X out.ln<<l MSB-et:s LSB et: u ffse MSB : s LSB : U 0ff:2 Out=ln>>l Of f:l mt=in>>l Off, ERROR LSB:U Of f:o Figure 6: Eample of a software alignment tree neee to go from ali-in =(msb 3 u) to ali-out=(msb 3 s u), if only an aition, shift-up an shift-own are available. he otte line inicates the best solution. he solution consists of 3 upshifts followe by 3 ownshifts. We buil our tree starting from the consumption alignment an we work backwars towar the prouction alignment. he resulting tree is shown in figure 6. Since the tree is built from consumption to prouction, the alignment consume by a P term is written on the right sie of the branch an the alignment prouce at the left sie. his way for a ownshift operation (which increments the offset), the lowest offset is locate at the right sie of the branch. Note that in the tree we assume that every branch, an therefore every P term has only one output etension. In reality most P terms will have more than one possible resulting alignment. his however oes not affect the number of branches in the tree. t each level in the tree we have 3 branches corresponing to each of the possible P terms. t the tirst level we see that the resulting alignment in one branch is the same as the alignment at the beginning of the branch. his branch can be prune, because any solution foun in this branch will be more costly than the final solution, because it contains a useless first step. Each new level m the tree represents a new P term in the equation, an therefore also an etra cycle neee in the SP-algorithm to eecute the alignment change. he prouction alignment is reache when the tree is 6 levels eep. t this point we ha to eamine 24 branches. Note that we can stop when the first solution is foun. ny other solution woul be more costly to implement, because it woul be foun at eeper levels in the tree an thus woul require more cycles to eecute in the SP-algorithm. By following the path in the tree from prouction alignment to consumption alignment (i.e from right to left), we can see that in orer to implement the alignment change we nee 3 upshifts followe by 3 ownshifts, in that orer. In the tree we can also fin the intermeiate alignments of the signal uring the transformation. he operations can now be ae to the SP-algorithm. 6 Conclusions he problem of bit-alignment is important in retargetable coe generation for SP, since SP algorithms contain signals of many ifferent types. he problem has been largely ignore in literature up till now. he purpose of the software alignment algorithm is to fin a bit-true mapping of the esign while minimizing the require number of etra operations. he algorithm is heuristic in nature. Future work will inclue the combination of the software alignment algorithm with the type-optimisation technique presente in [3]. References [1] K. Schoofs, G, Goossens, H. e Man, Bit-lignment in Harware llocation for Multiplee SP rchitectures, Proc EC 1993, p [2] [3] [4] [5] [6]. Lanneer, et al, rchitectural Synthesis for Meium an High hroughput Signal Processing with the new CHERL Environment, publishe in High-Level VLSI Synthesis, eite by R.Camposano an W.Wolf, Kluwer, K. Schoofs, G. Goossens, H. e Man, Signal ype Optimisation in the esign of time-multiplee SParchitectures., Proc. EC V. ho, R. Sethi, J.. Unman, Compilers, echniques an ools, ison-wesley Publishing Company, p , , Genin, J. e Moortel,. esmet, E. Van e Vele, System esign, Optimization an Intelligent Coe Generation for Stanar igital Signal Processors., Proc. ISCS 1989, p J. Van Praet, G. Goossens,. Lanneer, H. e Man, Instruction Set efinition an Instruction Selection for SIPs, Proc of High Level Synthesis Workshop, Ontario