Bit-Alignment for Retargetable Code Generators

Transcription

1 Bitlinmnt fo Rtatabl Co Gnatos Kon choofs Gt Goossns Huo Man y IMEC, Kaplf 75, B3 Luvn, Blium bstact Whn builin a bittu tatabl compil, vy sinal typ must b implmnt xactly as spcifi, vn whn th wolnth of th sinal os not match th lnth of th availabl hawa. Exta opations must b intouc in th aloithmic sciption in o to nsu that th mainin bits o not influnc th atabits an to assu that sinal typs a coctly convt fom on typ to anoth. n aloithm will b psnt which nats co to assu bittunss, optimis fo th availabl hawa. Intouction In Paloithms vy sinal has a ctain sinal typ, inicatin th numb of bits in th sinal, th numb of bits bhin th binay point, an th way of ncoin (sin, unsin, two s complmnt, tc..). In most aloithms, sinals with a la vaity of typs a psnt. h wo lnth of ths sinals (th numb of bits thy contain) can b iffnt fom th siz of th functional unit thy a mapp upon. If th wo lnth is bi, multipl pcision aithmtic must b us; if thy a small, it must b ci to what valu th mainin bits (call nonata bits) must b st. W hav to ci how many nonata bits w allow at th most sinificant bit (msb) an last sinificant bit (lsb) si of th ata wo, an to what valus ths bits can b st in o to avoi that thy coupt th ata bits uin aithmtic opations (th opations must main bittu). h allow valus of ths bits will pn stonly on what kin of opations will b on with ths sinals. (s fiu ) h psnc of iffnt typs nomally implis that uin computation, ctain sinals must b convt fom on typ to anoth. his is spcifi by mans of cast opations in th aloithmic spcification. cast opation may imply that ctain bits must b mov fom th oiinal sinal, that xta copis of th sinbit must b a at th msbsi of th ata wo, o that xta zos must b a at th lsbsi of th ata wo. In this pap, w psnt a tchniqu call softwa alinmnt, which taks ca of th bittunss of a sin an coctly implmnts typ chans by ain opations to th sin. his is n bcaus tatabl co natos can not simply a icat hawa his sach was sponso by EPRI pojct 226 "PRIE" of th E.. y Pofsso at th Katholik nivsitit Luvn + Nonata bits Fiu : two possibl alinmnts fo th aition of two 4bit ata wos on an 8bit a to a an mov bits uin typ chans, sinc thy nat co fo pfin pocsso cos. h bittu chaact is an impotant quimnt fo tatabl co nation, bcaus fo ctain applications (lik filts) th actual typ of th sinals is impotant to obtain th si sult. Moov poammabl Pcos a oftn us to o api pototypin whn sinin custom application spcific achitctus (ICs). In o to valuat what th influnc of th actual hawa bitwiths in th IC will b on th sin, w n to hav a bittu co nato. In this pap w assum an achitctu mol in which complx poammabl ata paths, consistin out of multipl functional units (Fs), a intconnct by busss. n xampl with two ata paths, a multipli/accumulato (MPY/CC) with a ownshift bank an an L with an upshift, is shown in fiu 2. I N P E L ata path << B M P Y C +/ + Functional nit (F) Fiu 2: Exampl achitctu of a poammabl pocsso co. >> O P Publish in : Poc. 7th IEEE/CM Int. ymp. on HihLvl ynthsis, NiaaaonthLak, May 994. Copyiht IEEE 994.

2 h sach scib in this pap is pat of th CHE tatabl co nation pojct. 2 Litatu suvy In litatu not much attntion has bn pai to th subjct scib abov. aitional, softwa compils a only concn with typ chans btwn int an floatin point, an btwn ints an oubls [4], an o not ally suppot convsions btwn ints with a iffnt numb of bits. Nomally th only kins of ints suppot hav a lnth which is a multipl of th ist siz of th tat pocsso. Fo tatabl P co natos th impotanc of sinal typs an th poblm of bittunss in simulation an implmntation was conis in [5], but no cla solution was psnt. om lat wok, concnin hawa alinmnt, can b foun in []. Hawa alinmnt solvs th sam poblm as softwa alinmnt but fo customisabl IC pocssos insta of pfin poammabl pocssos. In [] it is assum that th intconnction ntwok btwn th ata paths is not fully fin so that th compil can nat xta wiin to chan th alinmnt an typs of th sinals. n aloithm is psnt which minimiss th amount of xta wiin n fo this task. h sultin achitctu xactly implmnts th sinal typs as ivn in th oiinal spcification. In [3] an aloithm is psnt which moifis th sinal typs fom th spcification, without ain th pcision of th calculations, also in o to minimis th xta wiin n to implmnt th iffnt mainin typ chans. his os not ncssaily man that th numb of iffnt typs is uc, but only that th numb of iffnt wo lnths in th iffnt typs is uc. h aloithm is quit usful as a ppocssin stp bfo solvin th softwa alinmnt poblm iscuss in this pap. It can not, howv, tak th plac of this aloithm, bcaus it only optimiss th typs, an os not nat a st of opations which actually implmnt th typ chans. 3 finitions 3. inal typ Evy sinal in a P aloithm has a sinal typ which inicats how th ata bits must b intpt. If w assum that th sinal is always in two s complmnt psntation, th sinal typ can b not: < wl; fp > wh wl is th numb of ata bits of th sinal, an fp inicats th position of th binay point, countin fom th last sinificant ata bit, as illustat in fiu 3. nb nots th numb of bits of th wi o ist cayin th sinal. 3.2 inal alinmnt h alinmnt of a sinal is fin if w know th numb of nonata bits at th msb an lsb si of a ata wo an if w know th valu of ths nonata bits. h alinmntattibut of a sinal thfo consists of: h offst: this int inicats th numb of nonata bits ith at th msb o at th lsb si of th ata wo (th siz of th lsb o msb xtnsion). If on siz is known, th oth on can b iv sinc w know th lnth of th ata wo an th siz of th hawa cayin th ata. Offst FP WL MB si LB si MB si LB si MBxtnsion LBxtnsion Fiu 3: sinal with typ <6,4> mapp on a cai with nb=2. o th iht is a mo schmatic psntation in which th ata bits a psnt by a thick lin. h alinmnt si: his inicats whth th offst is spcifi fo th msb o lsb xtnsion. h msb xtnsion: his inicats th valu of th nonata bits at th msb si of th ata wo. h lsb xtnsion: his inicats th valu of th nonata bits at th lsb si of th ata wo. Fo both xtnsions, many iffnt bit pattns a possibl. In pactic, only th followinbitpattns a usful: Zoxtnsion: all nonata bits a st to zo. Onxtnsion: all nonata bits a st to on. inxtnsion: all nonata bits a st qual to th sin bit of th ata wo. Fo aloithmic asons w also fin th followin: on t ca xtnsion (xxt): o b us whn th numb of nonata bits is zo, o whn th contnts of th nonata bits a ilvant fo th coct xcution of th opation. nfin xtnsion (uxt): o b us if th contnts of th nonata bits can not b classifi in any oth catoy, (fo xampl, bcaus not vy bit in th xtnsion is st to th sam valu) o can not b tmin at compil tim (bcaus of, fo instanc, cayipplin). his os not man that ths bits a ilvant to th opation that uss thm. his xtnsion can b nat fo instanc at th msb si of th sult of an aition. s an xampl, th followin alinmnt attibut: ali = (msb s ) mans that th sinal has nonata bit at th msb si of th ata wo, of which th valu is qual to th sin bit of th ata wo, whil all nonata bits at th lsb si of th ata wo a st to zo. Publish in : Poc. 7th IEEE/CM Int. ymp. on HihLvl ynthsis, NiaaaonthLak, May 994. Copyiht IEEE 994.

3 3.3 linmnt popaation Most opations o not allow vy possibl alinmnt fo thi input opans. lso fo vy typ of opation th xists a lationship btwn th input an output offsts an btwn th input an output xtnsions. hs lationships a paamtis xpssions, which mol th fom availabl in th slction of th actual alinmnt of th sinals. his infomation is sto onc an fo all in th libay of th compil, in which all suppot opations a cla. In th cas of th CHE libay, ov opations a suppot in this way. linmnt popaation thn mans tminin th allow alinmnts fo vy sinal in th Paloithm bas on th allow alinmnts of th opans of th sinal. Mo tail infomation about alinmnt popaation is mol can b foun in [] Offst popaation h lation btwn input an output offsts can b xpss in mathmatical quations. In all pactical xampls ths quations a lina. Each opation contibuts a numb of quations qual to o lss than th numb of inputs of th opation, of th fom: #outputs i= #inputs a i x i + j= a j x j = Cst with a i an a j int an in most pactical cass qual to on o zo. Cst is also always int Extnsion popaation Whn w want to xpss th lation btwn th input an output xtnsions fo a ctain opation, w can not us a mathmatical fomulation of th sam simplicity as fo offsts. Insta w us lookup tabls. Fo ach opation possibl on ach functional unit, w qui 2 tabls p output (on fo th msb an on fo th lsb si). In ach tabl th valu of th output xtnsion is ivn fo ach allow combination of input xtnsion valus. s an xampl th lsb xtnsion tabl fo th th aition (tabl ) is ivn. POR POR B s x s x s s x x abl : lsb xtnsion tabl fo th aition h alinmnt tabls of sval opations happnin on aft th oth can b combin into la alinmnt tabls as is xplain in []. 3.4 linmnt conflicts W hav an alinmnt conflict if th alinmnt of on opation is unaccptabl as input fo th nxt opation. h softwa alinmnt aloithm will ty to fin alinmnts fo ach opation which minimis th numb of alinmnt conflicts. Howv, somtims alinmnt conflicts can not b avoi. In th st of th pap tchniqus a psnt to solv alinmnt conflicts. Fo xampl, xta opations can b inst in th Paloithm, call softwa alinmnt opations, which can solv th alinmnt conflicts. 3.5 oftwa alinmnt opations om Fs can xcut opations, which can moify th alinmnt of a sinal without affctin th valus of its ata bits. Fo instanc, a shift opation can b us to chan th offst, whil loicalor an N opations can chan th xtnsions. In a libay ths softwa alinmnt opations fo th most common functional units (a supst of th hawa availabl fo th paticula pocsso w a natin co fo) a sto. Ou libay cuntly suppots th followin softwa alinmnt opations: out = in + zo; out = in + in; out = in _ zo; out = in? zo; out = in ^ on; out = in zo; out = in n : c in ; out = in n : c in ; out = in 2 m ; wh ^, _,, n : c in an n : c in psnt loical N, OR, EOR, an up an ownshift spctivly. zo an on a constants with ata bits qual to an spctivly, an with xtnsions that can b chosn such to st th si xtnsions of th sult. n is th shift valu, which can b chosn to st th sult s offst; c in is th valu of th bits shift in. It can b chosn to st th sult s xtnsion. On ctain functional units (lik an asubtacto) a numb of iffnt softwa alinmnt opations a possibl. W can inicat this by combinin th xtnsion tabls, of th possibl softwa alinmnt opations into a multipl output tabl. his is an alinmnt tabl with fo ach combination of input valus at most n outputvalus, ach cosponin to a iffnt mo of th functional unit. Whn th most intstin output xtnsion is finally ci upon, th functional unit is st in th cosponin mo. In th st of th pap w will ino this option, in o to uc th complxity of th xampls, without any loss of nality. 4 oftwa alinmnt aloithm oftwa alinmnt has to assu th bittu chaact of a sin. It os this by tminin th coct alinmnt fo vy sinal in th Paloithm, by mans of alinmnt popaation. h popaation minimiss th numb of alinmnt conflicts, but in most pactical cass, still som conflicts will main. Rmainin conflicts can b solv in two ways by th softwa alinmnt aloithm. It can ty solv thm by placin xistin pass opations by softwa alinmnt opations (softwa alinmnt without intouction of xta cycls). If this is not sufficint it can Publish in : Poc. 7th IEEE/CM Int. ymp. on HihLvl ynthsis, NiaaaonthLak, May 994. Copyiht IEEE 994.

4 intouc xta opations (softwa alinmnt with intouction of xta cycls). Finally softwa alinmnt must also implmnt typ chans, wh a sinal typ chans fom on typ to anoth. hs th aspcts of softwa alinmnt will b xplain spaatly, an will b illustat by mans of a small xampl, mapp upon th achitctu psnt in fiu oftwa alinmnt without th intouction of xta cycls If w want to implmnt th aloithm z = (a b) + c on th hawa psnt in fiu 2, w can fist xcut tmp = a b on th multipli an sto tmp in c. Nxt, w can comput z = tmp + c. In o fo a sinal to t to its stination, it must tavl thouh a numb of Fs which a in pass mo. If w xplicitly wit th pass mos (this is on alay by th instuction slction tool of th tatabl co nato [6]) th aloithm bcoms : pass z = ((a b) + ) pass + c pass If w now plac th pass mos by softwa alinmnt opations, w can at th sam tim implmnt alinmnt chans qui to solv possibl alinmnt conflicts an tanspot th sinal fom on opation to th nxt. z = (a b + zo {z } alin ) n : c in {z } alin + c 2 m alin his quation must now b solv in th offst an xtnsions of th iffnt sinals an constants. If it can not b solv, xta vaiabls can b intouc, by ain xta softwa alinmnt opations, which causs an ovha of xta cycls to b xcut by th Paloithm. his can happn whn th ata path os not hav any Fs in passmo, o if ths Fs in passmos a unusabl to solv th alinmnt conflict. How oftn this happns, pns upon th hawa th Paloithm is mapp upon. 4.2 oftwa alinmnt with th intouction of xta cycls If vn with th tchniqus scib in th pvious sction som alinmnt conflicts main, xta opations must b a in btwn to convt th alinmnt to somthin that is accptabl. If th conflict occus btwn two opations that xcut on iffnt ata paths, th sinal is simply out thouh a numb of oth ata paths, uin which th alinmnt of th sinal is moifi. o tmin an fficint solution, th compil has to look up th possibl softwa alinmnt opations on ths aitional ata paths. Fo ach possibl path btwn an ata path input an output, ths opations a psnt in a spaat tm, call ata path tm (P tm). P tms a nat by th compil in a ppocssin stp. Fo xampl, fo th MPYCC ata path in fiu 2, a possibl P tm woul b: out = (in (2 n ) + zo) m : c () (2) to inicat that an incomin sinal can b multipli with a pow of 2 (which moifis th offst), thn a with zo an thn ownshift ov m positions, with th shiftin bits st to c. h contnts of ths P tms can b pun to tak into account ncoin stictions impos by th pocsso s contoll, if two iffnt functional units can not b in a ctain mo at th sam tim. Fom th list of tms, a tm is chosn which can tansfom th conflictin alinmnt into an accptabl fom. If no such tm xists a concatnation of ths tms must b us, inicatin that th sinal must pass thouh sval ata paths in o to t th coct alinmnt. If it can b povn that no combination of P tms will yil a vali solution, w hav an o conition. his is th cas whn th aition of any P tm to all chains of P tms alay comput, os not yil a nw tm which maks nw combinations of input an output alinmnts possibl. If th conflict os not happn at ata path bos, th conflictin sinal must fist b xpot out of th ata path (usin passmos, which can b plac by softwa alinmnt opations). hn th tms scib abov must b us, an finally th sinal must b impot aain to th position wh th oiinal conflict occu, aain usin passmos which can b plac by softwa alinmnt opations. ssum, in th xampl of quation that a conflict occus aft th multiplication"ab", an that th conflict can b solv with th upshift softwa alinmnt opation: out = in n : c in W us th followin P tm on th Lhift ata path: out = (in + zo) n : c in W woul thn t th followin quation: z = ((((a b) + zo) n : c {z in } xpot an alin +zo) n 3 : c {z in3 } alin via P tm 2 m + zo) n 2 : c in2 {z } impot an alin + c 2 m2 : h mo softwa alinmnt opations w a, th mo s of fom w hav to st ou xtnsions an offst coctly. If mo than on solution is possibl, claly th solution which quis th last numb of xta cycls must b chosn. 4.3 Cast opations implmnt with softwa alinmnt Fiu 4 shows th cast of a sinal of typ < 4; 3 > with alinmnt =(msb 4 x x) to a typ < 5; 2 > with alinmnt =(msb ) W can split up ach cast opation in a numb of lmntay tansfomations usin th followin pocu (s fiu 4): tp : Fist w intify which of th bits in th oiinal typ will still b psnt in th final typ. Publish in : Poc. 7th IEEE/CM Int. ymp. on HihLvl ynthsis, NiaaaonthLak, May 994. Copyiht IEEE 994.

5 MB LB B C C E E Fiu 4: castopation, split up in a numb of lmntay tansfomation; tps B, C an E a to b implmnt with softwa alinmnt. hown h is th cast of a < 4; 3 > ali=(msb 4 x x) to a < 5; 2 > ali=(msb ) on a 8 bit achitctu. "" inicats a atabit. tp B: hs bits a thn shift to th position in which thy must appa in th final sult. tp C: hn th sinxtnsion at th msb si an zoxtnsion at th lsbsi a nat, if ncssay. tp : h bits n in th sinal of th nw typ a now ay. W still hav to intify which bits a now psnt in th nw typ. tp E: Finally th coct alinmnt xtnsions a nat if n tps an o not qui any spcial opations, thy a just intnal bookkpin. tp B is an offst chan, an stps C an E a quivalnt to th sttin of alinmnt xtnsions. tps B, C, an E can b implmnt usin th softwa alinmnt tchniqus psnt in sction Outlin of th softwa alinmnt pocu Bas on infomation about th availabl pocsso hawa, an xhaustiv list of all possibl P tms is compil. Wh possibl, th pass opations in a sin a plac by softwa alinmnt opations. linmnt attibuts a chck thouhout th sin fo conflicts. ft this stp th alinmnt of vy sinal in th sin is known. h mtho fo oin this is intical to th alinmnt popaation tchniqu psnt in []. ll mainin alinmnt conflicts a solv usin th tchniqus scib in sction 4.2. h cast opations a implmnt usin th tchniqus scib in sction 4.3. uin th latt two stps, a combination of softwa alinmnt opations has to b foun which allow us to o fom on pfin alinmnt to anoth. his is on with a banch an boun mtho. Fo ach alinmnt conflict which quis th intouction of xta cycls an fo th stps B, C an E of ach cast opation, a t is built statin fom th consumption alinmnt (alicons) an builin towas th pouction alinmnt (alipo). t ach lvl of th t all availabl P tms a appli to th alinmnts sultin fom th pvious lvl of th t. Banchs which o not contain any nw combinations of offst an xtnsion a pun. h t has a finit pth bcaus th a only a limit numb of combinations of xtnsions an offsts. In, th numb of xtnsions is limit, an th offst is an int numb which has as an upp boun th siz of th hawa. If finally non of th banchs of th t match th initial alinmnt, w hav an alinmnt conflict which can not b solv on th availabl hawa. It is not qui to calculat th nti t. h t is built lvl by lvl. s soon as on of th banchs of th t at a ctain lvl contains th initialalinmnt, no futh lvls n to b comput. his is bcaus th cost function which is us to valuat th quality of iffnt solutions is simply th numb of cycls thy n whn implmnt in th Paloithm. his numb of cycls cospons to th pth of th t at which th solution was foun. If mo than on banch contains th tat initialalinmnt, that banch is slct which contains th opations that nats th last numb of souc conflicts uin schulin. ll this is illustat in fiu 5. a a3 {ali} {ali2, ali ali3} a2 cons {ali4,... ali7} a a2 a3 a a2 a3 a a2 a3 {ali2} {ali8} {ali2,ali3} a {ali3,ali4} {ali9} {ali} {ali6} a a2 a3 a2 a3 a a2 a3 {alipo} {ali3,ali5} {ali2..ali7} {alipo} {ali} {ali2} {ali3..ali6} Fiu 5: softwa alinmnt t fom alicons to alipo, fo th cas wh th softwa alinmnt opation a, a 2 an a 3 a availabl. coss inicats a an banch, a cicl inicats w hav ach th stination alinmnt. 5 Expimnts In this sction a small xpimnt is psnt to show how a softwa alinmnt t is nat. Fom th cast xampl in sction 4.3 w will nat th opations ncssay fo stp C. W assum only th followin P tms a availabl: out = in + zo out = in : sin bit out = in : an ach P tm can b us aft itslf an aft ach oth P tm. W also o not concn ouslvs with impotin an xpotin th sinal to an fom ata paths. W stat with a sinal with alinmnt ali=(msb 3 x u), an w want to nat a sinal with ali=(msb 3 s u). Publish in : Poc. 7th IEEE/CM Int. ymp. on HihLvl ynthsis, NiaaaonthLak, May 994. Copyiht IEEE 994.

6 MBxt:s LBxt:u offst:3 out=in<< out=in>> Off:4 off:3 out=in<< out=in>> out=in<< out=in>> Off:5 Off:4 off:3 off:3 Off: out=in<< out=in>> out=in<< out=in>> MB:x Off: Off:ERROR Off:5 Off:4 Off: out=in<< out=in>> MB:x Off: out=in<< out=in>> MB:x Off: MB:x Off:ERROR MB:x MB:x Off: MB:x Off: MB:x out=in>> MB:x Off: out=in<< MB:x Off:3 Fiu 6: Exampl of a softwa alinmnt t n to o fom aliin =(msb 3 x u) to aliout=(msb 3 s u), if only an aition, shiftup an shiftown a availabl. h ott lin inicats th bst solution. h solution consists of 3 upshifts follow by 3 ownshifts. W buil ou t statin fom th consumption alinmnt an w wok backwas towa th pouction alinmnt. h sultin t is shown in fiu 6. inc th t is built fom consumption to pouction, th alinmnt consum by a P tm is wittn on th iht si of th banch an th alinmnt pouc at th lft si. his way fo a ownshift opation (which incmnts th offst), th lowst offst is locat at th iht si of th banch. Not that in th t w assum that vy banch, an thfo vy P tm has only on output xtnsion. In ality most P tms will hav mo than on possibl sultin alinmnt. his howv os not affct th numb of banchs in th t. t ach lvl in th t w hav 3 banchs cosponin to ach of th possibl P tms. t th fist lvl w s that th sultin alinmnt in on banch is th sam as th alinmnt at th binnin of th banch. his banch can b pun, bcaus any solution foun in this banch will b mo costly than th final solution, bcaus it contains a uslss fist stp. Each nw lvl in th t psnts a nw P tm in th quation, an thfo also an xta cycl n in th Paloithm to xcut th alinmnt chan. h pouction alinmnt is ach whn th t is 6 lvls p. t this point w ha to xamin 24 banchs. Not that w can stop whn th fist solution is foun. ny oth solution woul b mo costly to implmnt, bcaus it woul b foun at p lvls in th t an thus woul qui mo cycls to xcut in th Paloithm. By followin th path in th t fom pouction alinmnt to consumption alinmnt (i. fom iht to lft), w can s that in o to implmnt th alinmnt chan w n 3 upshifts follow by 3 ownshifts, in that o. In th t w can also fin th intmiat alinmnts of th sinal uin th tansfomation. h opations can now b a to th Paloithm. 6 Conclusions h poblm of bitalinmnt is impotant in tatabl co nation fo P, sinc P aloithms contain sinals of many iffnt typs. h poblm has bn laly ino in litatu up till now. h pupos of th softwa alinmnt aloithm is to fin a bittu mappin of th sin whil minimisin th qui numb of xta opations. h aloithm is huistic in natu. Futu wok will inclu th combination of th softwa alinmnt aloithm with th typoptimisation tchniqu psnt in [3]. Rfncs [] K. choofs, G. Goossns, H. Man, "Bitlinmnt in Hawa llocation fo Multiplx P chitctus", Poc EC 993, p [2]. Lann, t al, "chitctual ynthsis fo Mium an Hih houhput inal Pocssin with th nw CHERL Envionmnt", publish in "HihLvl VLI ynthsis", it by R.Camposano an W.Wolf, Kluw, 99. [3] K. choofs, G. Goossns, H. Man, "inal yp Optimisation in th sin of timmultiplx Pachitctus.", Poc. EC994. [4]. V. ho, R. thi, J.. llman, "Compils, chniqus an ools", isonwsly Publishin Company, p344247, [5]. Gnin, J. Mootl,. smt, E. Van Vl, "ystm sin, Optimization an Intllint Co Gnation fo tana iital inal Pocssos.", Poc. IC 989, p [6] J. Van Pat, G. Goossns,. Lann, H. Man, "Instuction t finition an Instuction lction fo IPs", Poc of Hih Lvl ynthsis Wokshop, Ontaio, 994. Publish in : Poc. 7th IEEE/CM Int. ymp. on HihLvl ynthsis, NiaaaonthLak, May 994. Copyiht IEEE 994.