Semi-Logarithmic Number Systems

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1 IEEE TRANSACTIONS ON COMPUTERS VOL 47 NO FEBRUARY SmiLogarithmi Numbr Sstms JaMihl Mullr Mmbr IEEE Aladr Shrba ad Araud Tissrad Abstrat W prst a w lass of umbr sstms alld SmiLogarithmi Numbr Sstms that ostitut a famil of various ompromiss btw floatigpoit ad logarithmi umbr sstms This allows trad btw th spd of th arithmti opratios ad th siz of th rquird tabls W giv arithmti algorithms (additio/subtratio multipliatio divisio) for th SmiLogarithmi Numbr Sstms ad w ompar ths umbr sstms to th lassial floatigpoit or logarithmi umbr sstms Id Trms Logarithmi umbr sstms floatigpoit arithmti INTRODUCTION T HE floatigpoit umbr sstm [6] is widl usd for rprstig ral umbrs i omputrs but ma othr umbr sstms hav b proposd Amog thm o a it: th logarithmi ad siglogarithm umbr sstms [9] [5] [4] [7] [8] [3] [0] th lvlid umbr sstm [3] [0] [] som ratioal umbr sstms [] ad som modifiatios of th floatigpoit umbr sstm [] [] Thos sstms hav b dsigd to ahiv various goals g to avoid ovrflows ad udrflows to improv th aura or to alrat som omputatios For ista th siglogarithm umbr sstm itrodud b Swartzladr ad Alpoulos [5] was dsigd i ordr to alrat th multipliatios As poitd out b th authors it aot rpla ovtioal arithmti uits i gral purpos omputrs; rathr it is itdd to ha th implmtatio of spialpurpos prossors for spializd appliatios That umbr sstm is itrstig for problms whr th rquird prisio is rlativl low ad whr th ratio of multiplis (or divids or squar roots) to adds is rlativl high Roughl spaig i suh sstms th umbrs ar rprstd b thir radi logarithms writt i fidpoit rprstatio Th multipliatios ad divisios ar prformd b addig or subtratig th logarithms ad th additios ad subtratios ar prformd usig tabls for th futios log ( + ) ad log ( ) si: % &K 'K log05 B log05 A9 log05 B log05 A log A + B = log A + log log A B = log A + log Th major drawba of th Logarithmi Numbr Sstm ariss wh a high lvl of aura is rquird If th omputatios ar prformd with bit umbrs th a straightforward implmtatio rquirs a tabl otaiig lmts Itrpolatio thiqus allow th us of JM Mullr ad A Tissrad ar with CNRS Laboratoir LIP Éol Normal Supériur d Lo 46 Allé d Itali Lo Cd 07 Fra jmmullr@lipslofr A Shrba is with Stat Thial Uivrsit Cathdral SAPU 3 Povitroflotsi prospt Kiv 5037 Urai Mausript rivd 6 Mar 996; rvisd 7 Jul 997 For iformatio o obtaiig rprits of this artil plas sd mail to: t@omputrorg ad rfr IEEECS Log Numbr smallr tabls (s [6] [] [8]) so that 3bit logarithmi umbr sstms bom fasibl with urrt VLSI thologis Our purpos i this papr is to prst a w umbr sstm that allows th us of v smallr tabls That umbr sstm will b a sort of ompromis btw th logarithmi ad th floatigpoit umbr sstms Mor atl w show a famil of umbr sstms paramtrizd b a umbr ad th sstms obtaid for th two trmal valus of ar th floatigpoit ad th logarithmi umbr sstms With som of ths umbr sstms multipliatio ad divisio will b almost as as to prform as i th logarithmi umbr sstm whras additio ad subtratio will rquir smallr tabls THE SEMILOGARITHMIC NUMBER SYSTEMS Lt b a itgr lt b a ral umbr diffrt from 0 ad dfi as th multipl of satisfig < W immdiatl fid Dfi m as: m log = = If s is th sig of w obviousl hav = s m + () () whr is a bit approimatio of log ad m is a multipliativ orrtio fator From () w ddu: m = < Now lt us boud th valu This valu is qual to O a asil show that for a Œ (0 ) +a > a l /98/$ IEEE

2 46 IEEE TRANSACTIONS ON COMPUTERS VOL 47 NO FEBRUARY 998 Usig this rsult with a = / w gt + (3) for 0 As a osqu m < + This lads to th followig two dfiitios Th diffr btw th two is th ormalizatio of m : Th boud o m rquird b th gral form is asir to h DEFINITION (Caoial Form) Lt b a positiv itgr Evr ozro ral umbr is rprstd i th Caoial form of th SmiLogarithmi Numbr Sstm (SLNS for short) of Paramtr b thr valus s m ad satisfig: s = ± is a multipl of m < = s m DEFINITION (Gral Form) Lt b a positiv itgr Evr ozro ral umbr is rprstd i th Gral form of th SLNS of Paramtr b thr valus s m ad satisfig: s = ± is a multipl of m < + = s m Th aoial form is a id of floatigpoit rprstatio with pots that ar multipls of ad a orrspodig ormalizd matissa Th rprstatio of with matissa bits i th smilogarithmi umbr sstm of paramtr will b ostitutd b s ad a fratioal bit roudig of m I prati si m < + m has a biar rprstatio of th form: bits zros Si th first + bits of m ar ow i adva thr is o d to stor thm (this is similar to th hidd bit ovtio of som radi floatig poit sstms [6]) Eatl as for ormalizd floatig poit rprstatios a spial rprstatio must b hos for zro I th followig is osidrd impliit ad w writ m ad istad of m ad Som poits d to b mphasizd: If = 0 th th smilogarithmi sstm of ordr is rdud to a bit matissa floatigpoit sstm If th th smilogarithmi sstm of ordr is rdud to a logarithmi umbr sstm Th aoial form is a ordudat rprstatio I that form omparisos ar asil prformd: If th format of th rprstatio is from lft to right ostitutd b th sig th pot whih is a multipl of ad th th matissa th omparisos ar prformd atl as if w wr omparig itgrs Th gral form is a rdudat rprstatio For ista if = th has two possibl rprstatios aml º 0 th pot ad matissa ar writt i radi ad º 00 Although th omparisos ar slightl mor diffiult to prform with th gral form this is du to th rduda w will prfr that form baus th oditio m < + is asir to h tha th oditio m < ad baus th gral form lads to simplr arithmti algorithms Awa th ovrsio from th gral form to th aoial form is asil prformd: Assum s m is i gral form Compar m with r = If m < r th th umbr is alrad rprstd i aoial form If m r th add to ad divid m b r Th obtaid rsult will b th rprstatio of i aoial form So th paramtr mas it possibl to hoos various ompromiss btw th floatigpoit umbr sstm ad th logarithmi umbr sstm Eatl as i floatigpoit arithmti thr ar various possibl roudig mods For ista if w dfi =() as th umbr obtaid b roudig m (i aoial form) to zro th w gt: = 05 = s log log Similarl w a dfi roudig towards ± ad roudig to th arst 3 THE SLNS VIEWED AS A MIXEDBASE LOGARITHMIC SYSTEM Wh implmtig a logarithmi umbr sstm o has to hoos th bas (or radi) of th sstm I th itrodutio w assumd bas two (i a umbr is rprstd b its bas logarithm) Aothr atural hoi is bas Both sstms hav pros ad os Th mai advatag of bas two is that multiplig a fidpoit umbr b a itgr powr of two rdus to a shift (assumig that this umbr is rprstd i radi) This ma sav mmor ad tim wh prformig additios or ovrsios Th mai advatag of bas lis i th fat that if is small p() < + To sum up if is a itgr is asil omputd ad if is vr small is asil omputd As poitd out b o of th rfrs th smilogarithmi umbr sstms a b viwd as a midbas logarithmi umbr sstm that uss both bass: ad Lt

3 MULLER ET AL: SEMILOGARITHMIC NUMBER SYSTEMS 47 Algorithm 4 SLNS multipliatio bits = = + zros 7 whr is vr small (lss tha ) ad is a multipl of W hav + l = + ª = = whr = l( ) Thrfor th SLNS a b viwd as a miup of two logarithmi umbr sstms a bas sstm for ad a bas sstm for W will s latr that this mas it possibl to ta bfit from th prstd abov advatags of both bass 4 BASIC ARITHMETIC ALGORITHMS Now lt us prst basi algorithms for multipliatio divisio additio subtratio ad ompariso W must oti that as soo as is largr tha + ths algorithms ad spiall th multipliatio ad divisio algorithms bom vr simpl 4 Multipliatio Assum w wat to multipl s m b s m whr ths valus ar rprstd i th SmiLogarithmi Numbr Sstm of paramtr (gral form) Algorithm 4 dsribs th multipliatio mthod PROOF OF THE ALGORITHM From 0 5 a = log m * w asil ddu log m* a log m* + thrfor Algorithm 4 SLNS divisio m m a + m m* m* m m m* = + < + < + m* m* m* + This givs m + + m* + = < = = + + If m a < th (si m/m* ): thrfor 4 Divisio a m < a < m + Assum w wat to divid s m b s m whr ths valus ar rprstd i th Smi Logarithmi Numbr Sstm of paramtr (gral form) This a b do as dsribd i Algorithm 4 Th proof of this algorithm is vr similar to th proof of th multipliatio algorithm

4 48 IEEE TRANSACTIONS ON COMPUTERS VOL 47 NO FEBRUARY 998 most + bits of a si th iflu of its lss sigifiat bits is gligibl Th additio/subtratio algorithm is th ol algorithm that rquirs th us of a larg tabl (otaiig + valus) This should b ompard to th valus that ar rquird wh implmtig a Logarithmi Numbr Sstm without itrpolatios Of ours th us of itrpolatio thiqus a rsult i substatiall smallr tabls but th tabls rquird b th SLNS a b itrpolatd as wll If a tabl with + lmts aot b implmtd + o a us two tabls with + lmts ad dompos th omputatio of m i two stps: Dfi j = + I th first stp loo up i a tabl with (j + ) addrss bits (with m m º m j+ as addrss bits) th valus a ad a satisfig: log mmm j + a = ad omput m ( ) = m a O a show that m () is Algorithm 43 SLNS additio/subtratio 43 Additio ad Subtratio Assum w wat to omput ( s m ) ± ( s m ) whr ths valus ar rprstd i th SmiLogarithmi Numbr Sstm of paramtr (gral form) Eatl as i floatigpoit arithmti th basi mthod osists of aligig th matissas (i rwritig both umbrs with th sam pot) addig th aligd matissas ad rormalizig th rsult Algorithm 43 dsribs th additio/subtratio mthod Providd that > / + th ol larg multipliatio that appars i th arithmti algorithms is th alulatio of m = m a of th additio/subtratio algorithm (this is a multipliatio of two bit itgrs) It is possibl to avoid this multipliatio b slightl modifig th algorithm: If istad of ol rturig a ad a th tabl usd also rturs a th o a omput m as (m a ) a + It is as to show that m a < () thrfor th multipliatio (m a ) a is th multipliatio of a + bit umbr b a bit umbr If > / this lads to a sigifiat rdutio i th siz of th rquird multiplir ad th tim of omputatio Morovr this mthod dos ot iras th rquird amout of mmor: W ol d + bits of a (si its most sigifiat bits ar zrod wh th ar addd to m) ad w ol d at btw ad + j+ ( Loo up i a tabl with (j + ) addrss bits (with m ) j ( m ) a m as addrss bits) th valus a ad j+ + satisfig: a = log m j m j+ m ( ) a ad omput m = M ad = a a If m th m is th matissa of th rsult whil is its pot If m < th multipl m b ad subtrat from th w omputd valu : This givs th matissa ad th pot of th rsult + If tabls of siz + ar still too larg th both prvious stps a b domposd agai If w viw th SmiLogarithmi Numbr Sstm as a midbas logarithmi umbr sstm (s Stio 3) that is if w writ: = s = s with < th th algorithm uss (w assum ): + = + ª + + itgr part fratioal part = + + I this last approimatio th fat that w us two diffrt bass is sstial: Usig radi allows us to rdu # $#

5 MULLER ET AL: SEMILOGARITHMIC NUMBER SYSTEMS 49 itgr part( ) th multipliatio b to a mr shift Ol fratioal part( ) th multipliatio b rquirs a tabl Usig radi allows us to approimat b + 44 Comparisos Assum w wat to ompar = s m ad = s m whr ths valus ar rprstd i th SmiLogarithmi Numbr Sstm of paramtr (gral form) W assum that ad ar positiv (if thir sigs ar diffrt th th ompariso is straightforward ad if both umbrs ar gativ th rquird modifiatio of th algorithm is obvious) W also assum that (if this is ot tru hag ad ) Th ompariso a b do as follows: If > th > If = th if ad ol if m m If = th multipl m b th promputd valu if > / th this multipliatio a b rdud to a additio this givs a valu m * Th if ad ol if m m 45 Covrsios 45 From th FloatigPoit Rprstatio to th SLNS Rprstatio Lt b a positiv umbr (dalig with th sigs is obvious) E rprstd i biar floatigpoit as M that is E = M M < E is a itgr W wat to ovrt to th SLNS sstm of paramtr (gral form) i to fid m ad satisfig: = m m < + has fratioal bits Assum that th biar rprstatio of M is M M º M ad dfi * M = MM M+ Th ovrsio is similar to th last two stps of th additio algorithm: ) Loo up th valus a ad a dfid blow (i th tabl with ( + ) addrss bits alrad rquird b th additio algorithm with M M º M + as addrss bits): * * M log a = ) Comput m = M a ad = E a If m th m is th matissa of th rsult whil is its pot If m < th multipl m b ad subtrat from th w omputd valu : This givs th matissa ad th pot of th rsult if > / + this last multipliatio a b rdud to a additio 45 From th SLNS Rprstatio to th Floatig Poit Rprstatio Lt b rprstd i th SLNS sstm of paramtr b m ad W wat to fid th matissa M ( M < ) ad th (itgr) pot E of th biar floatigpoit rprstatio of This a b do as follows: ) Dfi f = fra( ) loo up for f i a tabl with addrss bits (or omput it) ; ) Multipl m b f (this is th multipliatio of a bit umbr b a bit umbr) this givs a umbr M* Dfi E* = Î ; 3) M* is btw o ad ( + ) If M* < th M = M* ad E = E* If M* (this as is vr ulil to our si th uppr boud o M* is vr los to two) th M is obtaid b shiftig M* b o positio to th right ad E is qual to E* + 5 STATIC ACCURACY OF THE SEMILOGARITHMIC NUMBER SYSTEM I this stio w valuat th Maimum Rlativ Rprstatio Error (MRRE) ad th Avrag Rlativ Rprstatio Error (ARRE) [5] of th smilogarithmi umbr sstms W assum that umbrs ar rprstd i th aoial form (si th gral form is rdudat it is muh mor diffiult to dfi th rprstatio rrors of that form) W prform th omputatios for th as of th roudigtozro mod I th othr ass th omputatios ar vr similar For th valuatio of th avrag rrors w assum Hammig s logarithmi distributio of umbrs [7] [4] [8] [9] That is w assum th dsit futio P 05 = < l whr W will ompar th SLNS of paramtrs ad with a floatigpoit sstm with matissa bits (th first is ot iludd); a LNS with bits i th fratioal part of th fidpoit rprstatio 5 Maimum Rlativ Rprstatio Error (MRRE) Assum is btw o ad two W hav: =05 = log Lt us dfi D as th domai whr Î + log log / =( ) quals That is D = [ ) I that domai qual to is

6 50 IEEE TRANSACTIONS ON COMPUTERS VOL 47 NO FEBRUARY 998 TABLE ARRE AND MRRE OF THE SEMILOGARITHMIC NUMBER SYSTEMS FOR DIFFERENT VALUES OF Roudig to zro Roudig to arst MRRE ARRE MRRE ARRE Floatig Poit From this w ddu: MRRE = ma Œ (4) ª ma ma = 0 + Œ =05 Lt us stimat m = ma + + Œ[ ) + From Œ[ ) w asil ddu Œ[ ) Th uppr boud + is approimatl qual to + l Thrfor: + If > th th itrval [ ) otais at last two itgrs h m = This givs MRRE ª ma ma = = 0 + Œ If = (i if w atuall us th logarithmi umbr sstm) th m < l This givs MRRE < l() As a osqu th floatigpoit sstm ad th smilogarithmi sstm (with < ) lad to th sam valu of th MRRE whil th radi logarithmi umbr sstm has a slightl bttr MRRE that is l() Th la of otiuit btw th ass < ad = (LNS) ma sm strag It is du to th diffr i th valu of m 5 Avrag Rlativ Rprstatio Error (ARRE) W wat to valuat =05 ARRE = I d (5) l Usig th domai D dfid i th prvious stio w fid : SLNS ( ) Logarithmi b its avrag valu This ap To gt this w rpla proimatio is valid if is small ompard to I prati this holds as soo as is lss tha I D 05 Z l d ª l d D ª l + Th trmal possibl valus for ar 0 (for = ) ad (for = ) This givs (b dfiig i as ): ARRE ª Â l i+ i= Thrfor I i ARRE ª Â l i= 0 ª l ª usig ª + l This approimatio is ot valid for small valus of (sa for ) For = 0 (i for th floatigpoit rprstatio) th ARRE is qual to ª 036 l For th radi logarithmi umbr sstm th ARRE is qual to l() Tabl sums up th diffrt valus of th maimum ad avrag rlativ rprstatio rror for various ass A immdiat olusio from this tabl is that although th floatigpoit ad th logarithmi umbr sstms ar slightl bttr tha th smilogarithmi umbr sstms all ths sstms lad to approimatl th sam aura: Th ratio of th ARRE of th SLNS sstm to th ARRE of th logarithmi sstm is /l() < 4 This orrspods to log (/l()) < / bit of aura 6 CONCLUSION W hav proposd a w lass of umbr sstms alld smilogarithmi umbr sstms Th ostitut a ompromis btw th floatig poit ad th logarithmi umbr sstms: If th paramtr is largr tha / + multipliatio ad divisio ar almost as asil prformd as i th logarithmi umbr sstms whras additio ad subtratio rquir muh smallr tabls For ista with = 3 = 5 ad j = 9 w would rquir a small 4 b

MULLER ET AL: SEMILOGARITHMIC NUMBER SYSTEMS 5 ROM Th bst valu for must rsult from a ompromis: If is larg th tabls rquird for additio ma bom hug ad if is small th algorithms bom ompliatd Valus of

domai of appliatio of th smilogarithmi umbr sstms is th sam as that of th logarithmi umbr sstms: Spial purpos prossors for solvig problms whr th ratio of multiplis (or divids or squar roots) to adds

IEEE Tras Computrs D 975 Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [6] FJ Talor A Etdd Prisio Logarithmi umbr Sstm IEEE Tras Aoustis Sph Sigal Prossig vol 3 p 3 983 [7] FJ

7 MULLER ET AL: SEMILOGARITHMIC NUMBER SYSTEMS 5 ROM Th bst valu for must rsult from a ompromis: If is larg th tabls rquird for additio ma bom hug ad if is small th algorithms bom ompliatd Valus of slightl largr tha / ar probabl th bst hoi Although th smilogarithmi umbr sstms ar slightl lss aurat tha th floatigpoit ad th logarithmi umbr sstms th diffr is vr small (roughl spaig / bit of aura) Th domai of appliatio of th smilogarithmi umbr sstms is th sam as that of th logarithmi umbr sstms: Spial purpos prossors for solvig problms whr th ratio of multiplis (or divids or squar roots) to adds is rlativl high ACKNOWLEDGMENTS This is a padd vrsio of a papr that was prstd at th th Smposium o Computr Arithmti (ARITH) Bath UK Jul 995 [5] EE Swartzladr ad AG Alpoulos Th SigLogarithm Numbr Sstm IEEE Tras Computrs D 975 Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [6] FJ Talor A Etdd Prisio Logarithmi umbr Sstm IEEE Tras Aoustis Sph Sigal Prossig vol 3 p [7] FJ Talor R Gill J Josph ad J Rad A 0 bit Logarithmi Numbr Sstm Prossor IEEE Tras Computrs vol 37 o pp Fb 988 [8] PR Turr Th Distributio of Ladig Sigifiat Digits IMA J Numrial Aalsis vol pp [9] PR Turr Furthr Rvlatios o LSD IMA J Numrial Aalsis vol 4 pp [0] PR Turr Implmtatio ad Aalsis of Etdd SLI Opratios Pro 0th IEEE Smp Computr Arithmti P Korrup ad D Matula ds pp 8 6 Ju 99 [] PR Turr Compl SLI Arithmti: Rprstatio Algorithms ad Aalsis Pro th Smp Computr Arithmti MJ Irwi EE Swartzladr ad G Julli ds pp 8 5 Ju 993 [] H Yooo Ovrflow/UdrflowFr FloatigPoit Numbr Rprstatios with SlfDlimitig VariablLgth Epot Filds IEEE Tras Computrs vol 4 o 8 pp Aug 99 REFERENCES [] SF Adrso JG Earl RE Goldshmidt ad DM Powrs Th IBM 360/370 Modl 9: FloatigPoit Eutio Uit IBM J Rsarh ad Dvlopmt Ja 967 Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [] MG Arold TA Bail JR Cowls ad JJ Cupal Rdudat Logarithmi Numbr Sstms Pro Nith Smp Computr Arithmti MD Ergova ad EE Swartzladr ds pp 44 5 Sata Moia Calif Spt 989 [3] MG Arold TA Bail JR Cowls ad MD Wil Applig Faturs of IEEE 754 to Sig/Logarithm Arithmti IEEE Tras Computrs vol 4 o 8 pp Aug99 [4] JL Barlow ad EH Bariss O Roudoff Error Distributios i FloatigPoit ad Logarithmi Arithmti Computig vol 34 pp [5] WJ Cod Stati ad Dami Numrial Charatristis of FloatigPoit Arithmti IEEE Tras Computrs vol o 6 pp Ju 973 [6] D Goldbrg What Evr Computr Sitist Should Kow About FloatigPoit Arithmti ACM Computig Survs vol 3 o pp 5 47 Mar 99 [7] RW Hammig O th Distributio of Numbrs Bll Sstms Thial J vol 49 pp Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [8] H Hl Improvd Additio for th Logarithmi Numbr Sstms IEEE Tras Aoustis Sph ad Sigal Prossig vol 37 pp [9] NG Kigsbur ad PJW Rar Digital Filtrig Usig Logarithmi Arithmti Eltroi Lttrs vol 7 pp Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [0] DM Lwis A Aurat LNS Arithmti Usig Itrlavd Mmor Futio Itrpolator Pro th Smp Computr Arithmti MJ Irwi EE Swartzladr ad G Julli ds pp 9 Ju 993 [] DW Matula ad P Korrup Fiit Prisio Ratioal Arithmti: Slash Numbr Sstms IEEE Tras Computrs vol 34 o pp 3 8 Ja 985 [] S Matsui ad M Iri A Ovrflow/Udrflow Fr Floatig Poit Rprstatio of Numbrs J Iformatio Prossig vol 4 o 3 pp Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [3] FWJ Olvr A Closd Computr Arithmti Pro Eighth IEEE Smp Computr Arithmti Ma 987 Rpritd i EE Swartzladr Computr Arithmti vol IEEE CS Prss Tutorial 990 [4] T Stouraitis ad FJ Talor FloatigPoit to Logarithmi Eodr Error Aalsis IEEE Tras Computrs vol 37 pp JaMihl Mullr rivd th Egir dgr i applid mathmatis ad omputr si i 983 ad th PhD i omputr si i 985 both from th Istitut Natioal Polthiqu d Grobl Fra I 986 h joid th CNRS (Frh Natioal Ctr for Sitifi Rsarh) From 986 to 989 h was postd to th Tim3Imag Laborator Grobl ad th to th LIP Laborator Lo H tahs omputr arithmti at th Eol Normal Supériur d Lo His rsarh itrsts ilud omputr arithmti ad omputr arhittur Dr Mullr srvd as gral hairma of th 0th Smposium o Computr Arithmti (Grobl Fra Ju 99) ad as oprogram hair of th 3th Smposium o Computr Arithmti (Asilomar Califoria Jul 997) H has b a assoiat ditor of IEEE Trasatios o Computrs si 996 H is a mmbr of th IEEE Aladr Shrba rivd th Egir dgr ad th PhD dgr i omputr si from th Kiv Istitut of Tholog i 977 ad 983 rsptivl H has b with th Kiv Thial Uivrsit si 980 whr h tahs CAD His rsarh itrsts ilud paralllism ad omputr arithmti Araud Tissrad rivd th MS dgr ad th PhD dgr i omputr si from th Éol Normal Supériur d Lo Fra i 994 ad 997 rsptivl H is with th Laboratoir d l Iformatiqu du Parallélism (LIP) i Lo Fra H tahs omputr arhittur ad VLSI dsig at th Éol Normal Supériur d Lo Fra His rsarh itrsts ilud omputr arithmti omputr arhittur ad VLSI dsig

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A Review of Complex Arithmetic /0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd