FloatingPoint Arithmetic Algorithms in the


 Ariel Booker
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1 IEEE TRANSACTIONS ON COMPUTERS, VOL. C3, NO.1, JANUARY 1974 Citroe, Paris, as a Research Egieer workig with the cotrol of machie tools. I 1956 he joied the Societe Sciak, Paris, as Head of the Electroics Laborator, to desig ad develop circuits for weldig machies. He spet 1959 ad 1960 with the Compagie Geerale de Telegraphie sas Fil (CSF), as Head of the Techolog Departmet. From 1961 to 1963 he was a Teachig Assistat at CaregieMello Uiversit. I 1963 joied the Uiversit of Alabama, Hutsville, as a Assistat Professor of Electrical Egieerig; he became Professor i He teaches ad coducts research i the area of commuicatios ad data processig. Dr. Polge is a member of the Societe Fracaise des Electricies ad Sigma Xi. iclude optimal cotrol, digital processig of image data, ad simulatio ad aalsis of radar sstems. James M. Carswell was bor i Bishop, Calff., B.K. Bhagava was bor i Msore, Idia, o o Jauar 19, 19. Februar 15, He received the B.E. degree His major field of iterest is mechaical i electrical egieerig from the Bagalore egieerig, although he has had cosiderable experiece i electrical egieerig. For the last Uiversit, Bagalore, Idia, i 1967, the M.E. 15 ears his work has icluded trajector degree i electrical power egieerig from the Idia Istitute of Sciece, Bagalore, i 1969, aalsis; i particular, digital simulatio of sixdegreeof freedom trajectories, Mote Carlo ad the Ph.D. degree from Southem Methodist aalsis of trajector variables, ad special Uiversit, Dallas, Tex., i problems i flight mechaics. Util Jue 1973 Presetl he is a Research Associate with he was with the Uiversit of Alabama, Hutsthe Research Isitute of the Uiversit of Alabama, Hustsville. His curret iterests ville. Curretl he is with North America Aviatio, Dowe, Calif. FloatigPoit Arithmetic Algorithms i the Smmetric Residue Number Sstem EISUKE KINOSHITA, HIDEO KOSAKO, MEMBER, IEEE, AND YOSHIAKI KOJIMA, SENIOR MEMBER, IEEE I. INTRODUCTION THE residue umber sstem is a iteger umber sstem. At preset, the techiques kow make it icoveiet to represet fractioal quatities. It is to be desired that umbers with fractioal parts ca be hadled as easil as itegers i the residue umber sstems. A few studies o the floatigpoit arithmetic i the residue Idex TermsCclic mixedradix sstem, expoet part, sstem have bee published [11, [1. I these reports a power floatigpoit arithmetic algorithms, floatigpoit represetatio, matissa, ormalized form, umber of precisio, smmetric residue of or 10 is used as a expoet. This paper deals with floatigpoit arithmetic with a umber sstem. expoet which is a product of moduli i the smmetric residue umber sstem. This umber sstem has the followig Mauscript received September 10, 1971; revised August 4, ) fidig the additive iverse of a residue digit is The authors are with the Departmet of Electroics, Uiversit of advatages: ) sig detectio b mixedradix coversio is eas, fairl Osaka Prefecture, Osaka, Japa. AbstractThe residue umber sstem is a iteger umber sstem ad is icoveiet to represet umbers with fractioal parts. I the smmetric residue sstem, a ew represetatio of floatigpoit umbers ad arithmetic algorithms for its additio, subtractio, multiplicatio, ad divisio are proposed. A floatigpoit umber is expressed as a iteger multiplied b a product of the moduli. The proposed sstem assumes existece of ecessar coversio procedures before ad after the computatio.
2 10 eas, ad 3) the result of scalig is rouded to the closest iteger. A cclic mixedradix sstem will be itroduced first. The, based o this sstem, a ew expressio of floatigpoit umbers ad algorithms for additio, subtractio, multiplicatio, ad divisio will be described i terms of ormalized operatios. II. NUMBER SYSTEM I order to provide the appropriate foudatios ad motivatio for the ormalized floatigpoit format proposed here, a cclic mixedradix sstem will be itroduced first. Cclic MixedRadix Sstem Cosider a weighted umber sstem i which a real umber is expressed i the form ± h E w.w. i= { ci } is ad a.h is the most sigificat digit of the umber. From a practical poit of view the series (1) should be approximated b its appropriate partial sum. Here we assume that there are give radices ml, m,, m. We deote b ii the least positive (iteger) remaider of the divisio i/, ad b [i/] the largest iteger less tha or equal to i/, i is a iteger. The the approximate value of a give real umber ma be represeted i the form (1) a set of permissible digits, I wil a set of weights, I+1 ± E iwi i=l with digits i succeedig positios, ai = ai i 1 ili +1 ) < a + i + 1 < mi i + 1 I wi= M /ml il mm () (3) (4) (lii 0) (I il =0) (5) ad et is the least sigificat digit, ad M = flh1 i. If a umber X is expressed i the form (), we call X a umber of "precisio." It should be oted that umbers i the biar or the decimal sstem are expressed b the form (). Number sstems i which the weights are ot powers of the same radix are called mixedradix sstems. A cclic mixedradix sstem is defied as a mixedradix sstem cosistig of all umbers of the form (). I the followig discussio it will be assumed that the radices are odd positive umbers so chose that ml < m <.. < m ad the digits a, are restricted so that Mlil +1M1 l l +(6) S t1il +1 6 IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 istead of (4). As a example, Table I lists the weights ad the permissible digits of the cclic mixedradix sstem with ml = 3,m = 5,adm3 = 7. Now cosider the umbers of precisio, i a cclic mixedradix sstem, expressed i the form Let e= [i/] I+1 z i=l a.w.. I I S= ill (9) a.=i±+j1i +1=ls+jlI +±1 I = 1,,,(). (10) The, these umbers ca be represeted i the followig form: kxme XM k is a iteger such that ki< (M1),M= rh1i, e is a iteger, ad s s = 11J MP. Ms= 1, i= 1 Proof: From (5) ad (9), ad moreover, usig (10), w1=m=m w. I=m .m im w I ji e=tiv i ma W0, v=l O the other had, b (3) ad (IO),5 Hece I11+j1=OL 1+;1 I + 1 l+1 p = aaj. E Oliwi = E rl+j_ 1Wl+j_l i=lj= Here cosider the expressio (11). Let The, sice the a0 (s #0) (s = 0). (j= 15,,, ). (7) (8) ±t+ E 'a( I mi)mems. (11) /ji k = ±a+ Ea( i0)a j= 1 v=l v are itegers such that, 3,,,, ). (1)
3 KINOSHITA et al.: FLOATINGPOINT ARITHMETIC TABLE I WEIGHTS AND PERMISSIBLE DIGITS OF CYCLIC MIXEDRADIX SYSTEM WITH ml = 3, M = 5, AND m3 = 7( = 3) i [ i/l i If < < 7x5x3 = 5= <a <43 7 x51 x3 = 4= <ct <1 7lx5lx3 = 3= <a_ = < 7 x <_a < <cL <1 1 = 0 = <ac < 3 03«a <3 5x <a <1 7 x 5 x <ca < 7 x 5 x 3 s 13_<a <3 7 x 5X 3 : 7x::3 we obtai kli< Therefore, m 1 a. mi a. a. I _1 )+ E ± ) mu) lk< (m m " ma 1I). a a1 a I view of (10) it is easil see that the sequece o1, a, is a permutatio of 1,,,. Hece, Ikl (M1). This completes the proof. It should be oted that (7) ca geerate egative umbers as well as positive umbers or zero because of the restrictio (6). Normalized FloatigPoit Format I floatigpoit operatio proposed here, the set of umbers cosists of 0 ad the set of all umbers of the form kxme XM k is a iteger, I(M/m8+ 1)S IkI.(M1), (s 0) 1(MM + 1)6 k <(M1) (s = ) w ag (13) ormalized b the coditio that the most sigificat digit aa i (1) is ot zero, ad e is a iteger ragig betwee E ad E, sa. We call the iteger k ad the pair (e, s) the matissa ad the expoet part, respectivel, of the floatig umber (13). Exceptioall, zero is defied as follows: O=OXM XMo. Theorem: The proposed ormalized floatigpoit format is uique. Proof: B cotradictio. It suffices to prove the theorem for positive umbers. For a positive umber A, let A = k1meims, ad A = kmems with k, # k, el 0 e, or s1 : s. Assume that the theorem is false; the it must be possible to fid a A = k Me, Ms, = kmems. with kl, el, si, k, e, ad s meetig the precedig restrictios. Without loss of geeralit we ma assume that e = el + Ae, Ae is a iteger such that Ae > 1. The it follows that which i tur implies Sice ad the Further, sice the I 11 (M/m + s i km =kmam I 1 8 k1m8 MAe =~ 1 kms S Me >M. (14) (i = 1, ) (M1l)Ms AfAe S M + m (Ml)MS m 8 1) < ki"im.<i( 1) +1 <m M AM MAe<M But this cotradicts the assumptio (14). Hece, I S<M, si el = e. Next we assume that s1 < s without loss of geeralit. From (15),  ki M1 k M + I m 8 11 (15)
4 1 Sice the m M ki <m k 8 O the other had, whether sl = O or #0, ki k = M1S +1 ms + ms but hece This cotradicts (16). Hece m 8al+ m~~~~~ 1 8sl+ 8s >m ki k 8 SI = S. 8~~~ (16) Cosequetl, k, =k. This completes the proof of the theorem. For a real x we deote b fl (x) either of the two umbers of the form (13) which miimize fl(x) x I, except that if the fl(lxx)= MMM<M I x I < MM (M+ m +l1) I1a(M + I MeM+, (si 1) +IMl)Me+,M0 (s=l) The value fl(x) will be called a ormalized floatigpoit represetatio of x. We defme the rage of floatigpoit umbers to be the iterval D= [M1 MEM, M MEMl ~~ 1~j Let fl(x) = kmem. be the ormalized floatigpoit represetatio of x for x e D, Ixl > ((M/m,) + IWE. The, sice m MeM < Ixl< MAme, (SOO) s MeM8. lxi< MMl M8, (S=#O) (17) IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 it is easil see that e, s, ad k ca be determied i the followig maer: e = [logmmf Ixl 1. The iteger s is foud as follows. First compute 1x1 Me+l This quatit is used i the relatioship Gi_ M m. to obtai G,, G,, etc. This iterative procedure is cotiued util [Gi] = 0. If this occurs o the ith (i = 0, 1, ), the Fiall k is defied b s = i. k= ( \MeMS Here the R refers to correct roudig i fixedpoit arithmetic. If e is outside the iterval E < e < E, we shall ol sa that a overflow or a uderflow has occurred ad shall ot proceed further. Coversio to the Residue Represetatio For the iteger k i (13) the least remaider i absolute value whe divided b mi ma be computed. This quatit, is referred to as the smmetric residue of k deoted b /k/mi, mod mi, ad the radices mi are called bases or modull. For a give set of moduli the residues of k ma be formed ito a tuple Ik/ml J, klm Iklm This tuple is called the smmetric residue represetatio of k. The iteger 1k/rm is called the ith smmetric residue digit M1 Ml to ma be uiquel represeted. Now cosider the coversio from a fixedradix sstem such as decimal or biar to the residue sstem. The iteger k is specified i a fixedradix sstem as k=d,r±+d, r1 + +dir+do r is the radix ad 0 < di 6 r 1. The, takig this expressio modulo mi, the followig equatios are obtaied: km =/dllr'lm + d_1 Ir''/mi+ +d,irlmi+ do/ml (i= 1,,,).
5 KINOSHITA et al.: FLOATINGPOINT ARITHMETIC Thus, if the powers of r modulo mi are directl available from the memor, lk/m, ma be computed b repetitive additio (modulo mi,) of those powers of r. Before goig ito the mai argumet, residue iteractig operatios of mixedradix coversio, baseextesio, ad scalig will be briefl described. (For detailed iformatio o these operatios, see [3].) MixedRadix Coversio Mixedradix coversio process is used to covert from the residue sstem to the mixedradix sstem. The particular mixedradix represetatio of iterest here is of the form k=am1m...m am1 + a1 the ai are the mixedradix digits which are to be determied b this procedure. A iteger i the rage M1 M1 to ma be represeted i this form ad hece this represetatio has the same rage as a residue sstem of moduli ml, m,, m Ṫhe mixedradix coversio is a fudametal operatio, from which other importat operatios such as base extesio, relative magitude compariso, ad sig determiatio ca be derived. I the smmetric residue sstem, the sig of a iteger is give b the sig of the most sigificat ozero digit of the mixedradix expressio of the umber. B successivel subtractig ai ad dividig b mi i residue otatio, all the oi ca be determied, startig with al. A simple example of this procedure is give i the Appedix. Base Extesio Base extesio is used to fid the residue digits for a ew set of moduli, give the residue digits relative to aother set of moduli. I most cases, oe or more moduli are added to the origial base. The procedure is a mixedradix coversio with a additioal fial step. A simple example of base extesio is give i the Appedix. Scalig I covetioal fixedradix arithmetic, scalig up or dow b a power of the radix is simpl a series of right or left shifts ad is a fast ecoomical operatio. I the residue umber sstem, because multiplicatio is a simple operatio, scalig up is o problem. Scalig dow is, i geeral, a difficult operatio. However, it is eas to scale dow b a product of the mi; for example, permissible divisors are mi1 X mi X m5 or m but ot mi1 X M X m5 or M. This restricted operatio is referred to as scalig. A simple example of scalig is give i the Appedix. Ill. FLOATINGPOINT ARITHMETIC Let X = kxmexmsx ad Y = kymeyms be two floatigpoit umbers i the rage D, the subscripts x ad represet X ad Y, respectivel. We defie b Z = fl(x * Y) the desired result of a floatigpoit operatio, Z is of ormalized form ad the * smbol represets additio, subtractio, multiplicatio, or divisio of two floatigpoit umbers. Adjustmet ad Overflow or Uderflow i Matissa Part I the proposed floatigpoit umber sstem, some adjustmet of a matissa part is eeded before computatio i multiplicatio or divisio as well as i additio or subtractio. Overflow or uderflow occurs a time the matissa of X * Y would fall, i absolute value, outside the iterval I1= +[f( I ),(M1)] (s * ) I m ) '(m 1)] (s= 0). The, special methods are eeded to detect the occurrece of overflow or uderflow ad to ormalize the result. Additio or Subtractio We assume, without loss of geeralit, that e e M XM >M YM sx S (18) The expoet parts of X ad Y must be made equal before additio or subtractio. To alig the expoet parts, we rewrite Y as ex Y=KM xm S x e e km Y xm K M The, we ca express the sum or differece ofx ad Y as e k M ZM z sz kz =kxkr± e5=e,s =s It should be oted that, i geeral, the floatigpoit umber is ot i the ormalized form at this poit. Now i order to kow how to get KR, cosider the relatioship betwee ex ad e, or sx ad s. 1) IfexZ>e +, I K I< I k IJMMs IMs < I k I(Mm ) < 1 /(m) Hecex Hece KR =0. ) If ex = e+ l ad sx >sy, x 13
6 14 Hece IKI= Ik 1M M /MI.I<IkI/M< 1/. SY KR =0. 3) If ex=e +ladsx <s, km1m K=k M1M8 /M8 = m S Sx m0ml.. ms m8 MIM "' M k m0m 1 s m s +1ms+ m x mo = 1. I the smmetric residue sstem the result of scalig is rouded to the closest iteger, hece KR is obtaied.... b scalig k b mom msxms+ims + m Here, b the assumptio that all the moduli are odd, KR K < 1/. 4) If ex = e ad sx > s, k K=k M /M_ =Y S x msy+1 ms + msx Hece KR is obtaied b scalig k b m,+1m8y+ m.sx 5) If e = e ad sx =s, the evidetl KR =k. Note that the ol remaiig case ex= e ad sx < s ever occurs b virtue of the assumptio (18). After makig ecessar arragemet for a aligmet of expoet parts as is previousl stated, we compute k. = kx + KR. The, sice 0 < Ikz8I M  1, a overflow or a uderflow ma have occurred, ad a test is required for overflow or uderflow detectio. Cosider the smmetric residue sstem with moduli mi, M,  mm ad a redudat odd modulus m +, m+ is pairwise relativel prime to all the other moduli ad satisfies the coditios M1.ia(Mm1+1), mi <M+1' (19) Suppose we have the residue represetatios of kx ad k for all moduli, icludig ml,l If k_ is expressed i its mixedradix form, we have kz a+1m1m m + tmim m1 +*±+ami1 +a 1. Defie au to be the most sigificat mixedradix digit which is ot equal to zero. The' subscript u will be equal to some iteger from + 1 through 1. The, u is equal to if ad ol if 4((M/m) + 1) < S 4(M  1). Hece a overflow has occurred if u = + 1. If u <, the (M ) 4(S ) which implies that a uderflow has occurred. The ol remaiig case is if u =. I this case it is ot possible to IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 kow from u aloe if a uderflow has occurred, ad aother test is required. A method for the uderflow detectio requires the availabilit of the quatities '((M/m,z) + l)(sz = 1,,,  1) modulo m1(i = 1,,, ). If s = 0, ((M/msz) + 1) < Ik I, which implies that o uderflow has occurred. The quatities :((M/m, ) + 1) are costats ad ca be permaetl stored i the residue form. If sz # 0, Ikz 1((M/msz) + 1) is formed i its residue code ad coverted to its mixedradix form. If the sig of the most sigificat ozero digit of this form is egative, a uderflow has occurred. After these tests, kz is replaced, if ecessar, b the iteger obtaied b scalig or multiplig kz b a factor, ad the expoet part (ez, sz) is corrected. I scalig kz, it is desirable to choose m,z+l as the factor, takig accout of the defiitio of MsZ, ad icrease sz b 1. If sz has reached, sz is set to zero ad ez is icreased b 1. It ca be show easil that kz falls ito the iterval I after beig scaled b m,z+i. I multiplig kz b a factor, if u <, sice kzmemzms be expressed as kz/ s + H mimz.=s +U+ 1 s +u z Hl M. j=1i ca we choose I=SZ+U+1 mi as the factor, decrease ez b 1 ad icrease sz b u, i is a modulo umber whe i >. If sz has reached, sz is set to zero ad ez is icreased b 1. The Sz quatities r1=.sz+u+1zi are costats determied b s ad u ad ca be permaetl stored i the residue form. If u =, we choose msz as the factor ad decrease sz b 1. The quatities msz(sz = 1,,,  1) are costats ad ca be permaetl stored i the residue form. It ca be show that kz is ormalized after a fiite umber of this multiplicative iteratios. I the case of u = 1 or u = ol oe iteratio is required. Thus we have the desired result fl(x ± Y) = kzmezms8 which is the floatigpoit represetatio of X ± Y. Multiplicatio It will be assumed, without loss of geeralit, that sx > s We first cosider the case of s = 0 or 1. I this case, the product of X ad Y ca be expressed i the form e k M zm z sz k =k k M, e =e +e, s =s. z x s z x s z x Obviousl this is a uormalized umber. If s = 0, i m + I) Ikz8<(M 1)
7 KINOSHITA et al.: FLOATINGPOINT ARITHMETIC ad if s = 15 If we suppose that (M+l)d(M ±+1) the multiplicative overflow occurs alwas i computig kz. To cope with this difficult, cosider the smmetric residue sstem with redudat odd moduli m+ M+3, m+p i additio torl,,, m+l, m.(i = +,+3, + p) are pairwise relativel prime to all the other moduli ad satisf the coditios p k(m l)i1.<(mm +1m+ + 1), m+ <m+ < < + (0) Before computatio of k5, we exted the base to iclude M+, m,3, , m+p for the residue represetatios of kx ad k1. The multiplier M1 required for gettig k5 whe s = I is costat ad ca be permaetl stored i the residue form. Now to ormalize k5, defie ao.to be the most sigificat ozero mixedradix digit of k_. The, if u is greater tha, overflow has occurred, ad therefore kz must be replaced b the result of scalig k_ b a factor. The scalig factor is chose as follows. For u such that u > +, we choose Il[j=ZzJ+.l as factor ad icrease sz b u , i is a modulo umber whe i >. If s has reached, sz is set to zero ad e_ is icreased b 1. It ca be show that the value of u decreases to u = + 1 after a fiite umber of this scalig iteratios whe u > +. If u has reached + I, kz is scaled b msz+1 ad sz is icreased b 1. If sz has reached, sz is set to zero ad ez is icreased b 1. It ca be easil show that ad I m m 1<1(, 4 + I + I m < Ik M m m 1 z 4 1, 11 ( M s +1 S (MRm +1 11(M 1< (>1) 1)M which implies a overflow ma have occurred. At this poit, to detect a overflow Ik_,I  (M  1) is formed i its residue code ad coverted to its mixedradix form. If the sig of the most sigificat ozero digit of this form is positive, a overflow has occurred, the kz is scaled b m, +I ad the expoet part is corrected. The quatit (M  1) ca be permaetl stored i the residue form modulo mp(i 1,, ; + 1). i Thus we have the desired result fl(x X Y) = kzmezm S5 the case of s = 0 or 1. A similar procedure is applicable to the case of s # 0 ad s # 1, usig the followig modificatios. I this case we ca express the product of X ad Y as e kzm ZM kek k = (= x1 R' e=e±+e±,is =S First of all, we exted the base to iclude m+, m+3, m,p pfor the residue represetatios of kx ad k. ad the compute kxk. Scalig kxk b ms +1msY+  m, we have kz. It ca be easil show that i ) + 1) < ki. Hece kz must be ormalized b the aforemetioed scalig algorithm. Divisio Cosider the divisio of X b Y. If Y = 0, divisio is ot defied. Otherwise, sice kx I, k <. (M 1), it is impossible to get the desired precisio of the quotiet kx/k b merel dividig kx b k. To icrease the precisio, we defie the followig fuctio A which is a multiplier to kx. 1) IfsX >s ad s = 0, X kxm e e 1 MX Y M. Hece A is defied b Hece ) IfsX > s ads *O, B= A =M. k BCM Xx XMex Y k Ms BC HI mi i=s+ 1 k kmcsx BCM e e kmc MX k BC e e  1M k MX A =BX C. s x X i=sxs+l Note that if sx = s ad s * 0, the A = M. Mi.. (1) () is
8 16 3) If sx<s, kxms B = x x e e Y km B s km B x s8 e e = km MX = Hece A is defied b km B x sx e e 1 X Mx kxms BD e e  k x~ Mx Y M D= f[ Mi. A=M i=+s S +1 x s x XBXD. +x  The quatit A is a fuctio of sx ad s ad is a product of moduli. It ca be easil see that A has a maximum Am = (Mm  1 * (+ 3)/ M(+1)/ ( is odd max (mm_ * m 1m )' ( is eve)  I (/)+1 whe SX = ad s = [1]. It follows that IkxIA < 4(M  l)amax. Therefore, the previous redudat moduli m+, m+3, +p are chose so that the further satisf the followig coditio: ~(Ml)Amm(Mx +1m+ im+p 1). (4) Before the computatio, we exted the base to iclude Im+p for the residue represetatios of kx m+, m+39 ad k. The we form the product of kx ad A ad divide kxa b k. The value of A is a costat determied b sx ad SY ad ca be permaetl stored i the residue form. It is to be desired that the iteger quotiet kz = (kxaik)r should be foud b applig the divisio algorithm proposed b the authors [4]. The equatios required for calculatio of expoet part is give as follows. Ifsx >s, from (1) ad (), ad if sx < s from (3), ez =ex e 1 sz =s, s, e5 =e e , sz =±s s. Thus we have a uormalized floatig umber IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 e kzmzms It ca be show that o uderflow has occurred ad a overflow ma have occurred i computig k.a/k. If a overflow has occurred, the same discussio ad procedure as described i the case of multiplicatio ma be applied to ormalizatio of kz. IV. SAMPLE RESIDUE NUMBER SYSTEM As a sample residue umber sstem, cosider the smmetric residue sstem cosistig of moduli ml = 13, m = 17,m3 = 19, m4 = 3, ad ms = 9 (M = ). The, for the matissa of the floatigpoit umber (13), the iterval of (3) defiitio is, i absolute value, [ 4889, ] [10771, ] [ 8375, ] [ 73704, ] [ 60886, ] (s = 0) (s= 1) (S= ) (s= 3) (s= 4). The redudat moduli which satisf (19), (0), ad (4) are chose asm6= 31,m7= 37,m8= 41, m9 = 43, ad mlo = 11. I floatigpoit add operatio, uderflow detectio requires a table of the quatities Lsz = T((M/is) + l)(sz = 1,, 3, 4). This ca be accomplished b storig, i a special memor, a table such as Table II. The smbol (( ))5 stads for the residue represetatio, for istace, ((L l))5 <> I/Li/ l,, / l /1m t  Normalizatio requires tables of HlsZ m. (for u K:) ad msi (for u = ) such as Tables III ad IV. For floatigpoit multiplicatio, the quatit M1 must be = 1. This ca be provided to get kz = kxkms whe s accomplished b storig ((ml ))1 0 i a special memor. Overflow detectio requires the residue represetatio of (0' (M  )) 6. To have the desired precisio of a quotiet i floatigpoit divisio, we must provide a table of multipliers A, such as Table V. V. ALGORITHMS From Sectio III, we ma summarize the algorithms for floatigpoit residue arithmetic as follows. Suppose kx ad k to be represeted b smmetric residue digits ((kx)) + 1 ad (() FloatigPoit Add or Subtract Operatio Assume that X = kxmexmsx ad Y = k3,meyms are the auged (miued) ad the added (subtrahed), respectivel, ad that 4eXMe > s,em 1) Compute Ae = exe. IfAe >, fl(x ± Y) = X. ) If Ae = 1, check whether SX > s. If this is true fl (X_ Y) = X. Otherwise, scale k b mom, * Ms Ms+lMs+ m ad let the result to be KR.
9 KINOSHITA et al.: FLOATINGPOINT ARITHMETIC 17 TABLE II TABLE COMPOSED OF LS = W/ms ) + 1) s z L S z 1 ((L1))5 ((L))5 3 ((L3))5 4 ((L4))5 TABLE III TABLE COMPOSED OF HiI=Zz++l m1 Z ((mm3m4m5 1)6 (( M34m5m1 ) 6 (m4m5m1m ))6 ((m5m1mm3 ))6 (( 1m m3m4 ))6 (( m3m4m5 ) 6 (Im4r5mr1 ) 6 ( m5m1m ) 6 r ml mr3 ) 6 ( mm3m4 ) 6 ((r3r4m5 6 (( m5r1 ) 6 r(( 1r ) 6 ( mr3 ) 6 r( 3m4 ) 6 4 (( m5 ( 6 ((m1 )6 ((r ( m 6 (( m3 ( ) 6 m((4 ( 6 4 TABLE IV TABLE COMPOSED OF MS s z i s z 1 ((ri1) F, 6 ((I ) 3 (( m3 )) 6 4 ((m4 )) 6 TABLE V TABLE COMPOSED OFA VALUES S X (( A )) ( 0 ( 0 m mm I m 1 (( r 3r4i5 ) 10 (( 10 ( r 3m4 5 ) 10 (t 3r4r5 )) 34m 10 (( 3 m m3m4m5 )) 10 ((1mm ( 1(mlm3 4mb )) 10 1( i )) mS ) 10 (( r3r4 mii )5 10 ((m3i4ms ) 10 (( m14ff5 )) 10 r 1 r4r5 )) ) 10 4 (( m m  m m mm34 5 ) 10 (( m (( (( :4 A 10 _ (~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3) If Ae =0, check whether sx = s. If so, set KR = k. Otherwise (if sx > s ), scale k b m +.m..+ m. ad let the result to be KR. 4) Compute kz = kx ± KR ad set ez = ex, sz = sx. 5) Fid the mixedradix digits cal, a,.',a+, of kz which are associated with the smmetric residue sstem ml, m, m+l1 If all the digits a(i= 1,,, + 1) are zero, set ez = 0, sz = 0. 6) Otherwise, deote b au the most sigificat ozero digit. If u = + 1, scale kz b m,z+l ad icrease sz b 1. If sz has reached, set sz = 0 ad icrease ez b 1. 7) If u <, multipl k b HIzs=! fl+u+imi which ca be
10 18 read from the table b meas of u ad sz, decrease ez b 1, ad icrease sz b u. The, if sz <, retur to Step 5). If sz >, after icreasig ez b 1 ad decreasig sz b, retur to Step 5). 8) If u =, check whether sz = 0. If so, kz has bee alread ormalized. If s, * 0, form kz  ((M/ms ) + 1) ad fid the sig of the result through the mixedradix coversio, ((M/ms3) + 1) is read from a memor. If the sig is egative, multipl kz b msz ad decrease sz b 1. Thus we ca get the desired result fl (X ± Y) = kzarzmszm. Example 1: I the same residue sstem i Sectio IV, suppose we add X= M'M4 ad Y = M1M. Examiig these two umbers we fid ex e = 0 ad sx > sy. The matissa of Y must be scaled b m3m4. This gives K KR=(19( X3 )R= Addig kx ad KR, we get kz = , the we set ez = 1, sz = 4. Replacig kz b (kz/m_)r ad correctig the expoet part, we get k = 48308, e =, s = 0. The desired result the is fl ( M1 M M1M) = 48308MMo0. FloatigPoit Multipl Operatio Assume that X = kxmexmsx ad Y = k1mems are the multiplicad ad the multiplier, respectivel, ad that sx > s,. 1) Perform the baseextesio operatio o kx ad k ad fid the smmetric residue digits /kxlm i ad /k/m i (i = +,+3,, +p). ) Set ez = ex + e, sz = sx. 3) If s = 0, compute kz = kxk, ad if s = 1, compute kz = kxkml, ml is fetched out from a memor. Otherwise, icrease ez b 1, compute the product kxk ad scale this result b m3 +1msY+  m to get kz = (kxk/(ms +IMs8+ )R 4) Fid the mixedradix digits  a1, a, a+p of k. If all the digits cxi are zero, set ez = 0, sz = 0. Otherwise deote b au the most sigificat ozero digit. 5) If it > +, replace kz b (kz/ily=z+lmi)r ad icrease sz b u , i is a modulo umber whe i >. If sz =, set s to zero ad icrease ez b 1. Retur to Step 4). 6) If u = + 1, replace kz b (kic/m+1 )R ad icrease sz b 1. If sz =, set sz to zero ad icrease ez b 1. Compute Ikz Ī(M  1) ad fid the sig of this result b meas of the mixedradix coversio, f(m  1) is fetched out from a memor. If the sig is positive, replace kz b (kz/msz+l )R ad icrease sz b 1. If sz =, set sz to zero ad icrease ez b 1. Thus we ca get the desired result fl (X X Y) = kimezm. IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 Example : I the precedig residue sstem, suppose we multipl X = MAM4 = fl (.9979 X 1010) b Y= M5MO = fl (6.656 X 107). First, we set e =0+(5)= 5,s =4. Sice s =0, k = k k = z x We ca easil kow that the value of u is 10. Hece, scalig kz b M, we have kz = 16549, ez =4, sz = 4. This matissa is i the ormalized form. Thus we have fl (310418AfM4 X M5MO) = 16549M4M4. FloatigPoit Divide Operatio Assume that X = kxmexmsx ad Y= kmem3 are the divided ad divisor, respectivel. 1) Perform the base extesio o kx ad k ad fid the smmetric residue digits /kx/l ad /k,/m (i= +, + 3,, + p). ) Multipl kx b A, which ca be read from a memor b meas ofsx ad s, ad set ez = exe 1, sz = S s. 3) If sx < s, decrease ez b 1 ad icrease sz b. 4) Compute kz = (kxa/k)r (check if k, = 0; if so, divisio is ot defied). Hereafter, follow from Step 4) to Step 6) i the multipl algorithm. Example 3: Let X= 16549M4M4 Y= 87955M4M4 = fl ( X 1016). The, sice sx = s= 4,A =M. ez 4(4)1=1, sz= 0. ki= ( A) = The subscript a for is 6. Hece, scalig kz b ml, we have kz = , ez =1 sz = 1. Thus we have fl (16549M4M4 /(87955M4M4)) = M'Ml. VI. CONCLUSION A ew residue floatigpoit umber expressio ad arithmetic algorithms based o it have bee proposed. The mai advatage of the residue sstem is that i additio, subtractio, ad multiplicatio a particular digit of the result is depedet ol o the correspodig operad digits. This propert elimiates carries from digit to digit for all three arithmetic operatios previousl metioed ad removes the eed for partial product formatio i multiplicatio. I cotrast to covetioal digital sstems,
11 KINOSHITA et al.: FLOATINGPOINT ARITHMETIC these three operatios ca be executed i the same time as required for a additio operatio, while the followig operatios have bee criticized as awkward i residue computers: detectig sig ad the occurrece of overflow, relativemagitude compariso of two umbers, extedig the rage of the umber sstem, ad shiftig the operad. For residue arithmetic, the use of matrix uits is most attractive for implemetig additio, subtractio, ad multiplicatio modulo mi. This method of implemetatio has the advatage of the absece of carries iside as well as amog residue digits. However, a disadvatage arises whe mechaizig the matrices. I geeral, Ioutofmi codig is used to drive the matrix; therefore, the umber of compoets required for each matrix is large. The use of the smmetric residue otatio permits foldig the arithmetic matrices, which results i a decrease i matrix compoet. A desirable form of logical implemetatio would be the use of the arithmetic matrices realized with LSI techiques. Such matrix uits would give extremel high speed to the three residue operatios. The proposed expressio is basicall differet from the other residue floatigpoit umber expressio. It represets a residue floatigpoit umber as a iteger multiplied b a product of the moduli. As a result the ewl proposed expressio has the advatage of efficiet applicatio of residue iteractig operatios to the floatigpoit arithmetics. These iteractig operatios are split ito base extesio, scalig, ad mixedradix coversio ad are closel coected with covetioal residue arithmetic operatios. There is much possibilit of this advatage beig regarded coversel as disadvatage because these iteractig operatios are usuall take as time cosumig. The proposed algorithms have the obvious advatage of etire performace of multiplicatio ad divisio as well as additio ad subtractio, uder a certai coditio of the moduli used. The algorithms make good use of the iteractig operatios which ca cope with the awkward operatios metioed previousl. The mai disadvatage of the residue iteractig operatios is that these operatios are relativel time cosumig because of the cascaded process used. This disadvatage, however, would be little worth cosideratio whe the aforemetioed LSI logic matrices are used, because the residue iteractig operatios cosist of residue additio, subtractio, ad multiplicatio. Coversio before or after the computatio will be discussed i aother paper uder preparatio. The material of this paper forms a ivestigatio of the applicabilit of residue umber sstem to the floatigpoit arithmetics. APPENDIX A. Example ofmixedradix Coversio For ml = 7,m = l1, m3 = 13, adm4 = 17, fid the smmetric mixedradix digits of X < ), 5,, 1. Solutio Residue represetatio of X Subtract cea Multipl b / I1/ri Subtract ax Multipl b /Al /m. I Subtract 13 Moduli a1 = a = l 03 = 6 At this poit, it should be apparet that the remaiig mixedradix digit a4 must be zero. Hece the process ca be termiated. Thus we have X= 6(7 X I1) + (1)(7) + (l) = 457. I the precedig, the quatit of the form /m is called the multiplicative iverse of a mod mi. The quatit //m exists if ad ol if (a, mi) = 1 ad lalm/ i 0 ad satisfies mi 16I/ a mr B. Example ofbaseextesio < ad /t/lm. X a/mi= 1. Give the residue represetatio X = 37 , 41 for the base with moduli 7 ad 11, fid /X/1 3 ad /X/l 7. Solutio Residue represetatio of X Subtract ail Multipl b /7/rm i Subtract a The Moduli a,1= = / X /X/l 3 + 4/13 = 0 / X/X11 7+ /17 =0. 19
12 0 IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 Hece REFERENCES I X /X/l 3/13 =4, [11 A. Svoboda, Digitale Iformatioswadler. Brauschweig, Germa: Vieweg ad Soh, A. Sasaki, "The basis for implemetatio of additive operatios i the residue umber sstem," IEEE Tras. Comput., vol. C17, pp , Nov [3] N.S. Szabo ad R.I. Taaka, Residue Arithmetic ad its Applicatios to Computer Techolog. New York: McGrawHill, [4] E. Kioshita, H. Kosako, ad Y. Kojima, "Geeral divisio i the smmetric residue umber sstem," IEEE Tras. Comput., vol. C, pp , Feb X /X/I 7/17 =. Cosequetl, /X/ 3 = /X/l 7 =3. C Example of Scalig For ml = 7, m =11, m3 = 13, a I m4 = 17, scalex= {1, 5, 1, 8 b the scale fa ctor 11 X 17. Deote the result b Z. Eisuke Kioshita was bor i Osaka, Japa, i 193. He received the B.S. degree i mathematics from Hiroshima Uiversit, Hiroshima, Japa, i He taught mathematics for two ears at a high school i Osaka, Japa, ad sice 196 has 1 bee a Research Assistat i the Departmet of Electroics, Uiversit of Osaka Prefecture, Solutio Moduli Residue represetatio of X Subtract/X/I1 =51 Multipl b /la /m. Subtract x/ =/8 / Multipl b /'m i Eter 0 ito missig colums for extesio of base Subtract Multipl b / 71/r  3 Subtract Osaka. His curret research iterests lie i the areas of digital sstems ad computer arithmetics Mr. Kioshita is a member of the Istitute of Electrical Egieers of Japa ad the Iformatio Processig Societ of Japa. dih?f sg, l ji'z'/ ~~~~Hideo Kosako (S'58M'6) was bor i Osaka, Japa, o November 15, He received the B.E. ad M.E. degrees i electrical egieerig ftom the Uiversit of Osaka Prefecture, Osaka, Japa, i 1953 ad 1957, respectivel, ad the Ph.D. degree i electrical commuicatio Zeigieerig from Osaka Uiversit, Osaka, i He joied the facult of Osaka Uiversit i ZZZZ g1960. Sice gsi ZgProfessor i 1961 he has bee a Associate the Departmet of Electroics, Uiversit of Osaka Prefecture. Durig he was a Visitig Professor of Electrical Egieerig at the Uiversit of Arizoa, Tucso. His teachig ad research iterests iclude hbrid computig sstems ad simulatios. Dr. Kosako is a member of the Istitute of Electroics ad Commuicatio Egieers of Japa. The / X /Z/II + 1/1 I = 0 ad / X/Z/I + /17 = 0. Hece /Z/, = 4 ad /Z/1 7 = 3. Therefore, the residue represetatio of 6873/(11 X 17) 37 is 1,4,,3[.Note that 686/(11 X 17) ad hece it was rouded to 37, the closest iteger;rather tha to ACKNOWLEDGMENT The authors would like to thak the referees for their ecouragemet ad ver helpful suggestios. Yoshiaki Kojima (SM'57) was bor i Osaka, Japa, o September 9, 193. He received the B.E. ad Ph.D. degrees i electrical egieerig from the Toko Istitute of Techolog, Toko, Japa, i 1946 ad 1961, respectivel. From 1946 to 1950 he was with the Sharp Corporatio, Japa. I 1950 he joied the staff of the Uiversit of Osaka Prefecture, Osaka, ad sice 1963 he has bee a Professor i the Departmet of Electroics. His preset research iterests iclude the areas of logical operatio sstems. Dr. Kojima is a member of the Istitute of Electroics ad Commuicatio Egieers of Japa, the Iformatio Processig Societ of Japa, ad the Istitute of Electrical Egieers of Japa.
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