Performing Arithmetic Operations on Round-to-Nearest Representations

Transcription

1 1 Performing Arithmetic Operations on Rond-to-Nearest Representations Peter Kornerp, Member, IEEE Jean-Michel Mller, Senior Member, IEEE Adrien Panhalex Abstract Dring any composite comptation there is a constant need for ronding intermediate reslts before they can participate in frther processing. Recently a class of nmber representations denoted RN-Codings were introdced, allowing an n-biased rondingto-nearest to take place by a simple trncation, with the property that problems with doble-rondings are avoided. In this paper we first investigate a particlar encoding of the binary representation. This encoding is generalized to any radix and digit set; however radix complement representations for even vales of the radix trn ot to be particlarly feasible. The encoding is essentially an ordinary radix complement representation with an appended rond-bit, bt still allowing ronding to nearest by trncation and ths avoiding problems with doble-rondings. Conversions from radix complement to these rond-to-nearest representations can be performed in constant time, whereas conversion the other way in general takes at least logarithmic time. Not only is ronding-to-nearest a constant time operation, bt so is also sign inversion, both of which are at best log-time operations on ordinary s complement representations. Addition and mltiplication on sch fixed-point representations are first analyzed and defined in sch a way that ronding information can be carried along in a meaningfl way, at minimal cost. The analysis is carried throgh for a compact (canonical) encoding sing s complement representation, spplied with a rond-bit. Based on the fixed-point encoding it is shown possible to define floating point representations, and a sketch of the implementation of an FPU is presented. Index Terms Signed-digit, rond-to-nearest, constant-time ronding and sign-inversion, floating-point representation, dobleronding. 1 Introdction In a recent paper [1] a class of nmber representations denoted RN-Codings were introdced, the RN standing for rond to nearest, as these radix-β, signeddigit representations have the property that trncation yields ronding to the nearest representable vale. PeterKornerp is with University of Sothern Denmark,Odense, Denmark, kornerp@imada.sd.dk; Jean-Michel Mller is with CNRS-LIP-Arénaire, Lyon, France, Jean-Michel.Mller@ens-lyon.fr; Adrien Panhalex is with École Normale Spériere de Lyon, Lyon, France Adrien.Panhalex@ens-lyon.fr Manscript received October 16, 009, revised Febrary 1, 010 They are based on a generalization of the observation that certain radix representations are known to posses this property, e.g., the balanced ternary (β = 3) system over the digit set{ 1, 0, 1}. Another sch representation is obtained by performing the original Boothrecoding [] on a s complement nmber into the digit set{ 1, 0, 1}, where it is well-known that the nonzero digits of the recoded nmber alternate in sign. To distingish between sitations where we are not concerned with the actal encoding of a vale, we shall here se the notation RN-representation. We shall in Section II (extracted from [1]) cite some of the definitions and properties of the general RN- Codings/representations. However, we will in particlar explore the binary representation, e.g., as obtained by the Booth recoding; the ronding by trncation property, inclding the featre that the effect of one ronding followed by another ronding yields the same reslt, as wold be obtained by a single ronding to the same precision as the last. Section III analyzes conversions between RNrepresentations and s complement representations. Conversion from the latter to the former is performed by the Booth algorithm, yielding a signeddigit/borrow-save representation in a straightforward encoding, which for an n-digit word reqires n bits. It is then realized thatn + 1 bits are sfficient, providing a simpler alternative encoding consisting of the bits of the trncated s complement encoding, with a rond-bit appended, termed the canonical encoding. Despite being based on a s complement encoding, it is observed that sign-inversion (negation) is a constant time operation on this canonical encoding. Conversion the other way, from RN-representation in this encoding into s complement representation (essentially adding in the rond-bit) is realizable by a parallel prefix strctre. Section IV generalizes the canonical representation to other radices and digit sets, showing that for even vales of the radix the encodings employing radixcomplement representations are particlarly feasible. TC , copyright IEEE

2 Section V then analyzes possible implementations of addition and mltiplication on fixed-point RNrepresented nmbers. [3] discssed implementations of these basic operations based on the signed-digit representation of RN-coded nmbers, whereas we here exploit the canonical encoding, which seems to be more convenient. Since it trns that there are two possible encodings of the reslt of an arithmetic operation, interpretations of the encodings as intervals may be sed to niqely define sms and prodcts in a consistent way. Section VI sketches how a floating point RN-representation may be defined and the basic arithmetic operations of an FPU may be realized. Then Section VII contains examples on some composite comptations where fast and optimal rondings are sefl, and may come for free when RN-representation in the canonical encoding is employed. Finally Section VIII concldes the paper. Definitions and Basic Properties (cited from [1]) Definition 1 (RN-representations): Let β be an integer greater than or eqal to. The digit seqence D =d n d n 1 d n (with β + 1 d i β 1) is an RN-representation in radixβ ofxiff 1) x = n i= d iβ i (that isd is a radix-β representation of x); ) for anyj n, j 1 i= d i β i 1 βj, that is, if the digit seqence is trncated to the right at any position j, the remaining seqence is always the nmber (or one of the two members in case of a tie) of the formd n d n 1 d n d n 3...d j that is closest tox. Hence, trncating the RN-representation of a nmber at any position is eqivalent to ronding it to the nearest. Althogh it is possible to deal with infinite representations, we shall first restrict or discssions to finite representations. The following observations on sch RN-representations for general β are then easily fond: Theorem (Finite RN-representations): ifβ 3 is odd, thend =d m d m 1 d l is an RN-representation iff i, β + 1 d i β 1 ; ifβ is even thend =d m d m 1 d l is an RN-representation iff 1) all digits have absolte vale less than or eqal to β ; ) if d i = β, then the first non-zero digit that follows on the right has the opposite sign, that is, the largestj<isch thatd j 0 satisfiesd i d j < 0. Observe that for oddβ the system is non-redndant, whereas forβ even the system is redndant, in the sense that some non-zero nmbers have two representations. In particlar note that for radix the digit set is { 1, 0, 1}, known by the names of binary signed-digit or borrow-save, bt here restricted sch that the nonzero digits have alternating signs. Theorem 3 (Uniqeness of finite representations): ifβ is odd, then a finite RN-representation ofxis niqe; ifβ is even, then some nmbers may have two finite representations. In that case, one has its least significant nonzero digit eqal to β, the other one has its least significant nonzero digit eqal to + β. Proof: If β is odd, the reslt is an immediate conseqence of the fact that the digit set is nonredndant. If β is even, then consider two different RN-representations representing the same vale x, and consider the largest positionj (that is, of weightβ j ) sch that these RN-representations differ, when trncated to the right of positionj. Letx a andx b be the vales represented by these digit strings. Obviosly, x a x b { β j, 0,β j }. Nowx a =x b wold contradict the way thatj was chosen. Withot loss of generality, then assmex b =x a +β j. This impliesx=x a +β j / = x b β j /, since the maximal absolte vale of a digit is β/. Hence, the remaining digit strings (i.e., the parts that were trncated) are digit strings starting from positionj 1, representing±β j /. The only way of representing β j / by an RNrepresentation starting from position j 1 is ) ( β This is seen as follows: if the digit at positionj 1 of a nmber is less than or eqal to β 1, then that nmber is less than or eqal to ( β 1 )β j 1 + ( β ) j β i <β j /, since the largest allowed digit is β. Also, the digit at positionj 1 of an RN-representation cannot be larger than or eqal to β Ifβ is even, then a nmber whose finite representation (by an RN-representation) has its last nonzero digit eqal to β has an alternative representation ending with β (jst assme the last two digits ared(β ): i=l

3 3 since the representation is an RN-representation, d < β, hence if we replace these two digits by (d + 1)( β ) we still have a valid RN-representation). This has an interesting conseqence: trncating a nmber which is a tie will rond either way, depending on which of the two possible representations the nmber happens to have. Hence, there is no bias in the ronding. Note that this ronding rle is different from the rond-to-nearest-even rle reqired by the IEEE standard [4]. Both rondings provide a rond-tonearest in the case of a tie, bt employ different rles when choosing which way to rond. Also note that this ronding is also deterministic, the direction of ronding only depends on how the vale to be ronded was derived, as the representation of the vale is niqely determined by the seqence of operations leading to the vale. Example: In radix 7, with digits{ 3,, 1, 0, 1,, 3}, all representations are RN-representations, and no nmber has more than one representation; in radix 10 with digits{ 5,..., +5}, 15 has two RN-representations: 15 and 5. Theorem 4 (Uniqeness of infinite representations): We now consider infinite representations, i.e., representations that do not ltimately terminate with an infinite seqence of zeros. ifβis odd, then some nmbers may have two infinite RN-representations. In that case, one is eventally finishing with the infinite digit string and the other one is eventally finishing with the infinite digit string β+1 β+1 β+1 β+1 β+1 β+1 ; if β is even, then two different infinite RNrepresentations necessarily represent different nmbers. As a conseqence, a nmber that is not an integer mltiple of an integral (positive or negative) power of β has a niqe RN-representation. Proof: If β is odd, the existence immediately comes from 1. β+1 β+1 β+1 β+1 = 0. = 1 Now, if for anyβ(odd or even) two different RNrepresentations represent the same nmber x, then consider them trncated to the right of some position j, sch that the obtained digit strings differ. The obtained digit strings represent valesx a andx b whose difference is±β j (a larger difference is impossible for obvios reasons). First, consider the case whereβ is odd. From the definition of RN-representations, and assmingx a < x b, we havex=x a +β j / =x b β j /. Sinceβ is odd, the only way of representingβ j / is with the infinite digit string (that starts from positionj 1) the reslt immediately follows. Now consider the case whereβ is even. Let s first show thatx a =x b is impossible. From Theorem 3, this wold imply that one of the corresponding digit strings wold terminate with the digit seqence β 00 00, and the other one with the digit string + β Bt from Theorem, this wold imply that the remaining (trncated) terms are positive in the first case, and negative in the second case, which wold mean (since x a =x b implies that they are eqal) that they wold both be zero, which is not compatible with the fact that the representations of x are assmed infinite. Hence x a x b. Assmex a <x b, which impliesx b =x a +β j. We necessarily havex = x a +β j / = x b β j /. Althoghβ j / has several possible representations in a general signed-digit radix-β system, the only way of representing it with an RN-representation is to pt a digit β at positionj 1, no infinite representation is possible. Example: In radix 7, with digits { 3,, 1, 0, 1,, 3}, the nmber 3/ has two infinite representations, namely and in radix 10 with digits { 5,..., +5}, the RNrepresentation of π is niqe. An important property of the RN-representation is that it avoids the doble ronding problem occrring with some ronding methods, e.g., with the standard IEEE rond-to-nearest-even. This may happen when the reslt of first ronding to a positionj, followed by ronding to positionk, does not yield the same reslt as if directly ronding to position k, as also discssed in [5]. We repeat from [1] the following reslt: Observation 5 (Doble ronding): Let rn i (x) be the fnction that ronds the vale ofx to nearest at positioniby trncation. Then fork>j, if x is represented in the RN-representation, then rn k (x) = rn k (rn j (x))

4 4 3 Converting to and from Binary RN- Representation 3.1 Conversion from s Complement to RN- Representation Consider an inpt valex= b m m + m 1 i=l b i i in s complement representation: x b m b m 1 b l+1 b l withb i {0, 1} andm>l. Then the digit string δ m δ m 1 δ l+1 δ l with δ i { 1, 0, 1} defined (by the Booth recoding []) fori =l,,m as δ i =b i 1 b i (withb l 1 = 0 by convention) (1) is an RN-representation of x with δ i { 1, 0, 1}. That it represents the same vale follows trivially by observing that the converted string represents the vale x x. The alternation of the signs of non-zero digits is easily seen by considering how strings of the form and are converted. Ths the conversion can be performed in constant time. Actally, the digits of the s complement representation directly provides for an encoding of the converted digits as a tple:δ i (b i 1,b i ) fori =l,,m where 1 (0, 1) 0 (0, 0) or (1, 1) () 1 (1, 0), where the vale of the digit is the difference between the first and the second component. Example: Let x = be a signextended s complement nmber and write the digits of x above the digits ofx: x x x in RN-repr where it is seen that in any colmn the two pper-most bits provide the encoding defined above of the signeddigit below in the colmn. Since the digit in position m+1 will always be 0, there is no need to inclde the most significant position otherwise fond in the two top rows. Ifxis non-zero andb k is the least significant nonzero bit of the s complement representation of x, then δ k = 1, confirmed in the example, hence the last non-zero digit is always 1 and ths niqe. However, if an RN-represented nmber is trncated for ronding somewhere, the reslting representation may have its last non-zero digit of vale 1. As mentioned in Theorem 3 there are exactly two finite binary RN-representations of any non-zero binary nmber of the form a k for integralaand k, bt reqiring a specific sign of the last non-zero digit makes the representation niqe. On the other hand withot this reqirement, ronding by trncation makes the ronding nbiased in the tie-sitation, by randomly ronding p or down, depending on the sign of the last non-zero digit in the remaining digit string. Example: Ronding the vale ofxin Example 1 by trncating off the two least significant digits we obtain rn (x) rn (x) rn (x) in RN-repr where it is noted that the bit of vale 1 in the pper rightmost corner (in boldface) acts as a rond bit, assring a rond-p in cases there is a tie-sitation as here. The example shows that there is another very compact encoding of RN-represented nmbers derived directly from the s complement representation, noting in the example that the pper row need not be part of the encoding, except for the rond-bit. We will denote it the canonical encoding, and note that it is a kind of carry-save in the sense that it contains a bit not yet added in. The same idea have previosly been prsed in [6] in a floating-point setting, denoted packet-forwarding. Definition 6 (Binary canonical RN-encoding): Let the nmberxbe given in s complement representation as the bit stringb m b l+1 b l, sch that x = b m m + m 1 i=l b i i. Then the binary canonical encoding of the RN-representation of x is defined as the pair x (b m b m 1 b l+1 b l, r) where the rond-bit is r = 0 and after trncation at positionk, form k>l rn k (x) (b m b m 1 b k+1 b k,r) with rond-bitr =b k 1. If (x,r x ) is the binary canonical ( s complement) RN-representation ofx, thenx=x +r x whereis the weight of the least significant position, from which it follows that X = x r x = x+ r x = x+(1 r x ) = x+ r x. Observation 7: If (x,r x ) is the canonical RNrepresentation of a valex, then ( x, r x ) is the canonical RN-representation of X, where x is the 1 s complement of x. Hence negation of a canonically encoded vale is a constant time operation. The signed-digit interpretation is available from the canonical encoding by pairing bits, (b i 1,b i ) sing the encoding () fori>k and (r,b k ), when trncated at position k.

5 5 There are other eqally compact encodings of RNrepresented nmbers, e.g., one cold encode the signeddigit string simply by the string of bits obtained as the absolte vales of the digits, together with say the sign of the most (or least) non-zero digit. De to the alternating signs of the non-zero digits, this is sfficient to reconstrct the actal digit vales. However, this encoding does not seem very convenient for arithmetic processing, as the correct signs will then have to be distribted over the bit string. 3. Conversion from Signed-Digit RN- Representation to s Complement The example of converting into its s complement eqivalent shows that it is not possible to perform this conversion in constant time, information may have to travel an arbitrary distance to the left. Hence a conversion may in general take at least logarithmic time. Since the RN-representation is a special case of the (redndant) signed-digit representation, this conversion is fndamentally eqivalent to an addition. If an RN-represented nmber is in canonical encoding, conversion into ordinary s complement representation may reqire a non-zero rond-bit to be added in, it simply consists in an incrementation, for which very efficient methods exists based on parallel prefix trees with AND-gates as nodes. 4 Canonical Representation, the General Case The binary canonical representation of RNrepresentation is specified by x = (a,r a ), which is a pair of a nmber and a bit. We cold decide to represent the vale ofaof that pair in something else than binary, say sing a higher radixβ and/or a digit set different from the set{0,...,β 1}. Definition 8 (Canonical encoding: general case): Let b be a nmber in radix β sing the digit set D, sch thatb = m 1 i=l b i β i withb i D, and the ronding bit r b {0, 1}. The pair (b,r b ) then represents the vale b +r b, whereis the nit in the last place ( =β l ). The definition is very general, as the representation doesn t necessarily allow ronding by trncation. We mst redefine the ronding operation so that we avoid problems with doble-rondings, basically by trying to convert the encoding into an RN-representation satisfying Definition Even Radix The definition seems to make particlar sense when a (non-negative) nmber is represented in an even radix with the reglar digit-set{0,...,β 1}. In that representation the link between the canonical encoding and RN-representation is trivial enogh, so that rondingto-nearest can be done by trncation of the valebin the pair (b,r b ). Consider an inpt vale in radix-β with 0 d i β 1 (x,r x ) = (d m d m 1 d n d l,r x ), and define variablesc k as { 1 if d k β c k+1 = 0 if d k < β (3) Withc l =r x the digitsδ k of an RN-representation can be obtained sing δ k =d k +c k βc k+1. The conversion for an even radix and digit set {0,...,β 1} into RN-representation gives s a way to easily perform ronding-to-nearest by trncation of (x,r x ) in the canonical encoding. Fork>l: rn k (x,r x ) (d n d n 1...d k+1 d k,r) { 1 if dk 1 β/ with rond-bitr = 0 if d k 1 <β/ so in RN-representation the vale can also be expressed by the digit stringδ n δ n 1...δ k { β,,β }. Example: With radix β = 10 and the reglar digit-set{0,..., 9}, for the vale represented by (9.5450, 1), we can trncate sing the previos algorithm: rn 3 (9.5450, 1) = (9.54, 1), the ronding-bit being 1 becased 4 = 5 c 3 = 1. We only need to generate one carry (c 3 ) to obtain the ronding bit. To confirm that the ronding is correct, we may represent the vale in RN-representation, by generating all the carries: c k d k RN-representation With this RN-representation, trncating at position 3 gives: c k d k RN-representation corresponding to the previos canonical encoding (9.54, 1). Note that to represent the given positive

6 6 vale,d 1 was set to zero. Had the vale been a (negative) 10 s complement represented nmber, thend 1 shold by sign extension have been set to Other Representations If we se other representations ofb(say binary borrowsave/signed-digit, odd radices,...), the ronding may take time O(log(n)): Example: [Borrow-save:] When trying to rond x in a general borrow-save representation to the nearest integer, we have for any rond bitr x {0, 1}: borrow-save ronded ( ,r x ) ( , 1) ( ,r x ) (1 1 10, 0) hence we may have to look arbitrarily far to the right when ronding the vales. However, borrow-save cold be interesting, since addition then can be performed in O(1), instead of O(log(n)) for binary canonical encoding sing the reglar digit-set{0, 1}. It is important to recall that borrowsave is not an RN-representation, even thogh it ses the same digits. To have an RN-representation, the non-zero digits mst alternate in signs, and translating an arbitrary nmber from borrow-save to RNrepresentation may take a O(log(n)) time. Example: [Odd radices:] When trying to rond to the nearest integer, we have similarly: radix 3 ronded ( ,r x ) (10, 1) ( ,r x ) (10, 0) which means we may have to look arbitrarily far to the right when ronding the vales. It is de to the fact that the mid-point between two representable nmbers needs to be represented with an infinite nmber of digits. If we were to redefine arithmetic from scratch, odd radices cold be a choice to be considered; bt since we have to keep simple conversion to conventional nmber systems in mind, we decided in the following to focs on radix. 5 Performing Arithmetic Operations on Canonically Represented Vales The fndamental idea of the canonical radix- RNrepresentation is that it is a binary representation of some vale sing the digit set{ 1, 0, 1}, bt sch that non-zero digits alternate in sign. We then introdced an encoding of sch nmbers, employing s complement representation, in the form (a,r a ) representing the vale (a +r a ) a =a +r a, whereis the weight of the least significant position of a. Note that there is then no difference between (a, 1) and (a +, 0), both being RN-representations of the same vale: a,v(a, 1) =V(a +, 0), where we se the notationv(x,r x ) to denote the vale of an RN-represented nmber. 5.1 An Interval Interpretation Considered as intervals as described below, the two representations (a, 1) and (a+, 0) describe different intervals. Since different representations of the same nmber can give different ronding reslts when trncated, it is then important to choose careflly the representation of the reslt when performing arithmetic operations like addition and mltiplication. Hence when defining the reslt it is essential to choose the encoding of it to reflect the domains of the operands. Consider a valeato be ronded at some position of weightwhere the rond bit is 1, shown in boldface: x x } {{ } } {{ } a + t x x } {{ } } {{ } a + t 0 and similarly when the rond bit is 0: x x } {{ } } {{ } a 0 t x x } {{ } } {{ } a 0 t expressing bonds on the tail t thrown away dring ronding by trncation. Observe that the right-hand ends of the intervals are closed, corresponding to a possibly infinite seqence of nits having been thrown away. We find that the valeabefore ronding into

7 7 V(a,r a ) mst belong to the interval: A = { [ ] } a ;a + [ forr a = 0 a + ;a +] forr a = 1 [ a +r a ;a + (1 +r a) ] =I(a,r a ). (4) In the following we shall sei(a,r a ) to denote the interval, the idea being to remember where the real nmber was before ronding } {{ }} {{ }} {{ }} {{ }} {{ }} {{ } (100,0) (100,1) (101,0) (101,1) (110,0) (110,1) Fig. 1. Example of interpreting RN representations as intervals with = 1 We may interpret the representations of an encoding as an interval of length /, as in Fig. 1. In the figre, any nmber between 101 and (for example ), when ronded to the nearest integer, will give the RN representation (101, 0). So we may say that (101, 0) represents the interval [101; 101.1] and in particlar I(a, 1) = [ a + ;a +], I(a +, 0) = [ ] a + ;a + 3. Hence even thogh the two encodings represent the same vale (a+), when interpreting them as intervals according to what cold have been thrown away, the intervals are essentially disjoint, except for sharing a single point. In general we may express the interval interpretation as pictred in Fig. We do not intend to define an interval arithmetic, bt only reqire that the interval representation of the I(a, 0) I(a, 1) I(a +, 0)I(a +, 1) [ ][ ][ ][ ] a a + a + a + 3 a + Fig.. Binary Canonical RN-representations as Intervals reslt of an arithmetic operation satisfies 1. I(A B) I(A) I(B) ={a b a A,b B}. To simplify the discssion, we will in this section only consider fixed-point representations for some fixed vale of. We will not discss overflow problems, as we assme that we have enogh bits to represent the reslt in canonically encoded representation. 5. Addition of RN-represented Vales Employing the vale interpretation we have for addition: V(a,r a ) = a +r a +V(b,r b ) = b +r b V(a,r a ) +V(b,r b ) = a +b+(r a +r b ) The reslting vale has two possible representations, depending on the ronding bit of the reslt. To determine what the ronding bit of the reslt shold be, Table 1 shows the interval interpretations of the two possible representations of the reslt, depending on he ronding bits of the operands. Since we wanti(v(a,r a ) +V(b,r b )) I(a,r a ) + I(b,r b ), and (a,r a ) + (0, 0) = (a,r a ), and in order to keep the addition symmetric, we define the addition of RN encoded nmbers as follows. Definition 9 (Addition): Ifis the nit in the last place of the operands, let: (a,r a ) + (b,r b ) = ((a +b+(r a r b )),r a r b ) 1. Note that this is the reverse inclsion of that reqired for ordinary interval arithmetic, e.g. [7]. r a = r b = 0 r a r b = 1 r a = r b = 1 I(V(a,r a) +V(b,r b )) I(a,r a) +I(b,r b ) I(a +b, 1) = [ a +b ;a +b] I(a +b, 0) = [ [a +b ;a +b+] ] a +b ;a +b+ [a +b ;a +b+] I(a +b, 1) = [ } a +b+ ;a +b+] I(a +b+, 0) = [ ] [ a +b+ a +b+ ;a +b+ ;a +b+ ] 3 3 I(a +b+, 1) = [ a +b+ 3 ;a +b+] I(a +b+, 0) = [ [a +b+ ;a +b+] ] a +b+ ;a +b+ 5 [a +b+ ;a +b+] TABLE 1 Interpretations of additions as intervals

8 8 Recalling that (x,r x ) = ( x, r x ), we observe that sing this definition, (x,r x ) (x,r x ) = (, 1), with V(, 1) = 0. It is possible alternatively to define addition on RN-encoded nmbers as (a,r a )+ (b,r b ) = ((a +b+(r a r b )),r a r b ). Using this definition, (x,r x ) (x,r x ) = (0, 0), bt then the netral element for addition is (, 1), i.e., (x,r x ) + (, 1) = (x,r x ). Example: Let s take two examples adding two nmbers that were previosly ronded to the nearest integer. Addition not ronded Addition on ronded canonical representations a (01011, 1) b (01001, 1) a 1 +b (010101, 1) a (01011, 1) b (01001, 1) a +b (010101, 1) Using the definition above, rn 0 (a 1 +b 1 ) = rn 0 (a 1 ) + rn 0 (b 1 ) holds in the first case. Obviosly, since some information may be lost dring ronding, there are cases like in the second example where rn 0 (a +b ) rn 0 (a ) + rn 0 (b ). Also note that de to that information loss,a +b is not ini((a,r a ) + (b,r b )) [ When interpreted as an interval, I(x,r x ) = x +rx ;x + (1 +r ] x) then its mirror image interval of negated vales is I(x,r x ) = [ x + rx ; x + (1 + r x) ] = I( x, rx ). Ths consider for sbtraction I ((a,r a ) (b,r b )) =I ( (a,r a ) + ( b, r b ) ) =I(a + b + (r a r b ),r a r b ) =a + b +(r a r b ) + (r a r b ) + [ ] 0 ; =a b+(r a r b ) (r a r b ) + [ ] 0 ; =a b+ (r a r b ) ( r a r b ) + [ 0 ; ]. On the other hand, I(a,r a ) I(b,r b ) = [ a + r a ;a + (1 +r a) ] [ b + r b ;b + (1 +r b) ] =a b+ (r a r b ) + [ ; ], hence for all pointsx I ((a,r a ) (b,r b )),xis in I(a,r a ) I(b,r b ). Hence sbtraction of (x,r x ) can be realized by addition of the bitwise inverted tple ( x, r x ). 5.3 Mltiplying Canonically Encoded RNrepresented vales By definition we have for the vale of the prodct V(a,r a ) = a +r a V(b,r b ) = b +r b V(a,r a )V(b,r b ) = ab + (ar b +br a ) +r a r b, noting that the nit of the reslt is, assming that 1. Considering the operands as intervals we find sing (4): I(a,r a ) I(b,r b ) = [ a +r a ;a + (1 +r a) ] [ b + r b ;b + (1 +r b) ] [ ] 0 ;a +b + for a>0,b>0 =a b [a ;b ] + for a<0,b>0 [b ;a ] for a>0,b<0 [ a +b + ; 0] for a<0,b<0 (5) wherea =a+r a andb =b+r b, and it is assmed thataanda share the same sign and similarly forb andb (assmed satisfied ifa 0 andb 0). Bt for a<0andb<0, withr a =r b = 0 we wold expect the reslt (ab, 0) as intervali(ab, 0) =ab + [ 0 ; ], to be in the appropriate interval defined by (5), which is NOT the case! However, since negation of canonical ( s complement) RN-represented vales can be obtained by constant-time bit inversion, mltiplication of sch operands can be realized by mltiplication of the absolte vales of the operands, the reslt being spplied with the correct sign by a conditional inversion. Ths employing bit-wise inversions, mltiplication in s complement RN-representations becomes eqivalent to sign-magnitde mltiplication, hence assming that both operands are non-negative, the interval prodct is I(a,r a ) I(b,r b ) = [ a+r a ;a+ (1+r a) ] [ b+r b ;b+ (1+r b) ] = [ (a+r a )(b+r b ) ; (a+ (1+r a))(b+ (1+r b)) ] = (a+r a )(b+r b )+[ 0 ; (a+r a )+(b+r b ] )+ =ab+ ( ar b +br a + rar b ) +[ 0 ;a+b+(r a +r b +1) ] ab + [ 0 ;a+b+ ] for r a = r b = 0 ab+a = +[0 ;a+b+] for r a = 0, r b = 1 ab+b +[0 ;a+b+] for r a = 1, r b = 0 ab+(a+b+ ) +[ 0;a+b+ 3 ] for r a = r b = 1 (6) It then follows that I ((ab + (ar b +br a ),r a r b ) I ((a,r a ) (b,r b ))

9 9 with nit, since the lefthand RN-representation where we note that ( , 1) corresponds to the corresponds to the interval interval: [ ] (ab+(ar b +br a ))+(r a r b ) ; (ab+(ar [ ; ] b+br a ))+(1+r a r b ) and its lower endpoint is greater than or eqal to the lower endpoint from (6): ab+(ar b +br a )+(r a r b ) ab+(ar b +br a + rar b ) together with the pper endpoint being smaller than or eqal to that from (6): ab+(ar b +br a )+(1+r a r b ) ab+(ar b +br a + rar b ) +(a+b+(r a+r b +1) ) both satisfied fora,b (i.e., non-zero) and all permissible vales ofr a,r b. Definition 10 (Mltiplication): Ifis the nit in the last place, with 1, we define for non-negative operands: (a,r a ) (b,r b ) = (ab +(ar b +br a ),r a r b ), and for general operands by appropriate sign inversions of the operands and reslt. If<1 the nit is < and the reslt may often have to be ronded to nit, which can be done by trncation. For an implementation some modifications to an nsigned mltiplier will handle ther a andr b rond bits, we jst have to calclate the doble length prodct with two additional rows consolidated into the partial prodct array. However, we shall not here go into the details of the consolidation. The mltiplier (b +r b ) may also be recoded into radix (actally it is already so when interpreted as a signed-digit nmber) or into radix 4, and a term (bit) d i r a may be added in the row below the row for the partial prodctd i a, whered i is the recodedi th digit of the mltiplier. Hence only at the very bottom of the array of partial prodcts will there be a need for adding in an extra bit as a new row. The doble length prodct can be retrned as (p,r p ), noticing that the nit now is =, bt the reslt may have to be ronded, which by simple trncation will define the ronded prodct as some (p,r p). Example: As an example with = 1: Not ronded Canonical Representation a (01011, 1) b (01001, 1) a b ( , 1) The mltiplication in canonical representation was done according to the definition: ab + (ar b +br a ) = ( ) = = , clearly a sbset of the interval [ ; ] = [ ; ]. It is obvios that ronding reslts in larger errors when performing mltiplication. Similarly, some other arithmetic operations like sqaring, sqare root or even the evalation of well behaved transcendental fnctions may be defined and implemented, jst considering canonical RNrepresented operands as s complement vales with a carry-in not yet absorbed, possibly sing interval interpretation to define the reslting rond bit. 6 Floating Point representations For an implementation of a floating point arithmetic nit (FPU) it is necessary to define a binary encoding, which we assme is based on the canonical s complement for the encoding of the significand part (say s encoded in p bits, s complement), spplied with the rond bit (sayr s ) and an exponent (sayein some biased binary encoding). It then seems natral to pack the three parts into a compter word (3, 64 or 18 bits) in the following order: e s r s with the rond bit in immediate contination of the significand part, ths simplifying the ronding by trncation. As sal we will reqire the vale being represented is in normalized form, say sch that the radix point is between the first and second bit of the significand field. If the first bit is zero, the significand field then represents a fixed point vale in the interval 1 s<1, if it is one then 1 s< 1. We shall now sketch how the fndamental operations may be implemented on sch floating point RN-representations, not going into details on overflow, nderflow and exceptional vales. 6.1 Mltiplication Since the exponents are handled separately, forming the prodct of the significands is precisely as described previosly for fixed point representations: sign-inverting negative operands, forming the doble-length prodct, normalizing and ronding it, and possibly negating the reslt, spplying it with the proper sign. Normalizing here may reqire a left shift, which is straight forward on the (positive) prodct before ronding by trncation.

10 10 6. Addition In effective sbtractions, after cancellation of leading digits there is a need to left normalize, so a problem here is to consider what to shift in from the right. Thinking of the vale as represented in signed digit, binary vale, obviosly zeroes have to be shifted in. In or encoding, say for a positive reslt (d,r d ) we may have a s complement bit pattern: d b k+ b p 1 and rond bitr d to be left normalized. { Here the least significant digit is encoded as rd b p 1 }. It is then fond that shifting in bits of valer d will precisely achieve the effect of shifting in zeroes in the signed-digit interpretation: k d 01b k+ b p 1 r d r d with rond bitr d, from (x,r x ) = (x,r x ) + (x,r x ) = (x +r x,r x ) Sbtraction, the "near case" Addition is traditionally now handled in an FPU as two cases [8], where the near case is dealing with effective sbtraction of operands whose exponents differ by no more than one. Here a significant cancellation of leading digits may occr, and ths a variable amont of left-normalization is reqired. As by the above this left shifting is handled by shifting in copies of the rondbit. 6.. Addition, the "far case" The remaining cases dealt with are the far case, where the reslt at most reqires normalization by a single right or left shift. Otherwise addition/sbtraction takes place as for the similar operation in IEEE, sign magnitde representation. There is no need in general to form the exact sm/difference when there is a great difference in exponents. 6.3 Division As for mltiplication we assme that negative operands have been sign-inverted, and that exponents are treated separately. Employing or interval interpretation, we mst reqire the reslt of division of (x,r x ) by (y,r y ) to be in the interval: [ x +r x y + (1 +r y ) ; x + (1 +r x ) y +r y After some algebraic maniplations it is fond that the exact rational q = x +r x y +r y ]. belongs to that interval. Hence any standard division algorithm may be sed to develop an approximation to the vale of q to (p + 1)-bit precision, i.e., inclding the sal rond bit where the sign of the remainder may be sed to determine if the exact reslt is jst below or above the fond approximation. 6.4 Discssion of Floating Point Representations As seen above it is feasible to define a binary floating point representation where the significand is encoded in the binary canonical s complement encoding, together with the rond-bit appended at the end of the encoding of the significand. An FPU implementation of the basic arithmetic operations is feasible at abot the same complexity as one based on the IEEE-754 standard for binary floating point, with a possible slight overhead in mltiplication de to extra terms to be added. Bt since the rond-to-nearest fnctionality is achieved at mch less hardware complexity, the arithmetic operations will generally be faster, by avoiding the sal log-time ronding. The other (directed) rondings can also be realized at minimal cost. Benefits are obtained throgh faster ronding and sign inversion (both constant time), also note that the domain of representable vales is symmetric 7 Applications in Signal Processing Let s here consider fixed-point RN representations in high-speed digital signal processing applications, althogh there are similar benefits in floating point. Two particlar applications needing freqent rondings come to mind: calclation of inner prodcts for filtering, and polynomial evalations for approximation of standard fnctions. For the latter application, a very efficient way of evalating a polynomial is to apply the Horner Scheme. Letf(x) = n i=0 a ix i be sch a polynomial approximation, then f(x) is efficiently evalated as f(x) = ( ((a n ) x+a n 1 ) x +a 1 ) x+a 0, where to avoid a growth in operand lengths, rondings are needed in each cycle of the algorithm, i.e., after each mltiply-add operation. Bt here the rond-bits can easily be absorbed in a sbseqent arithmetic operation, only at the very end a reglar conversion may be needed, bt normally the reslt is to be sed in some calclation, hence a conversion may be avoided. For inner prodct calclations, the most accrate reslt is obtained if accmlation is performed in doble precision, it will even be exact when performed in fixedpoint arithmetic. However, if doble precision is not available it is essential that a fast and optimal ronding is employed dring accmlation of the prodct terms.

11 11 8 Conclsions and Discssion We have analyzed a general class of nmber representations for which trncation of a digit string yields the effect of ronding to nearest. Concentrating on binary RN-represented operands, we have shown how a simple encoding, based on the ordinary s complement representation, allows trivial (constant time) conversion from s complement representation to the binary RN-representation. A simple parallel prefix (log time) algorithm is needed for conversion the other way. We have demonstrated how operands in this particlar canonical encoding can be sed at hardly any penalty in many standard calclations, e.g., addition and mltiplication, with negation even being a constant time operation, which often simplifies the implementation of arithmetic algorithms. The particlar featre of the RN-representation, that ronding-to-nearest is obtained by trncation, implies that repeated rondings ending in some precision yields the same reslt, as if a single ronding to that precision was performed. In [5] it was proposed to attach some state information ( bits) to a ronded reslt, allowing sbseqent rondings to be performed in sch a way, that mltiple rondings yields the same reslt as a single ronding to the same precision. It was shown that this property holds for any specific IEEE-754 [4] ronding mode, inclding in particlar for the rondto-nearest-even mode. Bt these rondings may still reqire log-time incrementations, which is avoided with the proposed RN-representation. The fixed point encoding immediately allows for the definition of corresponding floating point representations, which in a comparable hardware FPU implementation will be simpler and faster than an eqivalent IEEE standard conforming implementation. Ths in applications where many rondings are needed, and conformance to the IEEE-754 standard is not reqired, when employing the RN-representation it is possible to avoid the penalty of intermediate log-time rondings. Signal processing may be an application area where specialized hardware (ASIC or FPGA) is often sed anyway, and the RN-representation can provide faster arithmetic with rond to nearest operations at redced area and delay. References [1] P. Kornerp and J.-M. Mller, RN-Coding of Nmbers: Definition and some Properties, in Proc. IMACS 005, Jly 005, Paris. [] A. Booth, A Signed Binary Mltiplication Techniqe, Q. J. Mech. Appl. Math., vol. 4, pp , [3] J.-L. Bechat and J.-M. Mller, Mltiplication Algorithms for Radix- RN-Codings and Two s Complement Nmbers, INRIA, Tech. Rep., Febrary 005, available at: [4] IEEE, IEEE Std. 754 TM -008 Standard for Floating-Point Arithmtic, IEEE, 3 Park Avene, NY , USA, Agst 008. [5] C. Lee, Mltistep Gradal Ronding, IEEE Transactions on Compters, vol. 38, no. 4, pp , April [6] A. M. Nielsen, D. Matla, C. Ly, and G. Even, An IEEE Compliant Floating-Point Adder that Conforms with the Pipelined Packet-Forwarding Paradigm, IEEE Transactions on Compters, vol. 49, no. 1, pp , Janary 000. [7] R. E. Moore, Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ, [8] P. Farmwald, On the Design of High Performance Digital Arithmetic Units, Ph.D. dissertation, Stanford, Ag Peter Kornerp received the Mag. Scient. degree in mathematics from Aarhs University, Denmark, in After a period with the University Compting Center, from 1969 involved in establishing the compter science crriclm at Aarhs University, he helped fond the Compter Science Department there in 1971 and served as its chairman ntil in 1988, when he became Professor of Compter Science at Odense University, now University of Sothern Denmark. Prof. Kornerp has served on program committees for nmeros IEEE, ACM and other meetings, in particlar he has been on the Program Committees for the 4th throgh the 19th IEEE Symposim on Compter Arithmetic, and served as Program Co-Chair for these symposia in 1983, 1991, 1999 and 007. He has been gest editor for a nmber of jornal special isses, and served as an associate editor of the IEEE Transactions on Compters from 1991 to He is a member of the IEEE Compter Society. Jean-Michel Mller was born in Grenoble, France, in He received his Ph.D. degree in 1985 from the Institt National Polytechniqe de Grenoble. He is Directer de Recherches (senior researcher) at CNRS, France, and he is the former head of the LIP laboratory (LIP is a joint laboratory of CNRS, Ecole Normale Spériere de Lyon, INRIA and Université Clade Bernard Lyon 1). His research interests are in Compter Arithmetic. Dr. Mller was co-program chair of the 13th IEEE Symposim on Compter Arithmetic (Asilomar, USA, Jne 1997), general chair of SCAN 97 (Lyon, France, sept. 1997), general chair of the 14th IEEE Symposim on Compter Arithmetic (Adelaide, Astralia, April 1999). He is the athor of several books, inclding "Elementary Fnctions, Algorithms and Implementation" (nd edition, Birkhäser Boston, 006), and he coordinated the writing of the "Handbook of Floating-Point Arithmetic (Birkhäser Boston, 010). He served as associate editor of the IEEE Transactions on Compters from 1996 to 000. He is a senior member of the IEEE. Adrien Panhalex was born in Robaix, France, in He received his master degree in 008 from the École Normale Spériere de Lyon. He is now preparing his Ph.D. at École Normale Spériere de Lyon, France, nder the spervision of Jean-Michel Mller and Nicolas Lovet.