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 7Y}aZL 
 
 uiucDcs-R-Ta-^o 
 
 RADIX 16 EVALUATION OF SOME 
 ELEMENTARY FUNCTIONS 
 
 by 
 
 Milos D. Ercegovac 
 
 August, 1972 
 
UIUCDCS-R-72-5^0 
 
 RADIX 16 EVALUATION OF SOME 
 ELEMENTARY FUNCTIONS* 
 
 by 
 Milos D. Ercegovac 
 
 
 August, 1972 
 
 Department of Computer Science 
 
 University of Illinois at Urb ana-Champaign 
 
 Urb ana, Illinois 6l801 
 
 The research on which this paper is based was supported in part by the 
 National Science Foundation under contract GJ813 and by the Department of 
 Computer Science of the University of Illinois. 
 
 ^Presented at the IEEE-TCCA Symposium on Computer Arithmetic, College Park, 
 Maryland, May 15-16, 1972. 
 
Radix 16 Evaluation of Some 
 Elementary Functions 
 
 "by MLlos D. Ercegovac* 
 
 Abstract 
 
 This paper describes an approach for obtaining a class of similar 
 algorithms for evaluation of some elementary functions. The main objective 
 is to show feasibility of higher radix implementations, in particular radix 
 l6, as they of f er better performance than radix 2. The emphasis is not on 
 optimality of a single algorithm but rather on the optimality of the whole 
 class of algorithms. An attempt to implement a much wider class of functions 
 than is done in conventional arithmetic units, would be encouraged by the 
 present level of technology and by the existence of suitable algorithms. 
 Besides the definitions of the algorithms, which are based on continued 
 products (sums), some details related to implementation are discussed. 
 
 The research on which this paper is based was supported in part by the 
 National Science Foundation under contract GJ813 and by the Department of 
 Computer Science of the University of Illinois. 
 
 ^Department of Computer Science, University of Illinois, Urbana, Illinois. 
 
Acknowle dgment 
 
 The author wishes to thank Professor James E. Robertson for 
 highly valued guidance, discussions and support. Thanks are due also 
 to fellow- student Mr. Klshor S. Trivedi for many help full discussions 
 as well as to Mrs. June Wingler for an excellent gob of typing. 
 
1. Introduction 
 
 The use of continued products in the calculation of some elementary 
 functions appears as early as 1959, in Voider' s CORDIC technique [5]. The 
 main results of this approach, based on coordinate system transformations, 
 have been recently summarized in the form of a unified algorithm by 
 Walther [6]. Without using the notion of coordinate rotation, Specker [7] 
 derived a class of algorithms using the concept of continued products. 
 DeLugish [1] has defined powerful algorithms for a wide class of elementary 
 functions. Since the subject of this paper is motivated by some ideas and 
 is based upon results obtained by DeLugish, a brief overview of some of 
 his ideas and results follows. 
 
 The available technological possibilities could justify hardware 
 implementation of a wide class of functions without essentialy affecting 
 the relative cost of the arithmetic unit, given similarity of the proposed 
 algorithms . 
 
 One effective way to derive such a class of algorithms is to use 
 continued products (CP) or continued sums (CS) during function evaluation. 
 The CP's or CS's are closely related to the process of normalization. The 
 main idea is then to replace each required operation by two processes, 
 which utilize only the simplest operations: addition/ subtraction, shifting 
 and possibly access to precomputed constants in a read-only memory. 
 
 One of the processes is normalization, iteratively performed in 
 one arithmetic unit giving as a result digits of the CP (CS) representation, 
 one at a time. Using these digits, the second arithmetic unit performs 
 result evaluation at the same time. The apparent disadvantage of this 
 approach, namely use of at least two separate arithmetic units, can be 
 eliminated without affecting the speed significantly, as we propose later. 
 
Another remark applies to the fact that digit -by- digit evaluation is not a 
 consequence of inherent properties, but reflects realization strategy, which 
 attempts to achieve reasonably fast implementation for all functions under 
 consideration, retaining at the same time simplicity. 
 
 The use of redundant representations [3] results in simpler selection 
 procedures, which are essential in normalization, and, in the case of radix 
 2, increases speed by increasing the probability of a zero. 
 
 DeLugish has shown that for a class of functions, which includes 
 division, multiplication, square root, logarithm, exponential, trigonometric 
 and inverse trigonometric functions, operation times are from 1 to 3 
 multiplication cycle times. 
 
 For the radix l6 case, we will present algorithms without complete 
 derivations, which are given in [9]« 
 
 In what follows, we consider only algorithms for fractional parts 
 of floating point numbers, containing m radix 16 digits and sign, assuming 
 that exponent arithmetic can be done in a conventional manner. Also, only 
 positive initial values are considered. 
 
 By normalization of a given number X e[l/2, l) we mean a sequence of 
 transformation such that either 
 
 X .tt(Ml) - 1 (1.1) 
 
 o 
 
 i=o 
 or 
 
 X - S(Ai) - (1.2) 
 
 o 
 
 i=o 
 
 where multipliers are of the form 
 
 M= 1 + l6" k S , o < k < m; 
 
 and 
 

 summands are of the form 
 
 A^ = l6" k S k , o < k < m; 
 
 and S e{10, 9, . .., 1, 0, 1, . .., 9> 10} are step dependent constants, 
 (l.l) is called multiplicative normalization and uses continued products 
 while (1.2) is additive normalization using continued sums. Then constants 
 S are, in other words, the digits of the corresponding CP (CS) representation. 
 As defined by Robertson in [2], the choice of the set of values for S 
 corresponds to the redundancy ratio of 2/3, allowing efficient multiple 
 formation as well as low precision selection rules. 
 
 2. Multiplicative Normalization 
 
 The multiplicative normalization is performed recursively as 
 
 X. +1 = X,(l+l6~ k S ), o < k < m (2.1) 
 
 so that X , = X ir (l+l6~ 1 S.) is the normalized X . S. can he determined 
 m+1 0.1 ok 
 
 1=0 
 
 on the basis of X , but to keep selection dependent on the same register 
 positions, it is convenient to define the scaled remainder as 
 
 P^ = l6 k " 1 (X k -l), o < k < m (2.2) 
 The recursion now becomes 
 
 R^ +1 = 161^ + S k + l6" k+1 S k R k , o < k < m (2.3) 
 
 Since |S =10, the selection rules should preserve bounds of scaled 
 1 kmax 
 
 remainders, namely -2/3 < K < 2/3. This will guarantee that the error of 
 
 normalization |E L . I = 1 1-X . \ < 2/3 l6" m and that for all k there exist 
 1 m+1 m+1 — 
 
 intervals of R for each of which a particular S fc is a valid choice. 
 
 Those 
 
intervals should be overlapped (due to redundancy) so that the continuity of 
 representation is preserved. This condition has to be satisfied by the 
 selection procedure. To have practical selection rules, those overlaps 
 should contain numbers, simple in the binary sense, so that low precision 
 operations can be used in implementation. The results of the complete 
 derivation of the selection rules, given in [9], are used in the following 
 algorithms. The selection rules become simple after the first three steps, 
 since the correspondence between intervals and S ' s is given by 
 
 (-2S -1) < 321^ < (-2S +1), 3 < k < 
 
 m 
 
 (2.45 
 
 indicating that selection can be performed by rounding the scaled remainder 
 to one non-sign digit. Knowing this, we specify rules for k = 0, 1 and 2 
 through modified rounding rather than using a table look-up or a direct 
 combinational approach. 
 
 We now give the following definitions, relevant to the description 
 of the algorithms. 
 
 Sign and magnitude representation of the constants S^: 
 
 S, 
 
 (l-2s, )l s.2 , s.e{o, 1} for all i; 
 i=o 
 
 Two's complement representation of scaled remainders: 
 
 km 
 R, = -ro + Z r.2~ , r.e[o, 1} for all i; 
 i=l 
 
 Truncated scaled remainder: 
 
 £ 6 -i 
 R = -ro + 2 r.2 
 
 i=l 
 
 (2.5) 
 
 (2.6) 
 
 (2.7) 
 
 Non-sign part of ll : 
 
 \=< 
 
 f 6 -i 
 
 Z r.2 " if r = 0; 
 . -, 1 o 
 
 i=l 
 
 i=l 
 
 r.2- 1 
 1 
 
 if r = 1: 
 o 
 
 (2.8) 
 
Step-dependent rounding constant: 
 
 U = Z u.2~ , u.e[o,l) and 
 
 i=l 
 
 (2.9) 
 
 \ - * A> 
 
 Algorithm N (Multiplicative normalization) : (2.10) 
 
 Step HI. [Initialize] 
 
 k <- 
 
 o; 
 
 
 
 
 
 S 
 o 
 
 <- 1 : 
 
 If 
 
 1/2 < X 
 ' — o 
 
 < 
 
 5/8; 
 
 S 
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 Lf 
 
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 ' — o 
 
 < 
 
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 o 
 
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 i 
 
 
 for 
 
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 m 
 
 perform: 
 
 
 
 k ♦- 
 
 k + 
 
 1; 
 
 
 
 
 Step W2. [Loop] 
 
 S k *" L ( T k +U k )l6j; Slgn Sk^Sign^; 
 if k < k then 
 
 \ + i - 16 \ + \ + l6 ~ k+1 W 
 
 else : 
 
 where k - (m+3)/2. [ YJ denotes largest integer not larger than Y, and the 
 rounding constant U, is defined as follows : 
 
 U l = U 2 = ° 5 
 u 5 = K r r o .? 2 
 
 U 5 = VVSV + K 2 [r o +? l (? 2 + S ) + r 6 ] + K 
 
 U 6 ■ K lVlf + K 2 r o (r l +r 2 +r 5 ) 
 where K , K, and K stand for k=l, k=2 and k > 3 respectively. This 
 simplification is due to finite precision and the decreasing effect of the 
 term 16 S^^R^ D n \ + y It is an interesting feature of this procedure that 
 
at every step an increasing number of constants S is known. This constitutes 
 the basis for a possible variable radix approach. 
 
 3- Additive Normalization 
 
 The additive normalization, as defined by (1.2) is nothing more than 
 a right directed recoding where one replaces the non-redundant digit set 
 [0, ...,15} with a redundant one [10, ...,10}. The procedure is simple and 
 exact. Rounding, as a selection rule, applies to all steps. The scaled 
 remainder is as before (2.2), and since 
 
 \ +1 = \ ' l6 "\> o < k < m (3.1) 
 the basic remainder recursion is 
 
 R^ +1 = 161^ - S k , o < k < m (3.2) 
 
 and I R | < 2/3. 
 
 Using the definitions (2.5-2.9) we give 
 
 Algorithm A (Additive normalization) (3»3) 
 
 Step Al. [Initialize] k «- o; 
 
 Step A2. [Loop] 
 
 s • 
 
 
 - i; 
 
 
 
 V 
 
 
 
 - 
 
 V 
 
 for 
 
 k < 
 
 m 
 
 perform: 
 
 k «- 
 
 k + 
 
 1: 
 
 i 
 
 s k *■ L (T k + u k )16j ; Sign s k *- Sign R k ; 
 
 \ + l - 16R k - s k' 
 
 6 -i 
 
 where U. = z u .2 and 
 k . , 1 
 i=l 
 
 u. ± = o, i f 5 
 
 . -1 
 
k. Division 
 
 Let Q = Y q /X , where X , Y e[l/2,l). Then consider 
 
 y Y Q ft Mi) 
 
 o o 7r(Mi) 
 i=o 
 
 where 
 
 ^ = 1 + S k .l6" k , o < k < m 
 
 If X Q 7r(Mi) -> 1, then Y^Mi) -> Q, indicating a possible algorithm. 
 Constants S^ can be obtained through multiplicative normalization 
 (Algorithm N) and, if one recursively defines partial result as 
 
 % + i - \ (1+s k l6 " k) ^- 2 > 
 
 o < k < m 
 
 the quotient Q, = Q, , with m correct digits can be simultaneously evaluated 
 in a second arithmetic unit. 
 
 Algorithm D (Division) (^«3) 
 
 (AU1: Normalization) (AU2: Result evaluation) 
 
 Step Dl. [Initialize] k <- o; 
 
 Step Nl of Alg N; Q Q «- Y q ; 
 
 Step D2. [Loop] for k < m perform: 
 
 Step N2 of algorithm N; 9^ +1 «- Q. + Q^S^" 1 * 1 ; 
 
 An example is given in Figure lj--l. The implementation is shown in Figure k-2, 
 without control details. 
 
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11 
 
 Conventional multiplication techniques have been made compatible 
 ■with continued sums approach, using additive normalization [9]« 
 
 5. Logarithm 
 
 E 
 Let X = X 2 ' be a floating point number with fractional part 
 
 X e[l/2, l) and exponent E f 
 
 Then 
 
 fo X = fo X +E fo 2 (5.1) 
 
 ox w / 
 
 To find an algorithm which computes feX , one can consider 
 
 X = X TM./f M. 
 o o. 1' . 1 
 
 1=0 1=0 
 
 where It = 1 + l6" k S k - 
 
 Then, 
 
 foX = fo (X ttM. )-§fo(M) (5.2) 
 
 o o. 1 . i v ' 
 
 1=0 1=0 - 1 - 
 
 Since j X ir M. - 1 |< 2/3 16 , as described in Section 2, 
 i=o 
 
 foX =-? fo (1 + S.l6 -i ) (5.3) 
 
 0.1 ^ 
 
 1=0 
 
 Therefore, to compute fo X one needs to perform a summation of precomputed 
 
 constants of the f orm fo (l + S 16" ) stored in a fast read-only memory (ROM). 
 
 S ' s, obtained through multiplicative normalization, serve as keys to access 
 
 corresponding constants. Clearly, the logarithm for any base can be 
 
 realized, depending on the stored constants. We consider the natural 
 
 logarithm, since the same set of stored constants can be used for the 
 
 exponential algorithm.0 The summation (5*3) is performed recursively: 
 
 \+l = \ ~ ^ ( 1+S k l6 ~ k )> o < k < m (5.*0 
 
 L = 
 o 
 
12 
 
 The result is correct to m digits, if one assumes that the stored constants 
 are exactly represented with m digits— which is not the case. Therefore, 
 either extended precision or smaller accuracy has to he accepted. On the 
 other hand, the actual requirement for ROM capacity is decreased due to 
 finite precision. From the power series expansion for the logarithm, it can 
 he shown that for 
 
 k > k = (2 foglO-l+V)/8 « (5.5+^m)/8 (5-5) 
 
 2m (1+S. l6" k ) = S. l6" k 
 k k 
 
 thereby reducing the required capacity by approximately one-half. Evaluation 
 of the term E 2m2 does not present a problem E being of short length and it 
 is not considered here. Constant 2m 2 can be kept in ROM together with other 
 constants. 
 
 Algorithm L (Logarithm) (5-6) 
 
 (AU1: Normalization) (AU2: Result evaluation) 
 Step LI. [Initialize] k «- o; 
 
 Step Nl of Alg. N; L *- o; 
 
 Step L2. [Loop] for k < m perform: 
 
 Step N2 if k < k- then: 
 
 i^^-Mi^ie-*); 
 
 else: 
 Implementation and an example are shown on Figures 5-2 and 5-1, respectively. 
 
13 
 
 
 
 
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 Figure 5-2 
 
15 
 
 6. Exponential 
 
 X 
 To evaluate e , some preliminary transformations are necessary [1]. 
 
 Consider 
 
 e = e (6.1) 
 
 Let Xfog^e = I + F where I and F denote the integer and fractional 
 parts, respectively. Then, 
 
 X JF 2m 2 Jxo 
 
 e 
 
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 X o T 
 
 and the problem is now evaluation of e , where X = F 2m 2. The factor 2 
 
 is easily incorporated into the exponent part of the result. To simplify 
 
 the initial step, we restrict F to negative values, assuming that I is 
 
 increased by one when F > o and F is replaced by 1 - F. Then Xe(-fe2, o] 
 
 Once again a convenient identity is used: 
 
 Xo Xo - 2m ( 7T M. ) + 2m ( f M. ) (6-3) 
 
 e=e l .1 
 
 i=o i=o 
 
 where M= 1 + S l6~ , o < k < m. 
 
 m 
 
 If X - in ( tt M. ) -* o, then 
 o . 1 
 i=o 
 
 m ., Xo 
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 l 
 
 1=0 
 
 The recursion for the result evaluation is simple 
 
 E.,- = E. (1+S. l6" k ), o < k < m (6.k) 
 
 k+1 k k ' — — 
 
 E = 1 
 o 
 
 while the scaled remainder recursion for additive normalization becomes: 
 E^ = 16R - l6 k 2m (1+S k l6" k ), o < k < m (6.5) 
 
16 
 
 The required set of precomputed constants is the same as the one 
 for logarithm. In this case also for k > k (6.5) no constants should be 
 stored and normalization is defined by Algorithm A. The derivation of the 
 selection rules, given in [9] will be omitted. The initial step is done 
 according to the following table: 
 
 TABLE 7.1 
 
 xo Mo ton. Mo 
 
 [-l/8,o] 1 o 
 
 [ -3/8, -1/8) -1/k -1/k 
 
 (-0*2,-3/8) -17/32 -17/32 
 
 Then by proper restrictions of possible PL - ranges the validity of 
 the rounding rule is preserved in all remaining steps. Values for Mo and 
 ton Mo should be stored in the ROM. 
 
 X 
 
 We summarize evaluation of e as follows: 
 Preliminary transformations : 
 
 a) I + F = xfa^e 
 
 b ) X = F ton 2 
 
 o 
 
 Algorithm E (Exponential) (6.6) 
 
 (AU1: Normalization) (AU2: Result evaluation) 
 
 Step El. [Initialize] k «- o; 
 
 R, «- x -ton Mo: E_ <- Mo; 
 
 1 o J. 
 
 Step E2. [Loop] for k < m perform: k <- k + 1; 
 
 if k < k x then E k+1 «- E k + iyy.6"* 
 
 S k *" L (T k +U k )l6j; 8ign S k *~ Sign V 
 
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 else: 
 
 Step A2 of Alg A; 
 
IT 
 
 where U = 1/32 and k is determined by (5*5) • 
 
 An example and implementation are shown on Figures 6-1 and 6*2 
 respectively. 
 
 7. Implementation 
 
 Here we discuss briefly some basic aspects of the radix l6 
 
 implementation, comparing them with those of the radix 2 case, and omitting 
 
 details related to a particular design. The general configuration for the 
 
 described algorithms contains two arithmetic units, operating simultaneously. 
 
 The main parts are the same: the adder structure with the multiple formation 
 
 networks, the shifting network and the argument register. The adder structure 
 
 for radix 16 requires two adders and two select-complement networks, which 
 
 represent a major increase in hardware compared to radix 2. The speed of 
 
 addition will be only slightly decreased if both adders are unified into 
 
 one three-digit adder. We estimate that this part will require twice as 
 
 much hardware as the corresponding part in the radix 2 case. If the add 
 
 time of the adder in radix 2 is t n , we assume that t n/ - < 1.2 t ^, for 
 
 a2' alb a2' 
 
 sufficiently large m. The shifting network, required to shift right/left 
 k-digits, for o < k < m-1, is simpler for higher radix. We assume that the 
 shifting network is realized using a "barrel switch" technique [7]« Namely, 
 shifting is performed in two or more levels so that the combination of level 
 shifts corresponds to the required shift. Left shift is performed using the 
 same data paths, only the two's complement of the number specifying the 
 shift is given as the control. We assume that radix 2 requires 30$ more 
 hardware than radix 16. (For example, if m = kQ, then level 1 provides 
 displacements of 0, 16 or 32 positions, level 2 provides displacements of' 
 0, h, 8, or 12 positions and, in the radix 2 case, level 3 would be necessary 
 with displacements 0, 1, 2, or 3 positions.) Speedwise, t , > 1.3 t ^n^« 
 
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19 
 
 AUl: 
 
 So -•{ 
 
 ri 
 
 k+i 
 
 Uk 
 
 T U 
 
 ri 
 
 { 
 { 
 
 A 
 
 k Rl 
 
 R k + 1 = 16R k -16 k C k , 
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 Rk + l = 16R k -S k , 
 
 C k =l. kj<k<m 
 
 AU2: 
 
 E k + 1 = Ek+EkS k 16 
 
 -k 
 
 ADDER 
 
 n — ? — 
 
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 Ek+l 
 
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 j [ SELECT-COMPLEMENT «— 1 S k 
 
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 k 
 
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 — s k 
 
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 . 
 
 
 
 
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 — s k 
 
 16- k E kl 
 
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 *- k 
 
 ' 
 
 r 
 
 
 
 
 
 
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 Figure 6-2 
 
20 
 
 Furthermore, the selection procedure for radix 2 requires only low precision, 
 k bits long comparison, while in radix l6 five simple equations and 
 comparisons of 7 hits are required. In both cases then, the selection 
 block requirements are neglected. 
 
 To obtain some approximate comparisons between those two implementations, 
 we first define the measure of the performance as the number of bits of the 
 result per gate level delay. In general, for radix r, the performance 
 P = fo^r/Tr, where Tr is the total delay necessary to evaluate 8o^r bits of 
 the result [k] . Assuming that the add time is dominant over control, S 
 select and shift time in all steps, then T-,/- - 3T p , since in radix 2 the 
 probability of a zero (po=2/3) is exploited by providing an adder bypass. 
 Therefore, P .//P = h/3 on the average. 
 
 To estimate the efficiency of the implementation we consider the ratio 
 between performance and cost per bit. With previous assumptions, we obtain 
 that E-^/E = 1. If ROM capacity is taken into account then radix 2 offers 
 more efficient design, but radix l6 maintains better performance. It should 
 be noted that radix 16 implementation offers constant execution time, while 
 radix 2 has a variable one. 
 
 The need for two arithmetic units might be considered as a 
 disadvantage. We outline here how the performance can be maintained even 
 if only one arithmetic unit is used. We first recall that there are two 
 processes, namely normalization and result evaluation, and that for both 
 appearance of the result is essentially determined by the carry propagation. 
 The sequential organization, shown in Figure 7-1, eliminates one arithmetic 
 unit by interleaving two processes. But the performance is also reduced, 
 i.e., the execution is around two times slower. One way to preserve double 
 unit performance while using only one arithmetic unit, Is to "pipeline" two 
 
21 
 
 OUT. 
 
 k+1 
 
 ADDER; 
 
 SELECT & COMPLEMENT; 
 
 SHIFTING NETWORK 
 
 R 
 
 R' 
 
 t 1 - SELECT TIME 
 t - CALCUIATE TIME 
 t, - TRANSFER TIME 
 
 OUT. 
 
 k+1* 
 
 I I *1 
 
 R: x 
 
 s I 
 o 1 
 
 O'- 
 
 t 
 
 R«: q I 
 o , 
 
 i q ii i i 
 
 Q. I x_ i 
 
 °l I 1 ! 
 
 X, 
 
 1*1 
 
 •x^ 
 
 I I 
 
 j 1 1 — i — -i — i 1 i-4 
 
 \ t 2 t t 2 t 5 t x t 2 tj, 
 
 Figure 7-1 
 
22 
 
 processes by partitioning adder structure and data paths into separately 
 controlled groups and then to overlap processes. For example, let there be 
 n groups of m/n digits, and let superscript denote the part processed by 
 group. Then, as soon as the group G. has finished generating the result 
 
 part x, ,, it can be transferred into register R., (Figure 7-2) and part 
 
 i i-1 
 
 q can be processed while group gives processes part x , etc. In other 
 
 words, result parts for two processes x, and q are simultaneously 
 
 obtained. After part x, _ has been computed, S can be determined and 
 
 a new cycle initiated. The "pipeline" will work with maximal efficiency, 
 
 since both processes are always present. Although there will be extra 
 
 control, the reduction to one arithmetic unit with a slight decrease in 
 
 speed may yield the result that the "pipeline" approach is the optimal one. 
 
 Conclusion 
 
 The algorithms based on continued products (sums) provide simplicity 
 and uniformity, important for efficient implementation of a wider class of 
 elementary functions. The available technology could easily justify 
 realization of arithmetic units also evaluating the elementary functions, 
 usually implemented in software. This approach, without significantly 
 affecting relative cost of arithmetic unit design, might have an advantage 
 in multiprogramming and multiprocessing systems. 
 
 The radix l6 approach, described here, offers faster implementation 
 than the radix 2. The selection rules, the major problem in a higher radix, 
 remain relatively simple. A useful property of the recursion based on the 
 continued products could be exploited in even higher radix implementation, 
 thus yielding further speed-up. Namely, after performing initial steps 
 (k < 3), not only constant S fc but also constants S k+1 , •••> s 2k _3 can be 
 
 
I 1 IH 
 
 n I 
 
 n 
 
 G. 
 
 :.*L. 
 
 G. 
 
 ii- 
 
 R 
 
 23 
 
 n 
 
 Ri 
 
 n 
 
 r 
 
 Figure 7-2 
 
2k 
 
 determined at the step k. One could also think about a variable radix 
 approach, performing the initial, most difficult steps in some low radix 
 and, as the selection process becomes easier, increasing the radix at every 
 step. 
 
 Whether square root, trigonometric and inverse trigonometric 
 functions can be easily included in radix 16 approach, remains to be 
 determined by finding corresponding selection rules, but it is believed 
 that this is possible. 
 
25 
 
 References 
 
 [1] B. G. Delugish, "A class of algorithms for automatic evaluation of 
 certain elementary functions in a binary computer," Report No. 399, 
 Department of Computer Science, University of Illinois, Urbana, 
 June 1970. 
 
 [2] J. E. Robertson, "A new class of digital division methods," IRE 
 Transactions on Electronic Computers, vol. EC-7, pp. 218-222, 
 September 1958^7 
 
 [3] , Lecture Notes for Computer Science Courses 39^ 
 
 (Fall 1970) and k&2 (Spring 1971), Department of Computer Science, 
 University of Illinois, Urbana. 
 
 [h] D. E. Atkins, "A study of methods for selection of quotient digits 
 during digital division," Report No. 397, Department of Computer 
 Science, University of Illinois, Urbana, June 1970* 
 
 [5] J« E. Voider, "The CORDIC trigonometric computing technique," IEEE 
 Transactions on Electronic Computers, vol. EC-8, No. 5, PP» 330-33^+, 
 September 1959 • 
 
 [6] J. S. Walther, "A unified algorithm for elementary functions," AFIPS 
 
 Conf. Proc, vol. 38, pp. 379-385, Spring Joint Computer Conference 1971* 
 
 [7] W. H. Specker, "A class of algorithms for LnX, ExpX, SinX, CosX, Tan" X, 
 
 and Cot"^X, " IEEE Transactions on Electronic Computers, vol. EC-l^i-, No. 1, 
 pp. 85-86, February 1965* 
 
 [8] R. L. Davis, "The ILLIAC IV processing element," IEEE Transactions on 
 Computers, vol. C-l8, No. 9, pp. 8OO-816, September 1969. 
 
 [9] M. D. Ercegovac, "Radix l6 division, multiplication, logarithm and 
 exponential algorithms based on continued product representations, " 
 M.S. Thesis, Department of Computer Science, University of Illinois, 
 Urbana, in preparation. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 I. Title and Subtitle 
 
 1. Report No. 
 
 UIUCDCS-R-72-5^0 
 
 Radix l6 Evaluation of Some Elementary Functions 
 
 Author(s) 
 
 Milos D. Ercegovac 
 
 Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 August, 1972 
 
 6. 
 
 8. Performing Organization Rept. 
 No. 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSF GJ 813 
 
 2. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D . C . 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 5. Supplementary Notes 
 
 6. Abstracts 
 
 This paper describes an approach for obtaining a class of similar 
 algorithms for evaluation of some elementary functions. The algorithms 
 for division, logarithm and exponential evaluation are developed, using 
 continued products (sums) in radix 16. Some aspects of the proposed 
 implementation are discussed and comparisons between radix l6 and 
 radix 2 approaches are given. 
 
 7. Key Words and Document Analysis. 17o. Descriptors 
 
 Digital computer arithmetic 
 Redundant number representation 
 Continued products 
 Continued sums 
 Radix l6 
 
 Division 
 Logarithm 
 Exponential 
 Pipelining 
 
 7b. Identifiers/Open-Ended Terms 
 
 7c. COSAT1 Field/Group 
 
 8. Availability Statement 
 
 Release Unlimited 
 
 19.. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 25 
 
 22. Price 
 
 ORM NTIS-38 ( 10-70) 
 
 USCOMM-DC 40329-P7! 
 
*e fi . 
 
 ^1 
 
 '#%> 
 
i.