Svoboda-Tung Division With No Compensation

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1 Svoboda-Tung Division With No Compensation Luis MONTALVO (IEEE Student Member), Alain GUYOT Integrated Systems Design Group, TIMA/INPG 46, Av. Félix Viallet, Grenoble Cedex, France. Abstract The development of a new general radix-b division algorithm, based on the Svoboda-Tung division, suitable for VLSI implementation is presented. The new algorithm overcomes the drawbacks of the Svoboda-Tung techniques that have prevented the VLSI implementation. First of all, the proposed algorithm is valid for any radix b 2; and next, it avoids the possible compensation due to overflow on the iteration by re-writing the two most significant digits of the remainder. This simplifies both the generation of the multiples of the divisor and the quotient digit selection function. An analysis of the algorithm shows that a known radix-2 and two recently published radix-4 division algorithms are particular cases of this general radix-b algorithm. Finally, since the new algorithm is valid only for a reduced range of the IEEE normalised divisor, a pre-scaling technique, based on the multiplication of both the operands by a stepwise approximation to the reciprocal of the divisor is also presented. 1: Introduction As the capabilities of IC technologies improve, hardware implementation of all basic arithmetic operations is becoming important in the design of processors. Whereas the design of fast and efficient adders and multipliers is well understood, division and square root still remain a serious design challenge [1]. There are mainly two techniques for performing division. The digitrecurrence approach that uses addition/subtraction and shifting in a manner similar to the paper-and-pencil approach, and the Newton-Raphson method that uses multiplication and addition to develop increasingly accurate approximations to the desired quotient [2]. The division algorithm to be presented in this paper uses the first approach. Digit-recurrence algorithms obtain the quotient digitwise. In the very well known SRT division [3] the quotient digit is selected by inspecting a few of the most significant digits of both the remainder and the divisor. In 1963, Svoboda [4] published a division algorithm where the quotient digit is estimated without considering the divisor, if the estimate is not accurate an overflow occurs and a compensation is carried out; Tung in [5] and [6] investigated the implementation of the Svoboda division with a signed digit-set. The Svoboda-Tung algorithm however has two main drawbacks that have prevented the VLSI implementation: a) It is valid for radixes greater than 4, and b) Because of the possible compensation due to overflow on the iteration, the quotient digit is actually selected from an over-redundant digit-set (i.e. the quotient digit can be greater than the radix-b [7]), hence the generation of some of the multiples of the divisor is not straightforward. The purpose of this paper is to describe the development of a new division algorithm based on the Svoboda-Tung techniques that overcomes the drawbacks. First of all, it is valid for any radix b 2; and next, the possible compensation due to overflow on the iteration is avoided by re-writing the two most significant digits, as a result of this, the quotient digit is selected from the same digit-set as of the remainder. This simplifies both the generation of the multiples of the divisor and the quotient digit selection function. An analysis of the algorithm shows that a known radix-2 and two recently published radix-4 division algorithms [8], [9], [10], [11] are particular cases of this general radix-b algorithm. The paper is structured as follows. Section 2 reviews very briefly the principles of the digit-recurrence and the Svoboda-Tung divisions. In section 3, the drawbacks of the Svoboda-Tung division that have prevented the VLSI implementation are studied. Sections 4 and 5 describe the development of the new Svoboda-Tung division with no compensation. In section 6, the new algorithm is analysed for several combinations of radixes and digit-sets. Section 7 discusses the pre-scaling technique that lets the algorithm conform to the IEEE 754 standard. Finally, section 8 presents the conclusions.

2 2: Preliminaries Division can be mathematically defined by the following equation: X = Q*Y + R. The dividend X and the divisor Y are IEEE normalised numbers (1 X < Y < 2) [12], Q is the quotient, and R is the remainder. Digit-recurrence division algorithms obtain Q digitwise, based on the recurrence: R (j+1) = b * R - q j+1 * Y. (2.1) R is the remainder after the jth iteration, R (0) is the dividend X, R (n-1) is the final remainder R (n is the number of digits of the quotient Q), b is the radix, and q j+1 is the quotient digit selected at the (j+1)th step [3]. n-1 Q = j=0 q j * b -j and R n-1 = i=0 (r i * b -i ) * b -j. In these equations: j = 0, 1,, n-1 is the recursion index, i = 0, 1,, n-1 is the digit index of the remainder, and r i stands for the ith digit of the jth remainder. On each step of the recurrence, q j+1 is selected from a given digit-set D, so that the remainder stays bounded by: -Y < R (j+1) < Y. The SRT division [3] uses a redundant signed digit-set to represent the quotient Q and selects the quotient digit q j+1 by inspecting a few of the most significant digits of both the remainder R and the divisor Y. In 1963, Svoboda [4] published a division algorithm, where q j+1 is estimated independent of the divisor Y. The estimate of q j+1 is the most significant digit r 1 of every jth remainder, if the estimate is not accurate, an overflow occurs and a compensation is carried out. The Svoboda algorithm is valid for a divisor in the range 1 Y < 1 + 1/b and uses the conventional digit-set D = {0, 1, b - 1}. Tung in [5] and [6] investigated the implementation of the Svoboda division with a signed digit-set D<b.> = { _ 1 _, 0, 1 } (b is the radix, _ stands for -, and 5b/2 + 1 b - 1) [13]. The algorithm uses the recurrence (analogous to 2.1): R (j+1) = b * R - (b * t 0 + r 1 ) * Y (2.2) where, t 0 {1 _, 0, 1} is the overflow from r 1 to r 0. The Svoboda-Tung algorithm is valid for a divisor Y in the range [13]: (b - 1) * b ( + 1) < Y < (b - 2) (b - 1) * ( - 1) (2.3) 3: Drawbacks of the Svoboda-Tung division The Svoboda-Tung division presents the following difficulties that have prevented the VLSI implementation: a) Since the remainder R and the divisor Y are represented using the signed digit-set D<b.>, carry propagation free addition/subtraction is only possible for radixes b 4 [13]. Moreover, if radix-4 were used, according to Avizienis' definition would be 3 and from inequality (2.3) we would get 1 < Y < 1, which is impossible to achieve. In conclusion, the Svoboda-Tung algorithm is valid for radixes b > 4. b) Since a compensation has to be carried out when an overflow occurs due to a wrong estimate of the quotient digit, q j+1 is finally selected from an over-redundant digitset D<b.υ> = {υ _ 1 _, 0, 1 υ} (a digit-set where υ b) [7], in this case υ = For example, suppose drawback a) is overcome so that the Svoboda- Tung algorithm is valid for radixes b 2; if the radix-4 signed digit-set D<4.2> = {2 _, 1 _, 0, 1, 2} were used to represent the remainder, the quotient digit q j+1 would actually have to be selected from the over-redundant digitset D<4.5> = {5 _ 1 _, 0, 1 5}. It is obvious that some of the multiples of the divisor Y can not be easily generated. 4: The new Svoboda-Tung division with no compensation In order to overcome the drawbacks of the Svoboda- Tung division, the following modifications have been introduced into the original algorithm. a) A signed digit-set D<b.> = { _ 1 _, 0, 1 } with in the range 5b/2 b - 1 (Note the difference with Avizienis' definition) is used to represent the quotient Q, and the remainder R ; and the conventional digit-set D = {0, 1 b - 1} is used to represent the divisor Y. This choice still allows carry propagation free, then fast, addition and subtraction. b) The possible overflow on the iteration is avoided by re-writing the two most significant digits of every jth (j+1) remainder r 1 so that r 0 is always zero, therefore t 0 = 0 and q j+1 = r 1. Recurrence (2.2) then becomes: R (j+1) = b * R - r 1 * Y (4.1) c) The following tighter arithmetic bounds are imposed: - b - 1 < R(j+1) < b - 1 (4.2)

3 5: Range of the divisor and correctness of the algorithm In this section, the range of the divisor Y for which the algorithm is valid is deduced. The following inequality is obtained by replacing recurrence (4.1) in inequality (4.2): - b b * R < r 1 * Y < b b * R (5.1) Since inequality (5.1) must be satisfied for all R, the smallest of the upper bound and the greatest of the lower bound have to be considered. In view of this fact, inequality (5.1) can be re-written as: - b b *R max < r 1 * Y < b b *R min (5.2) R is maximal when all its digits to the right of the decimal point have the highest digit magnitude, i.e.: R max = 0.. Thus: b * R max * b b - 1 (5.3) The R minimal that could mislead to a wrong estimate of the prospective quotient digit q j+1 arises when R assumes the format: R min = 0. _. Hence: b * R min + b - b * (b - 1) (5.4) It should be pointed out that the symmetrical formats of the two just described lead to the same results. The following inequality is obtained by replacing relations (5.3) and (5.4) in inequality (5.2): 1 Y < 1 + δ, where δ = + * b (5.5) Notice that the algorithm is valid if and only if δ is greater than 0. This means that, if r 1 and have different signs, must be such that >. When these two conditions are not met, r 1 are rewritten using alternative digits r 1a a so that the following equation will hold: b * r 1 + = b * r 1a + a If r 1 is positive but not zero, r 1a is set to r 1a = r 1-1, and a is computed from the equation: a = + b. Else if r 1 is negative, r 1a is set to r 1a = r 1 + 1, and a is computed from the equation: a = r 2 - b. Since a must be such that r 2a, re-writing r 1 is possible if r 2 b -. It can be concluded that the worst misleading condition appears when: = b. 6: Analysis of the algorithm Table 1 lists 1/δ for all the possible combinations of and for radixes 2 and 4. It is worth noting that the only possible radix-2 division algorithm of this kind corresponds to Burgess' algorithm [8] (*), where 1 _ 1 is rewritten as 01 _ and 11 _ as 01; the minimally redundant radix- 4 algorithm = 2, = 1 _ corresponds to the algorithm presented in [9] (**), where r 1 are re-written whenever their signs are different and = 2; and the maximally redundant radix-4 algorithm = 3, = 1 _ corresponds to Srinivas et. al.'s algorithm [10] (***), where r 1 are rewritten whenever their signs are different and 2., Table 1: 1/δ for radixes 2 and 4. Radix-b 2 4 1, 0 _ * , 2 _ , 1 ** , 0 _ , 1 *** Let us continue the analysis of the algorithm for higher radixes with different combinations of the possible digitsets and re-writing conditions. Let "MRMR" be the Maximally Redundant Maximally Rewritten case, where: = b - 1, and = 0. Then: 1/δ = b. Let "MRmr" be the Maximally Redundant minimally re-written case where: = b - 1, and = 2 - b. Then: 1/δ = b(b-1). Let "mr" be the minimally redundant case where: = b/2, and = 1 - b/2. Then: 1/δ = b 2 /2. Finally, let "MROR" be the Maximally Redundant Optimally Re-written case where: = b - 1, and = 1 - b/2. Then: 1/δ = 2(b-1). This case is called Optimally Re-written because the implementation of its quotient digit selection function demands the examination of one radix-b digit plus only one binary signed digit {1 _, 0, 1}, while all other cases demand the examination of two full radix-b digits. Table 2 shows 1/δ in these special cases for radixes 2 b 64. A case not listed in Table 2 and deserving special attention concerns radix-16 with digit-set D<16.10>, because all digits in such a digit-set can be represented as

4 the sum of two digits from the radix-4 digit-set D<4.2> (i.e. D<16.10> = 4 * D<4.2> + D<4.2>) [14]. The minimally re-written version of this case is: b = 16, = 10, = 5 _ ; therefore 1/δ = 32. Table 2: 1/δ in the special cases. Radix - b Case δ MRMR b MRmr b(b-1) mr b 2 /2; MROR 2(b-1) ,5 0,4 0,3 0,2 0,1 d MRMR MROR mr MRmr MRMR: MRmr: mr: MROR: Maximally Redundant Maximally Re-written Maximally Redundant minimally re-written minimally redundant Maximally Redundant Optimally Re-written 0, Fig. 1: δ in the special cases. b Data from Table 2 is shown graphically in Fig. 1 for radixes 2 b 16. Assuming IEEE normalised operands, it is clear that greater the δ the simpler is the scaling unit, because less number of bits have to be considered in order to decide which constant K the operands should be multiplied by, so that the divisor is scaled into the proper range. It is also clear that the quotient digit selection unit is simpler if less number of bits have to be examined. In light of these two remarks the following conclusions can be made: a) the "MRmr" case is not suitable at all for VLSI implementation because it has the smallest δ and its quotient digit selection function requires the examination of two full radix-b digits, b) the "MRMR" case is not convenient either because its quotient digit selection function also requires the examination of two full radix-b digits, although it has the greatest δ, c) the "mr" case is comparable to the "MROR" case for radixes b 4, and d) the "MROR" case is the most suitable for radixes b > 4, but intermediate solutions such as the digit-set D<16.10> could be more performing. 7: Pre-scaling of the operands Whereas on one hand, the IEEE 754 standard [11] specifies a normalised range for the operands (1 X < Y < 2), on the other hand, the proposed division algorithm is valid for a scaled divisor in the range 1 Ys < 1 + δ, where 0 < δ < 1; therefore pre-scaling of the dividend and the divisor is required. The pre-scaling technique to be described is based on the idea of finding a stepwise approximation K to the reciprocal 1/Y [15], and then multiplying both the operands by this approximation. In this way, a scaled divisor "Ys" close to and greater than 1 is obtained. Since the divisor Y is normalised, K lies in the range 1/2 K < 1. The accuracy of the approximation to the reciprocal 1/Y depends on δ and has to be such that the following inequality holds: 1 K * Y < 1 + δ (7.1) The technique is graphically illustrated by Fig. 2, where the reciprocal 1/Y and the stepwise approximation K are plotted for δ = 1/4. Notice that the distances between two consecutive steps of the approximation K are equal and notice also that the error of the approximation is always positive and less than δ = 1/4. The transition points from one step to the next have been placed in such a way that a minimum number (chosen to be 5) of the most significant digits of the divisor Y is examined.

5 1,0 0,9 0,8 0,7 0,6 0,5 1/Y, K 1,0 1,2 1,4 1,6 1,8 2,0 Fig. 2: The prescaling technique for δ = 1/4. In general, the pre-scaling technique can be mathematically formulated as follows: Since the divisor Y is normalised and K is an approximation to its reciprocal 1/Y, they can be written as: Y = 1 + d, and K = 1 - e, where "d" is within the range (0, 1], and "e" lies within the range (0, 1/2]. Consequently inequality (7.1) can be re-written as: 1 (1 - e) * (1 + d) < 1 + δ (7.2) Inequality (7.2), in turn can be expressed as: e 1 - e d < δ + e 1 - e Y (7.3) In order to simplify the circuitry of the pre-scaling unit, the full precision variables "δ", "d", and "e" in inequality (7.2) are replaced appropriately by their truncated versions that can be expressed as a sum of few terms of the form 2 -h, "h" being an integer. This allows the use of a simple full adder, or at the most a tree of full adders, as a multiplier [16]. The pre-scaling technique can be summarised as follows: a) A truncated δt = 2 -s δ is found. s = log2 (1/δ). b) The full precision fraction "e" is replaced by a stepwise approximation of the form e i = i * δt/2, where 0 i < 1/δt; thus the approximation to the reciprocal 1/Y has the format K i = 1 - i * 2 -(s+1). In other words, the range of the reciprocal 1/Y is divided into 2 s equal intervals. c) Using inequality (7.3), the corresponding range of "d i ", for which "e i " is valid, is computed. d) The transition points "d i " are selected such that they lie inside the valid range computed in c) and are of the form d i = m * 2 -p, "m" and "p" being integers. 8: Conclusions A new digit-recurrence division algorithm, based on the Svoboda-Tung techniques, has been described and analysed. The new algorithm overcomes the drawbacks of the Svoboda-Tung division; it is valid for any radix b 2 and it avoids the possible compensation due to overflow on the iteration by re-writing the two most significant digits of the remainder. The analysis of the algorithm has shown that Burgess' [8] and Srinivas et. al.'s [10] algorithms are particular radix-2 and radix-4 cases of this general radix-b algorithm. The new algorithm is valid for a reduced range of the IEEE normalised divisor Y, namely 1 Y < 1 + δ, where: δ = + * b being the highest digit magnitude in the digit-set, and the highest non-re-writable magnitude of the second most significant digit of the remainder. Finally, a general and simple pre-scaling technique, based on the multiplication of the operands by a stepwise approximation to the reciprocal of the divisor 1/Y has been presented. References [1] M. D. Ercegovac and T. Lang, "Division and Square Root: Digit-Recurrence Algorithms and Implementations," The Netherlands: Kluwer Academic Publishers, [2] E. E. Swartzlander, Jr, "Computer Arithmetic," vol. 1, Los Alamitos - California, IEEE Computer Society Press, 1990, pp [3] D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders," IEEE Trans. Comp., vol. C-17, no. 10, pp , Oct [4] A. Svoboda, "An algorithm for division," Information Processing Machines (Prague, Czechoslovakia), no. 9, pp , [5] C. Tung, "A division algorithm for signed-digit arithmetic," IEEE Trans. Comp. (Short Notes), vol. C- 17, pp , Sept [6] C. Tung, "Signed-digit division using combinational arithmetic nets," IEEE Trans. Comp., vol. C-19, no. 8, pp , Aug [7] P. Montuschi and L. Cimiera, "Over-redundant digit sets and the design of digit-by-digit division units," IEEE

6 Trans. on Comp., vol. 43, no. 3, pp , March [8] N. Burgess, "A fast division algorithm for VLSI " in Proc. IEEE-ICCD International Conference on Computer Design" Cambridge MA, pp , Oct [9] L. Montalvo and A. Guyot, "A hybrid radix-4 divider with operands scaling," To appear in Proc. of ESSCIRC'94, Ulm - Germany, Sept [10] H. R. Srinivas and K. K. Parhi, "A fast radix-4 division algorithm" in Proc. IEEE International Conference on Circuits and Systems, London - Great Britain, May 30 - June [11] A. Guyot et. al., "Comparison of the layout synthesis of radix-2 and pseudo-radix-4 dividers" To appear in Proc. International Conference on VLSI Design, New Delhi, India, January 4-7, [12] IEEE 1987, "ANSI/IEEE Std , IEEE standard for binary floating point arithmetic," IEEE, [13] A. Avizienis, "Signed-digit number representations for fast parallel arithmetic," IRE Trans. Electron. Comp., vol. EC-10, pp , Sept [14] E. V. Krishnamurthy, "On range-transformation techniques for division," IEEE Trans. Comp. (Short Notes), vol. C-19, no. 2, pp , Feb [15] T. M. Carter and J. E. Robertson, "Radix-16 signeddigit division," IEEE Trans. Comp., vol. 39, no. 12, pp , Dec [16] D. Ferrari, "A division method using a parallel multiplier," IEEE Trans. on Comp., vol. EC-16, pp , April [17] C. S. Wallace, "A suggestion for a fast multiplier," IEEE Trans. on Comp., vol. EC13, pp , February 1964.

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