D. E. Atkins. Higher-Radix Division Using Estimates of the Divisor and Partial Remainders. IEEE Transactions on Computers, C-17:925-934, October 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....often viewed as a black art among system designers. Extensive literature exists describing the theory of division. Subtractive methods, such as nonrestoring SRT division which was independently proposed by and subsequently named for Sweeney, Robertson, and Tocher, are described in detail in [3], 4] 7] 17] 21] 22] Multiplication based algorithms such as functional iteration are presented in [1] 9] 11] 23] Various division and square root implementations have been reported in [1] 2] 6] 24] However, little emphasis has been placed on studying the effects of FP ....

D.E. Atkins, "Higher-Radix Division Using Estimates of the Divisor and Partial Remainders," IEEE Trans. Computers, vol. 17, no. 10, pp. 925-934, Oct. 1968.

....of Sweeney, Robertson [1] and Tocher [2] who developed the algorithm independently at approximately the same time. SRT division uses subtraction as the fundamental operator to retire a fixed number of quotient bits in each iteration. Two fundamental works on SRT division are those of Atkins [3], the first major analysis of SRT algorithms, and Tan [4] a derivation of high radix SRT division and an analytic method of implementing SRT look up tables. Ercegovac and Lang [5] provide a comprehensive treatment of the theory of SRT division and square root. Although division is typically an ....

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders," IEEE Trans. Computers, vol. C-17, no. 10, Oct. 1968.

....can be usefully applied to complex arithmetic circuits. Even though it is not feasible to verify the overall circuit functionality, just verifying one iteration can uncover many possible design errors. We demonstrate this by showing the desired behavior for one iteration of radix 4 SRT division [1], as used in the Pentium divider, can be specified and verified using BDDs. This verification will detect incorrect entries in the PD table, used to generate a quotient digit on each division step, such as occurred in the Pentium, as well as other potentially subtle design errors. Going on ....

....quotient digit. In our implementation, the table was created from a PLA description generated by the ESPRESSO logic optimizer and then translated automatically into a gate level equivalent. A specification for one iteration of the divider can be expressed readily, using the formulation by Atkins [1], modified for the particular numeric format. In our circuit, divisor D is always positive, with a leading 1 to the left of the binary point, while partial remainder words PS and PC are in two s complement form, with 3 bits, plus the sign bit to the left of the binary point. D can therefore be a ....

[Article contains additional citation context not shown here]

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainder," IEEE Transactions on Computers, Vol. C-17, No. 10 (October, 1968), pp. 925-- 934.

....the radix 8 conventional one, with the additional advantage of requiring about 25 less area per quotient bit. 1 Introduction Most implementations for the division operation are based on the SRT algorithm which involves a recurrence in which one digit of the quotient is produced per iteration [1]. Consequently, to reduce the number of iterations it is convenient to use a higher radix for the quotient digit. However, as the radix increases the added complexity of the quotient selection function increases the iteration delay and eliminates the advantage. Because of this, implementations ....

D.E. Atkins, "Higher-Radix Division Using Estimates of the Divisor and Partial Remainder," IEEE Trans. Computezs , Vol. C-17, pp. 925-934, Oct. 1968.

....of Sweeney, Robertson [1] and Tocher [2] who developed the algorithm independently at approximately the same time. SRT division uses subtraction as the fundamental operator to retire a fixed number of quotient bits in every iteration. Two fundamental works on SRT division are those of Atkins [3], the first major analysis of SRT algorithms, and Tan [4] a derivation of high radix SRT division and an analytic method of implementing SRT look up tables. Ercegovac and Lang [5] provide a comprehensive treatment of the theory of SRT division and square root. Although division is typically an ....

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders," IEEE Trans. Computers, vol. C-17, no. 10, Oct. 1968.

....converging algorithms, such as Newton Raphson, and linear converging algorithms, the most common of which is SRT [15] SRT division computes a quotient one digit at a time, with an iteration time independent of the operand length. The theory of SRT division is discussed thoroughly in Atkins [1], Ercegovac [7] Robertson [17] and Tan [20] Several SRT implementations have been reported, including radix 2 dividers by Knowles [11] Kuninobu [12] Vandemeulebroecke [22] and Zuras [23] radix 4 by Birman [3] and Fandrianto [8] radix 8 by Fandrianto [9] and Prabhu [16] and radix 16 by ....

....algorithmic and circuit tradeoffs in SRT divider design. There are many performance and area tradeoffs when designing an SRT divider. One metric for comparison of different designs is the minimum required truncations of the divisor and partial remainder for quotient digit selection. Atkins [1] and Robertson [17] provide such analyzes of the divisor Manuscript received November 12, 1995; revised August 20, 1997. This work was supported by the U.S. National Science Foundation under Grant MIP93 13701. The authors are with the Computer Systems Laboratory, Stanford University, Stanford, ....

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders," IEEE Trans. Comput.,, vol. C-17, Oct. 1968.

....converging algorithms, such as NewtonRaphson, and linear converging algorithms, the most common of which is SRT [14] SRT division computes a quotient one digit at a time, with an iteration time independent of the operand length. The theory of SRT division is discussed thoroughly in Atkins [1], Ercegovac [7] Robertson [16] and Tan [19] Several SRT implementations have been reported, including radix 2 dividers by Knowles [10] Kuninobu [11] Vandemeulebroecke [21] and Zuras [22] radix 4 by Birman [3] and Fandrianto [8] radix 8 by Fandrianto [9] and Prabhu [15] and radix 16 by ....

....presented in Ercegovac [6] Montuschi [13] and Srinivas [17] There are many performance and area tradeoffs when designing an SRT divider. One metric for comparison of different designs is the minimum required truncations of the divisor and partial remainder for quotient digit selection. Atkins [1] and Robertson [16] provide such analyses of the divisor and partial remainder precisions required for quotient digit selection. Burgess and Williams [4] present in more detail allowable truncations for divisors and both carry save and borrow save partial remainders. However, a more detailed ....

D. E. Atkins. Higher-radix division using estimates of the divisor and partial remainders. IEEE Transactions on Computers, C-17(10), October 1968.

....dividers will be a cost effective method for improving floating point performance. In the past, others have presented summaries of specific classes of division algorithms and implementations. Flynn [19] discusses the theory and methodology of multiplicationbased division algorithms. Atkins [1] is the first major analysis of SRT algorithms. Tan [38] derives and presents the theory of high radix SRT division, along with an analytic method of implementing SRT look up tables. Soderquist [35] presents performance and area tradeoffs in divider design in the context of a specialized ....

D. E. Atkins. Higher-radix division using estimates of the divisor and partial remainders. IEEE Transactions on Computers, C-17(10), October 1968.

....division in modern processors has been named SRT division by Freiman [4] taking its name from the initials of Sweeney, Robertson [5] and Tocher [6] who developed the algorithm independently at approximately the same time. Two fundamental works on division by digit recurrence are Atkins [7], which is the first major analysis of SRT algorithms, and Tan [8] which derives and presents the theory of high radix SRT division and an analytic method of implementing SRT look up tables. Ercegovac and Lang [9] present a comprehensive treatment of division by digit recurrence, and it is ....

D.E. Atkins, "Higher-Radix Division Using Estimates of the Divisor and Partial Remainders," IEEE Trans. Computers, vol. 17, no. 10, Oct. 1968.

....= fp rm1 fp rm0 = fp rm0 Delta fp le Delta fp To demonstrate the feasibility of this approach, a combined interval and floating point divider was designed and simulated at the behavioral level using VHDL. The significand divide unit, shown in Figure 4, is a minimallyredundant radix 4 SRT [15] divide unit with estimation on the quotient digit and the use of a redundant adder subtractor in the recurrence operation [16] The residual w j is stored in redundant form as sum and carry bits, while the divisor d is stored in conventional form. The recurrence takes the form: w j 1 = 4 Delta ....

D. E. Atkins, "Higher Radix Division Using Estimates of the Divisor and Partial Remainder," IEEE Transactions on Computer, vol. C-17, pp. 925--934, 1968.

....for several applications in digital signal and image processing, computer graphics, and scientific computing [1] 3] Most algorithms for performing these operations, however, have long latencies or large area requirements. Although digit recurrence algorithms, such as those presented in [4] [6] require less area than other methods, they exhibit linear convergence and often require a large number of iterations. In comparison, multiplicative reciprocal and divide algorithms, such as those presented in [7] 9] exhibit quadratic convergence, yet require large amounts of area. This ....

D. E. Atkins, "Higher Radix Division Using Estimates of the Divisor and Partial Remainder," IEEE Transactions on Computer, vol. C-17, pp. 925--934, 1968.

....contribute to clarity, generality, and reuse. 1 Introduction The SRT division algorithm is one of the most popular methods for implementing floating point division and related operations in high performance arithmetic units. Even though the theory of SRT division has been extensively studied [Atk68], the design of dividers still remains a serious challenge [OF94] and it is easy to make mistakes in its implementation as was illustrated by the much publicized FDIV error in the Intel Pentium chip. As Pratt [Pra95] points in his analysis, it is unlikely testing alone would have caught that ....

D.E. Atkins. Higher-radix Division Using Estimates of the Divisor and Partial Remainders. IEEE Transactions on Computers, C-17(10):925--934, October 1968.

....of hardware. But this is changing because of the increasing area budget and the growing interests in graphics applications. Unlike FP add and multiply, FP division has a much longer latency because of its low parallelism. Many modern machines use an iterative division algorithm known as SRT [45, 46]. In the algorithm, the number of bits obtained per iteration can be roughly considered as the radix of the algorithm. Current SRT algorithms use small radices (4 to 16) because of the cycle time and possibly perceived hardware limitation. This chapter presents a method to reduce approximately by ....

....radix 64 divider. In this divider, an IEEE double precision operation can be computed in 11 cycles at a competitive clock rate (e.g. a 54 bit ALU addition time) 50 CHAPTER 5. A RADIX 64 FP DIVIDER 51 5.2 Background Sweeney, Robertsons [47] and Tochner [48] independently developed SRT. Atkins [45, 49] extended it to higher radices. But the extension requires a large lookup table. More recently, Taylor [50] overlapped 2 simple radix 4 dividers to obtain a radix 16 divider. To reduce latency, Taylor s method requires expensive hardware duplication so that the second stage can operate in parallel ....

D. E. Atkins, "Higher-radix Division Using Estimates of the Divisor and Partial Remainders," IEEE Transactions on Computers, vol. C-17, pp. 925--934, October 1968.

....named SRT division by Freiman [41] taking its name from the initials of Sweeney, Robertson [42] and Tocher [43] who discovered 43 CHAPTER 4. DIVISION ALGORITHMS 44 the algorithm independently in approximately the same time period. Two fundamental works on division by digit recurrence are Atkins [44], which is the first major analysis of SRT algorithms, and Tan [45] which derives and presents the theory of high radix SRT division and an analytic method of implementing SRT look up tables. Ercegovac and Lang [22] is a comprehensive treatment of division by digit recurrence. The theory and ....

....implementation in commercial microprocessors is SRT division. There are many performance and area tradeoffs when designing an SRT divider. One metric for comparison of different designs is the minimum required truncations of the divisor and partial remainder for quotient digit selection. Atkins [44] and Robertson [42] provide such analyses of the divisor and partial remainder precisions required for quotient digit selection. Burgess and Williams [81] present in more detail allowable truncations for divisors and both carry save and borrow save partial remainders. However, a more detailed ....

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders," IEEE Transactions on Computers, vol. C-17, no. 10, pp. 925--934, October 1968.

....can be usefully applied to complex arithmetic circuits. Even though it is not feasible to verify the overall circuit functionality, just verifying one iteration can uncover many possible design errors. We demonstrate this by showing the desired behavior for one iteration of radix 4 SRT division [1], as used in the Pentium divider, can be specified and verified using BDDs. This verification will detect incorrect table entries such as occurred in the Pentium, as well as other potentially subtle design errors. Going on beyond verification, we show that a correct PD table can be generated ....

....values of the divisor and partial remainder, was created from a PLA description generated by the ESPRESSO logic optimizer and then translated automatically into a gate level equivalent. A specification for one iteration of the divider can be expressed readily, using the formulation by Atkins [1], modified for the particular numeric format. In our circuit, divisor D is always positive, with a leading 1 to the left of the binary point, while partial remainder words PS i and PC i are in two s complement form, with 3 bits, plus the sign bit to the left of the binary point. D can therefore ....

[Article contains additional citation context not shown here]

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainder, " IEEE Transactions on Computers, Vol. C-17, No. 10 (October, 1968), pp. 925--934.

....can contribute to clarity, generality, and reuse. 1 Introduction The SRT division algorithm is one of the most popular methods for implementing floating point division and related operations in high performance arithmetic units. Even though the theory of SRT division has been extensively studied [Atk68], the design of dividers still remains a serious challenge [OF94] and it is easy to make mistakes in its implementation as was illustrated by the much publicized FDIV error in the Intel Pentium chip. As Pratt [Pra95] points Appeared in the Proceedings of Computer Aided Verification ....

D.E. Atkins. Higher-radix Division Using Estimates of the Divisor and Partial Remainders. IEEE Transactions on Computers, C-17(10):925-- 934, October 1968.

....Arithmetic, Hardware Verification, SRT Division 1. Introduction The SRT division algorithm is one of the most popular methods for implementing floating point division and related operations in high performance arithmetic units. Even though the theory of SRT division has been extensively studied [2], the design of dividers still remains a serious challenge [23] and it is easy to make mistakes in its implementation as was highlighted by the much publicized FDIV error in the Intel Pentium chip. Pratt [27] points in his analysis that it is unlikely testing alone would have caught that error ....

D.E. Atkins. Higher-radix Division Using Estimates of the Divisor and Partial Remainders. IEEE Transactions on Computers, C-17(10):925--934, October 1968.

....to clock the system accordingly. Many asynchronous circuits sense computation completion, and will run as quickly as the current physical properties allow [1] The subtractive divide algorithms used in current microprocessors fall into the broad category of SRT (Sweeny, Robertson, Tocher) methods [2] [3] These algorithms use redundant representation of values and treat groups of consecutive bits as single higher radix digits in order to enhance performance [4] All known implementations are synchronous [5] Research in the asynchronous SRT divider subject is very important though, recursive ....

D. E. Atkins, " Higher radix division using estimates of the divisor and partial remainders ", IEEE Transactions on computers, Vol. C-17, No. 10, 1968.

.... history variables into the abstracted model that store the previous values of prem, proot, and b. 3.3 Division In this section, we discuss the verification of floating point division. Weitek WTL3170 3171 uses a radix 4 SRT division algorithm. The similar algorithms can also be found in [Fri61, Atk68] Since the SRT division algorithm is also iterative, the loop invariant verification technique introduced in the previous section also applies here. Several published papers [CKZ96, BC95] have also shown how to verify radix 4 SRT division circuits using model checking techniques and thus our ....

....have been computed. Suppose that the quotient digits are within the range f Gamman; Gamman 1; Gamma1; 0; 1; n Gamma 1; ng for some positive n. Then a radix 4 SRT division algorithm is guaranteed to be correct if both of the following properties are true in each division loop [Atk68] r i 1 = 4 Delta r i Gamma q i Delta b jr i j n Delta b 3 The loop invariant INV i that we want to verify with our verifier is the conjunction of the two properties above. We want to verify that the invariant INV 0 is true initially and also INV i ) INV i 1 . Since the quotient digits ....

D. E. Atkins. Higher-radix division using estimates of the divisor and partial remainders. IEEE Transactions on Computers, C-17(10):925--934, October 1968.

....of efficient digit by digit binary division, originally using multiples of 1, 0, 1 , were established by [Robertson 1958] and [Tocher 1956] summarised by [MacSorley 1961] and analysed by [Freiman 1961] who introduced the term SRT division . Robertson discusses Radix 4 SRT division and [Atkins 1970] extends the analysis to higher radix dividers (radices of 16, 64 and 256) Wilson and Ledley 1961] describe division with shifting over 0s and 1s. A good discussion of these early methods is also given by [Flores 1963] Waser and Flynn 1982] describe these as subtractive algorithms because ....

....exact, may be in error by 1, and never has an error of 2. The refinement is, however relatively expensive to implement and he does not discuss the necessary hardware. It does however have interesting connections with the more recent techniques for producing very accurate quotient estimates. [Atkins 1970] presents an extensive analysis of SRT division and its extension to higher radices. He discusses redundancy of the quotient representation (for example, 3 may be represented as either 2 1 or 4 1) and states that With redundancy, the quotient digit . need not be precise. Then from a detailed ....

D.E. Atkins, "Higher-Radix Division Using Estimates of the Divisor and Partial Remainders", IEEE Trans. Comp., August 1970, pp 720--733

....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 ....

....the recurrence: R (j 1) b R (j) q j 1 Y. 2. 1) R (j) 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]. Q = j=0 n 1 q j b j and R (j) i=0 n 1 (r i (j) 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 (j) stands for the ith digit of the jth remainder. On each step of the ....

[Article contains additional citation context not shown here]

D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders," IEEE Trans. Comp., vol. C-17, no. 10, pp. 925 - 934, Oct. 1968.

....converges to a result quadratically. This performance comes at the price of additional hardware, complexity, and accuracy. In this paper, a variety of SRT division schemes will be compared. SRT is often used in workstations because it provides reasonable performance and it is relatively simple [1, 2]. The tradeoffs of these schemes will be analyzed, and the design and performance of a new divider will be presented. It will be shown that by mapping a simple topology to an aggressive circuit style in conjunction with some logic optimization, a high performance CMOS divider can be implemented. ....

D. E. Atkins. Higher-radix division using estimates of the divisor and partial remainders. IEEE Transactions on Computers, C-17(10):925-934, October 1968.

No context found.

D. E. Atkins. Higher-Radix Division Using Estimates of the Divisor and Partial Remainders. IEEE Transactions on Computers, C-17:925-934, October 1968.

No context found.

Atkins, D.E. (1968). Higher-radix division using estimates of the divisor and partial remainder. IEEE Trans. of Computers 17, 925--934.

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Atkins, D. F. (1968). Higher-radix division using estimates of divisor and remainder. IEEE Trans. on Computers 17, 925--934.

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