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... Q( 1) R(n) = R(n 1) + P(n)
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14.6 SQUARE ROOTERS
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Synchronized registers ensure one digit per step. The cost and computation time of a sequential 2n-bit square rooter (n-bit square root), as shown in Figure 14.23, are given by C(p) Cadder=subtractor (2:n) Cindshifter (2:n) Cinv 6:Creg (2:n), T(p) n:(Tadder=subtractor (2:n) Tinv Tindshifter (2:n)): 14.6.3 Newton Raphson Square Rooter (Naturals) (14:49)
The iteration equation (7.96) xi 1 1xi (3 X x2 ), i 2 (14:50)
converges toward the inverse square root 1/X 1/2. A dedicated implementation is depicted in Figure 14.24. The iteration step involves one squaring unit, two multipliers, and one dedicated (3 s complement) subtractor. A pseudo-operator (shifter) stands for a right-shift operation, readily achieved by an appropriate loopconnection to the multiplexer represented in the general structure of Figure 14.24. The operators are designed according to the required precision p. A look-table (LUT), addressed by the truncated argument Xt, provides a rst t-bit evaluation of 1/X 1/2; its dimension is (roughly) t.2t bits. The outputs of the operators are rounded up at the required precision p; several bits may be added to cope with rounding and error propagation. The argument length is assumed not greater than 2p. Although the argument is a natural number, the
x(i) X
OTHER ARITHMETIC OPERATORS
p-bit square
precision-t LUT
1/X
pxp-bit multiplier 3
0 1 1: start
X p-bit subtractor 3-X.x2(i)
x(i)
ith iteration step
pxp-bit multiplier
x(i+1)
shifter
x(i+1)
iteration step
general structure
Newton Raphson iteration circuit: 2p-bit square rooter.
intermediate results, as well as the nal one, are not. Floating-point notation is recommended to optimize the overall precision. Example 14.2 illustrates this point.
Example 14.2
X 01011011 whose 16-bit precision root X 2 is 1001:1000101000011 1 , 2 13 :
Assume that the LUT value x(0) of 1/X1/2 is 11.225, 2-bit approximation corresponding to the inverse of X 1/2 root of the truncated 2-bit value Xt 01100000 t (rounded up). The following table shows the rst two steps of computation with rounding to 16 bits.
x(i 1) x(i).(3 2 X.x(i)2)
x(i)2
X.x(i)2
3 2 X.x(i)2
1100110011.2210 100011001101.2210 1101001100111.2216 0 1001.2210 1 1010111001000101.2222 1111011111001010.2216 1000001000001101.2214 1101011010011010.2219
14.7 BIBLIOGRAPHY
After step 2 (i 1), the approximation of X 1/2 is X:x(2) 1011011 1101011010011010:2 19 1001:1000100100010111110 showing 10-bit accuracy. The cost and computation time of the Newton Raphson square rooter ( p-bit square root), as shown in Figure 14.24, are given by C(p) CLUT (t 2t ) Csquare (p) 2:Cmultiplier (p) Csubtractor (p) Cmux2 (p) Cacc (p), T(p) TLUT (t 2t ) k:(Tsquare (p) 2:Tmultiplier (p) Tsubtractor (p) Tmux2 (p)), where k log2 (p/t). Comment 14.1 An important question about algorithms using look-up tables, and convergence algorithms in particular, is the evaluation of the exact (i.e., minimum) amount of bits necessary for any prescaling or intermediate calculations to ensure a correct result within the desired precision. Theoretically, the Newton Raphson algorithms provide a quadratic convergence rate, doubling the number of exact bits at each step. In practice, the accuracy of the look-up tables together with the rounding errors could slow down that rate, unless additional bits are provided to represent LUT data and intermediate results. Actually, the accurate calculus of rounding errors is not a straightforward matter. This mathematical problem has been treated extensively in the literature ([COR1999], [DAS1995], [TAN1991]). Using extra-bits is a safe and easy way to ensure correctness; nevertheless, a careful error computation can lead to signi cant savings. 14.7 BIBLIOGRAPHY (14:51)
[COR1999] M. A. Cornea-Hasegan, R. A. Golliver, and P. Markstein, Correctness proofs outline for Newton Raphson based oating-point divide and square root algorithms. Proceedings of the 14th IEEE Symposium on Computer Arithmetic, 1999, pp. 96 105. [DAS1995] D. DasSarma and D. W. Matula, Faithfull bipartite ROM reciprocal tables. Proceedings of the 12th IEEE Symposium on Computer Arithmetic, 1995, pp. 17 28. [GAR1959] H. L. Garner, The residue number system. IRE Trans. Electron. Comput., EC 8: 140 147 (1959). [SZA1967] N. S. Szabo and R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology. McGraw-Hill, New York, 1967. [TAN1991] P. K. Tang, Table look-up algorithms for elementary functions and their error analysis. Proceedings of the 10th IEEE Symposium on Computer Arithmetic, 1991, pp. 232 236.