High-radix iterative algorithm for powering computation

A high-radix composite algorithm for the computation of the powering function (X/sup Y/) is presented. The algorithm consists of a sequence of overlapped operations: (i) digit-recurrence logarithm, (ii) left-to-right carry-free (LRCF) multiplications, and (iii) online exponential. A redundant number system is used, and the selection in (i) and (iii) is done by rounding except from the first iteration, when selection by table look-up is necessary to guarantee the convergence of the recurrences. A sequential implementation of the algorithm is proposed, and the execution times and hardware requirements are estimated for single and double-precision floating-point computations, for radix r=128, showing that powering can be computed with similar performance as high-radix CORDIC algorithms.

[1]  Javier D. Bruguera,et al.  Very-high radix CORDIC vectoring with scalings and selection by rounding , 1999, Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336).

[2]  Jean-Michel Muller,et al.  Elementary Functions: Algorithms and Implementation , 1997 .

[3]  D. Harris,et al.  A powering unit for an OpenGL lighting engine , 2001, Conference Record of Thirty-Fifth Asilomar Conference on Signals, Systems and Computers (Cat.No.01CH37256).

[4]  Javier D. Bruguera,et al.  High-Speed Double-Precision Computation of Reciprocal, Division, Square Root and Inverse Square Root , 2002, IEEE Trans. Computers.

[5]  Stuart F. Oberman,et al.  Floating point division and square root algorithms and implementation in the AMD-K7/sup TM/ microprocessor , 1999, Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336).

[6]  M. Ercegovac,et al.  Division and Square Root: Digit-Recurrence Algorithms and Implementations , 1994 .

[7]  Javier D. Bruguera,et al.  On-line high-radix exponential with selection by rounding , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[8]  Javier D. Bruguera,et al.  High-radix logarithm with selection by rounding , 2002, Proceedings IEEE International Conference on Application- Specific Systems, Architectures, and Processors.

[9]  S. Moshier,et al.  Computer approximations , 1986 .

[10]  Naofumi Takagi.  Powering by a Table Look-Up and a Multiplication with Operand Modification , 1998, IEEE Trans. Computers.

[11]  Arnaud Tisserand,et al.  Reciprocation, square root, inverse square root, and some elementary functions using small multipliers , 1998, Optics & Photonics.

[12]  I. S. Gradshteyn,et al.  1 – ELEMENTARY FUNCTIONS , 1980 .

[13]  Tomás Lang,et al.  Fast Multiplication Without Carry-Propagate Addition , 1990, IEEE Trans. Computers.

[14]  Javier D. Bruguera,et al.  Faithful powering computation using table look-up and a fused accumulation tree , 2001, Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001.

[15]  Michael J. Schulte,et al.  Approximating Elementary Functions with Symmetric Bipartite Tables , 1999, IEEE Trans. Computers.

[16]  William M. Waite,et al.  Software manual for the elementary functions , 1980 .