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An Algorithm for Redundant Binary Bit-Pipelined Rational Arithmetic
August 1990 (vol. 39 no. 8)
pp. 1106-1115

The authors introduce a redundant binary representation of the rationals and an associated algorithm for computing the sum, difference, product, quotient, and other useful functions of two rational operands, using this representation. The algorithm extends R.W. Gosper's (1972) partial quotient arithmetic algorithm and allows the design of an online arithmetic unit with computations granularized at the signed bit level. Each input or output port can be independently set to receive/produce operands/result in either binary radix or the binary rational representation. The authors investigate by simulation the interconnection of several such units for the parallel computation of more complicated expressions in a tree-pipelined manner, with particular regard to measuring individual and compounded online delays.

[1] M. D. Ercegovac, "On-line arithmetic: An overview,"SPIE Vol. 495, Real Time Signal Processing VII, pp. 86-93, 1984.
[2] R. W. Gosper, "Item 101 in Hakmem," AIM239, MIT, Feb. 1972, pp. 37-44, further developed in an unpublished manuscript.
[3] C. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 5th ed. London, England: Oxford University Press, 1979.
[4] A. Y. Khinchin,Continued Fractions, 1935, Translated from Russian by P. Wynn and P. Noordhoff Ltd., Grooningen, 1963.
[5] D. E. Knuth,The Art of Computer Programming, Vol. 2, Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1981.
[6] P. Kornerup and D. W. Matula, "Finite precision rational arithmetic: An arithmetic unit,"IEEE Trans. Comput., vol. C-32, no. 4, pp. 378-387, Apr. 1983.
[7] P. Kornerup and D. W. Matula, "Finite precision lexicographic continued fraction number systems," inProc. 7th IEEE Symp. Comput. Arithmetic, 1985, pp. 207-214.
[8] P. Kornerup and D. W. Matula, "An on-line arithmetic unit for bit-pipelined rational arithmetic,"J. Parallel Distributed Comput., vol. 5, pp. 310-330, 1988.
[9] P. Kornerup and D. W. Matula, "LCF: A lexicographic binary representation of the rationals," submitted for publication.
[10] D. W. Matula and P. Kornerup, "Foundations of finite precision rational arithmetic,"Computing, Suppl., vol. 2, pp. 88-111, 1980.
[11] D. W. Matula and P. Kornerup, "An order preserving finite binary encoding of the rationals," inProc. 6th IEEE Symp. Comput. Arithmetic, 1983, pp. 201-209.
[12] R. B. Seidensticker, "Continued fractions for high-speed and high-accuracy computer arithmetic," inProc. 6th IEEE Symp. Comput. Arithmetic, 1983.
[13] K. S. Trivedi and M. D. Ercegovac, "On-line algorithms for division and multiplication,"IEEE Trans. Comput., vol. C-26, no. 7, pp. 681-687, July 1977.

Index Terms:
tree pipeline; Gosper; redundant binary bit-pipelined rational arithmetic; redundant binary representation; sum; difference; product; quotient; rational operands; partial quotient arithmetic algorithm; online arithmetic unit; signed bit level; binary radix; binary rational representation; simulation; interconnection; parallel computation; online delays; digital arithmetic; number theory; redundancy.
Citation:
P. Kornerup, D.W. Matula, "An Algorithm for Redundant Binary Bit-Pipelined Rational Arithmetic," IEEE Transactions on Computers, vol. 39, no. 8, pp. 1106-1115, Aug. 1990, doi:10.1109/12.57048
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