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Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field
August 1992 (vol. 41 no. 8)
pp. 1033-1039

A class of new floating-point representations of real numbers, based on representations of the integers, is described. In the class, every representation uses a self-delimiting representation of the integers as a variable length field of the exponent, and neither overflow nor underflow appears in practice. The adopted representations of the integers are defined systematically, so that representation's of numbers greater than one have both exponent-significant and integer-fraction interpretations. Since representation errors are characterized by the length function of an underlying representation of the integers, superior systems in precision can be easily selected from the proposed class.

[1] A. Apostolico and A. S. Fraenkel, "Robust transmission of unbounded strings using Fibonacci representations,"IEEE Trans. Inform. Theory, vol. IT-33, no. 2, pp. 238-245, 1987.
[2] C. W. Clenshaw and F. W. J. Olver, "Beyond floating point,"J. ACM, vol. 31, pp. 319-328, Apr. 1984.
[3] C. W. Clenshaw, F. W. J. Olver, and P. R. Turner, "Level-index arithmetic: An introductory survey," inNumerical Analysis and Parallel Processing, Lecture Notes in Mathematics, vol. 1397, P. R. Turner. Ed. Berlin, Germany: Springer-Verlag, 1989, pp. 95-168.
[4] J. W. Demmel, "On error analyis in arithmetic with varying relative precision," inProc. 8th IEEE Symp. Comput. Arithmetic, Como, Italy, 1987, pp. 148-152.
[5] P. Elias, "Universal codeword sets and representations of the integers,"IEEE Trans. Inform. Theory, vol. IT-21, no. 2, pp. 194-203, 1975.
[6] S. Even and M. Rodeh, "Economical encoding of commas between strings,"Commun. ACM, vol. 21, no. 4, pp. 315-317, 1978.
[7] R. G. Gallager,Information Theory and Reliable Communication. New York: Wiley, 1972, p. 80.
[8] H. Hamada, "URR: Universal representation of real numbers,"New Generation Comput., vol. 1, pp. 205-209, 1983.
[9] H. Hamada, "Data length independent real number representation based on double exponential cut,"J. Inform. Processing, vol. 10, no. 1, pp. 1-6, 1986.
[10] H. Hamada, "A new real number representation and its operation," inProc. 8th IEEE Symp. Comput. Arithmetic, Como, Italy, 1987, pp. 153-157.
[11] IBM,IBM System/370 Principles of Operation, GA22-7000-8, 1981.
[12] D. E. Knuth, "Supernatural numbers," inThe Mathematical Gardner, D. A. Klarner, Ed. Boston, MA: Prindle Weber and Schmidt, 1980, pp. 310-325.
[13] D. E. Knuth,The Art of Computer Programming, Vol. 2, Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1981.
[14] D. W. Lozier and F. W. J. Olver, "Closure and precision in level-index arithmetic,"SIAM J. Numer. Anal., vol. 27, no. 5, pp. 1295-1304, 1990.
[15] S. Matsui and M. Iri, "An overflow/underflow-free floating-point representation of numbers,"J. Inform. Processing, vol. 4, no. 3, pp. 123-133, 1981.
[16] D. W. Matula and F. Kornerup, "An order preserving finite binary encoding of the rationals," inProc. 6th IEEE Symp. Comput. Arithmetic, Aarhus, Denmark, 1983. pp. 201-209.
[17] R. Morris, "Tapered floating point: A new floating-point representation,"IEEE Trans. Comput., vol. C-20, no. 6, pp. 1578-1579, 1971.
[18] ANSI/IEEE Std. 754-1985: "IEEE Standard for Binary Floating Point Arithmetic," New York: ANSI/IEEE, 1985.

Index Terms:
integer representation; floating-point number representations; self-delimiting variable-length exponent field; real numbers; digital arithmetic; number theory.
H. Yokoo, "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field," IEEE Transactions on Computers, vol. 41, no. 8, pp. 1033-1039, Aug. 1992, doi:10.1109/12.156546
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