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Tapered floating point

## Summary

In computing, tapered floating point (TFP) is a format similar to floating point, but with variable-sized entries for the significand and exponent instead of the fixed-length entries found in normal floating-point formats. In addition to this, tapered floating-point formats provide a fixed-size pointer entry indicating the number of digits in the exponent entry. The number of digits of the significand entry (including the sign) results from the difference of the fixed total length minus the length of the exponent and pointer entries.[1]

Thus numbers with a small exponent, i.e. whose order of magnitude is close to the one of 1, have a higher relative precision than those with a large exponent.

## History

The tapered floating-point scheme was first proposed by Robert Morris of Bell Laboratories in 1971,[2] and refined with leveling by Masao Iri and Shouichi Matsui of University of Tokyo in 1981,[3][4][1] and by Hozumi Hamada of Hitachi, Ltd.[5][6][7]

Alan Feldstein of Arizona State University and Peter Turner[8] of Clarkson University described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions.[7]

In 2013, John Gustafson proposed the Unum number system, a variant of tapered floating-point arithmetic with an exact bit added to the representation and some interval interpretation to the non-exact values.[9][10]

## References

1. ^ a b Zehendner, Eberhard (Summer 2008). "Rechnerarithmetik: Logarithmische Zahlensysteme" (PDF) (Lecture script) (in German). Friedrich-Schiller-Universität Jena. pp. 15–19. Archived (PDF) from the original on 2018-07-09. Retrieved 2018-07-09. [1]
2. ^ Morris, Sr., Robert H. (December 1971). "Tapered Floating Point: A New Floating-Point Representation". IEEE Transactions on Computers. IEEE. C-20 (12): 1578–1579. doi:10.1109/T-C.1971.223174. ISSN 0018-9340.
3. ^ Matsui, Shourichi; Iri, Masao (1981-11-05) [January 1981]. "An Overflow/Underflow-Free Floating-Point Representation of Numbers". Journal of Information Processing. Information Processing Society of Japan (IPSJ). 4 (3): 123–133. ISSN 1882-6652. NAID 110002673298 NCID AA00700121. Retrieved 2018-07-09.`{{cite journal}}`: CS1 maint: url-status (link) [2]. Also reprinted in: Swartzlander, Jr., Earl E., ed. (1990). Computer Arithmetic. Vol. II. IEEE Computer Society Press. pp. 357–.
4. ^ Higham, Nicholas John (2002). Accuracy and Stability of Numerical Algorithms (2 ed.). Society for Industrial and Applied Mathematics (SIAM). p. 49. ISBN 978-0-89871-521-7. 0-89871-355-2.
5. ^ Hamada, Hozumi (June 1983). "URR: Universal representation of real numbers". New Generation Computing. 1 (2): 205–209. doi:10.1007/BF03037427. ISSN 0288-3635. Retrieved 2018-07-09. (NB. The URR representation coincides with Elias delta (δ) coding.)
6. ^ Hamada, Hozumi (1987-05-18). Irwin, Mary Jane; Stefanelli, Renato (eds.). "A New Real Number Representation and Its Operation". Proceedings of the Eighth Symposium on Computer Arithmetic (ARITH 8). Washington, D.C., USA: IEEE Computer Society Press: 153–157. doi:10.1109/ARITH.1987.6158698. ISBN 0-8186-0774-2. [3]
7. ^ a b Hayes, Brian (September–October 2009). "The Higher Arithmetic". American Scientist. 97 (5): 364–368. doi:10.1511/2009.80.364. S2CID 121337883. [4]. Also reprinted in: Hayes, Brian (2017). "Chapter 8: Higher Arithmetic". Foolproof, and Other Mathematical Meditations (1 ed.). The MIT Press. pp. 113–126. ISBN 978-0-26203686-3.
8. ^ Feldstein, Alan; Turner, Peter R. (March–April 2006). "Gradual and tapered overflow and underflow: A functional differential equation and its approximation". Journal of Applied Numerical Mathematics. Amsterdam, Netherlands: International Association for Mathematics and Computers in Simulation (IMACS) / Elsevier Science Publishers B. V. 56 (3–4): 517–532. doi:10.1016/j.apnum.2005.04.018. ISSN 0168-9274. Retrieved 2018-07-09.`{{cite journal}}`: CS1 maint: url-status (link)
9. ^ Gustafson, John Leroy (March 2013). "Right-Sizing Precision: Unleashed Computing: The need to right-size precision to save energy, bandwidth, storage, and electrical power" (PDF). Archived (PDF) from the original on 2016-06-06. Retrieved 2016-06-06.
10. ^ Muller, Jean-Michel (2016-12-12). "Chapter 2.2.6. The Future of Floating Point Arithmetic". Elementary Functions: Algorithms and Implementation (3 ed.). Boston, Massachusetts, USA: Birkhäuser. pp. 29–30. ISBN 978-1-4899-7981-0.

## Further reading

• Luk, Clement (1974-10-02) [1974-09-30]. "Microprogrammed significance arithmetic with tapered floating point representation". Proceeding MICRO 7 Conference Record of the 7th Annual Workshop on Microprogramming. Micro 7. Palo Alto, California, USA: 248–252. doi:10.1145/800118.803869.
• Azmi, Aquil M.; Lombardi, Fabrizio (1989-09-06). "On a Tapered Floating Point System" (PDF). Proceedings of the 9th IEEE Symposium on Computer Arithmetic (ARITH 9). Santa Monica, California, USA: IEEE: 2–9. doi:10.1109/ARITH.1989.72803. ISBN 0-8186-8963-3. Archived (PDF) from the original on 2018-07-13. Retrieved 2018-07-13.
• Yokoo, Hidetoshi (August 1992). "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field". IEEE Transactions on Computers. Washington, DC, USA: IEEE Computer Society. 41 (8): 1033–1039. doi:10.1109/12.156546. ISSN 0018-9340.. Previously published in: Yokoo, Hidetoshi (June 1991). Komerup, Peter; Matula, David W. (eds.). "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field". Proceedings of the 10th IEEE Symposium on Computer Arithmetic (ARITH 10). Washington, DC, USA: IEEE Computer Society: 110–117.
• Anuta, Michael A.; Lozier, Daniel W.; Turner, Peter R. (March–April 1996) [1995-11-15]. "The MasPar MP-1 As a Computer Arithmetic Laboratory". Journal of Research of the National Institute of Standards and Technology. 101 (2): 165–174. doi:10.6028/jres.101.018. PMC 4907584. PMID 27805123.
• Ray, Gary (2010-02-04). "Between Fixed and Floating Point". Electronic Systems Design Engineering incorporating Chip Design. Archived from the original on 2018-07-10. Retrieved 2018-07-09.
• Beebe, Nelson H. F. (2017-08-22). "Chapter H.8 - Unusual floating-point systems". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, Utah, USA: Springer International Publishing AG. p. 966. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. […] representation with a moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called tapered arithmetic) […]