Tapered floating point
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]
See also
References
- ↑ 1.0 1.1 "Rechnerarithmetik: Logarithmische Zahlensysteme" (in German). Friedrich-Schiller-Universität Jena. Summer 2008. pp. 15–19. https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf. [1]
- ↑ "Tapered Floating Point: A New Floating-Point Representation". IEEE Transactions on Computers (IEEE) C-20 (12): 1578–1579. December 1971. doi:10.1109/T-C.1971.223174. ISSN 0018-9340.
- ↑ "An Overflow/Underflow-Free Floating-Point Representation of Numbers". Journal of Information Processing (Information Processing Society of Japan (IPSJ)) 4 (3): 123–133. 1981-11-05. NAID 110002673298 NCID AA00700121. ISSN 1882-6652. https://www.researchgate.net/publication/243733790. Retrieved 2018-07-09. [2]. Also reprinted in: Swartzlander, Jr., Earl E., ed (1990). Computer Arithmetic. II. IEEE Computer Society Press. pp. 357–.
- ↑ Accuracy and Stability of Numerical Algorithms (2 ed.). Society for Industrial and Applied Mathematics (SIAM). 2002. pp. 49. 0-89871-355-2. ISBN 978-0-89871-521-7. https://books.google.com/books?id=epilvM5MMxwC.
- ↑ "URR: Universal representation of real numbers". New Generation Computing 1 (2): 205–209. June 1983. doi:10.1007/BF03037427. ISSN 0288-3635. https://www.researchgate.net/publication/220619145_URR_Universal_Representation_of_Real_Numbers. Retrieved 2018-07-09. (NB. The URR representation coincides with Elias delta (δ) coding.)
- ↑ "A new real number representation and its operation". 1987 IEEE 8th Symposium on Computer Arithmetic (ARITH). Washington, D.C., USA: IEEE Computer Society Press. 1987-05-18. pp. 153–157. doi:10.1109/ARITH.1987.6158698. ISBN 0-8186-0774-2. [3]
- ↑ 7.0 7.1 "The Higher Arithmetic". American Scientist 97 (5): 364–368. September–October 2009. doi:10.1511/2009.80.364. [4]. Also reprinted in: "Chapter 8: Higher Arithmetic". Foolproof, and Other Mathematical Meditations (1 ed.). The MIT Press. 2017. pp. 113–126. ISBN 978-0-26203686-3. https://books.google.com/books?id=E4c3DwAAQBAJ.
- ↑ "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. March–April 2006. doi:10.1016/j.apnum.2005.04.018. ISSN 0168-9274. https://www.researchgate.net/publication/223110157. Retrieved 2018-07-09.
- ↑ "Right-Sizing Precision: Unleashed Computing: The need to right-size precision to save energy, bandwidth, storage, and electrical power". March 2013. http://www.johngustafson.net/presentations/Right-SizingPrecision1.pdf.
- ↑ "Chapter 2.2.6. The Future of Floating Point Arithmetic". Elementary Functions: Algorithms and Implementation (3 ed.). Boston, Massachusetts, USA: Birkhäuser. 2016-12-12. pp. 29–30. ISBN 978-1-4899-7981-0.
Further reading
- "Microprogrammed significance arithmetic with tapered floating point representation". Conference record of the 7th annual workshop on Microprogramming - MICRO 7. Palo Alto, California, USA. 1974-10-02. pp. 248–252. doi:10.1145/800118.803869. ISBN 9781450374217.
- "On a tapered floating point system". Proceedings of 9th Symposium on Computer Arithmetic. Santa Monica, California, USA: IEEE. 1989-09-06. pp. 2–9. doi:10.1109/ARITH.1989.72803. ISBN 0-8186-8963-3. http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Azmi.pdf. Retrieved 2018-07-13.
- "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. August 1992. doi:10.1109/12.156546. ISSN 0018-9340. https://dl.acm.org/citation.cfm?id=141503.. Previously published in: "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. June 1991.
- "The MasPar MP-1 As a Computer Arithmetic Laboratory". Journal of Research of the National Institute of Standards and Technology 101 (2): 165–174. March–April 1996. doi:10.6028/jres.101.018. PMID 27805123.
- "Between Fixed and Floating Point". Electronic Systems Design Engineering incorporating Chip Design. 2010-02-04. http://chipdesignmag.com/display.php?articleId=3921.
- "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. 2017-08-22. p. 966. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. "[…] representation with a moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called tapered arithmetic) […]"
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