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词条 Tapered floating point
释义

  1. History

  2. See also

  3. References

  4. Further reading

In computing, tapered floating point (TFP) refers to 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]

History

{{anchor|Leveling}}The tapered floating-point scheme was first proposed by Robert Morris of Bell Laboratories in 1971, 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.[6][7][8]

Alan Feldstein of Arizona State University and Peter Turner[9] of Clarkson University described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions.[8]

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.[11][12]

See also

  • Logarithmic number system (LNS)
  • Level-index arithmetic (LI) and symmetric level-index arithmetic (SLI)
  • Posit (number format)
  • Level (logarithmic quantity)

References

1. ^1 {{cite journal |title=An Overflow/Underflow-Free Floating-Point Representation of Numbers |author-first1=Shourichi |author-last1=Matsui |author-first2=Masao |author-last2=Iri |date=1981-11-05 |orig-year=January 1981 |journal=Journal of Information Processing |issn=1882-6652 |volume=4 |issue=3 |pages=123–133 |publisher=Information Processing Society of Japan (IPSJ) |id={{NAID|110002673298}} {{NCID|AA00700121}} |url=https://www.researchgate.net/publication/243733790_An_overflowunderflow-free_floating-point_representation_of_numbers |access-date=2018-07-09 |dead-url=no}} [https://ci.nii.ac.jp/naid/110002673298/en]. Also reprinted in: {{cite book |editor-first=Earl E. |editor-last=Swartzlander, Jr. |title=Computer Arithmetic |volume=II |publisher=IEEE Computer Society Press |date=1990 |pages=357-}}
2. ^1 {{cite journal |title=URR: Universal representation of real numbers |author-first=Hozumi |author-last=Hamada |journal=New Generation Computing |issn=0288-3635 |date=June 1983 |volume=1 |issue=2 |pages=205–209 |doi=10.1007/BF03037427 |url=https://www.researchgate.net/publication/220619145_URR_Universal_Representation_of_Real_Numbers |access-date=2018-07-09}} (NB. The URR representation coincides with Elias delta (δ) coding.)
3. ^1 {{cite journal |title=A New Real Number Representation and Its Operation |author-first=Hozumi |author-last=Hamada |editor-first1=Mary Jane |editor-last1=Irwin |editor-first2=Renato |editor-last2=Stefanelli |date=1987-05-18 |journal=Proceedings of the Eighth Symposium on Computer Arithmetic (ARITH 8) |pages=153–157 |doi=10.1109/ARITH.1987.6158698 |location=Washington, D.C., USA |publisher=IEEE Computer Society Press |isbn=0-8186-0774-2 |url=https://ieeexplore.ieee.org/document/6158698/ |access-date=2018-07-09}} https://web.archive.org/web/20180709212237/http://www.acsel-lab.com/arithmetic/arith8/papers/ARITH8_Hamada.pdf -->
4. ^1 2 {{cite journal |title=The Higher Arithmetic |author-first=Brian |author-last=Hayes |journal=American Scientist |date=September–October 2009 |volume=97 |number=5 |pages=364-368 |doi=10.1511/2009.80.364 |url=https://www.americanscientist.org/article/the-higher-arithmetic |access-date=2018-07-09 |dead-url=no |archive-url=https://web.archive.org/web/20180709194903/https://www.americanscientist.org/article/the-higher-arithmetic |archive-date=2018-07-09}} [https://www.americanscientist.org/sites/americanscientist.org/files/20097301410207456-2009-09Hayes.pdf]. Also reprinted in: {{cite book |title=Foolproof, and Other Mathematical Meditations |chapter=Chapter 8: Higher Arithmetic |publisher=The MIT Press |author-first=Brian |author-last=Hayes |date=2017 |edition=1 |isbn=978-0-26203686-3 |id={{ISBN|0-26203686-X}} |pages=113-126 |url=https://books.google.com/books?id=E4c3DwAAQBAJ}}
5. ^1 {{cite journal |title=Gradual and tapered overflow and underflow: A functional differential equation and its approximation |author-first1=Alan |author-last1=Feldstein |author-first2=Peter R. |author-last2=Turner |journal=Journal of Applied Numerical Mathematics |issn=0168-9274 |date=March–April 2006 |volume=56 |number=3–4 |pages=517–532 |doi=10.1016/j.apnum.2005.04.018 |publisher=International Association for Mathematics and Computers in Simulation (IMACS) / Elsevier Science Publishers B. V. |location=Amsterdam, Netherlands |url=https://www.researchgate.net/publication/223110157_Gradual_and_tapered_overflow_and_underflow_A_functional_differential_equation_and_its_approximation |access-date=2018-07-09 |dead-url=no}}
6. ^1 {{cite web |author-first=John Leroy |author-last=Gustafson |author-link=John Leroy Gustafson |title=Right-Sizing Precision: Unleashed Computing: The need to right-size precision to save energy, bandwidth, storage, and electrical power |date=March 2013 |url=http://www.johngustafson.net/presentations/Right-SizingPrecision1.pdf |access-date=2016-06-06 |dead-url=no |archive-url=https://web.archive.org/web/20160606203112/http://www.johngustafson.net/presentations/Right-SizingPrecision1.pdf |archive-date=2016-06-06}}
7. ^1 {{cite book |author-first=Jean-Michel |author-last=Muller |title=Elementary Functions: Algorithms and Implementation |chapter=Chapter 2.2.6. The Future of Floating Point Arithmetic |pages=29-30 |edition=3 |publisher=Birkhäuser |location=Boston, MA, USA |date=2016-12-12 |isbn=978-1-4899-7981-0 |id={{ISBN|1-4899-7981-6}}}}
8. ^1 {{cite book |title=Accuracy and Stability of Numerical Algorithms |edition=2 |author-first=Nicholas John |author-link=Nicholas John Higham |author-last=Higham |publisher=Society for Industrial and Applied Mathematics (SIAM) |year=2002 |isbn=978-0-89871-521-7 |id=0-89871-355-2 |pages=49 |url=https://books.google.com/books?id=epilvM5MMxwC}}
9. ^1 2 {{cite web |title=Rechnerarithmetik: Logarithmische Zahlensysteme |type=Lecture script |date=Summer 2008 |author-first=Eberhard |author-last=Zehendner |language=German |publisher=Friedrich-Schiller-Universität Jena |pages=15-19 |url=https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf |access-date=2018-07-09 |dead-url=no |archive-url=https://web.archive.org/web/20180709202904/https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf |archive-date=2018-07-09}} [https://web.archive.org/web/20180806175620/https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.komplett.pdf]
[1][2][3][4][5][6][7][8][9]
}}

Further reading

  • {{cite journal |title=Microprogrammed significance arithmetic with tapered floating point representation |author-first=Clement |author-last=Luk |journal=Proceeding MICRO 7 Conference record of the 7th annual workshop on Microprogramming |pages=248-252 |location=Palo Alto, CA, USA |orig-year=1974-09-30 |date=1974-10-02 |doi=10.1145/800118.803869 |url=https://dl.acm.org/citation.cfm?doid=800118.803869}}
  • {{cite journal |title=On a Tapered Floating Point System |author-first1=Aquil M. |author-last1=Azmi |author-first2=Fabrizio |author-last2=Lombardi |date=1989-09-06 |journal=Proceedings of the 9th IEEE Symposium on Computer Arithmetic (ARITH 9) |isbn=0-8186-8963-3 |doi=10.1109/ARITH.1989.72803 |publisher=IEEE |location=Santa Monica, CA, USA |pages=2–9 |url=http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Azmi.pdf |access-date=2018-07-13 |dead-url=no |archive-url=https://web.archive.org/web/20180713190341/http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Azmi.pdf |archive-date=2018-07-13}}
  • {{cite journal |author-first=Hidetoshi |author-last=Yokoo |title=Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field |journal=Journal IEEE Transactions on Computers - Special issue on computer arithmetic |volume=41 |issue=8 |date=August 1992 |issn=0018-9340 |pages=1033-1039 |publisher=IEEE Computer Society |location=Washington, DC, USA |doi=10.1109/12.156546 |url=https://dl.acm.org/citation.cfm?id=141503}}. Previously published in: {{cite journal |author-first=Hidetoshi |author-last=Yokoo |title=Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field |work=Proceedings of the 10th IEEE Symposium on Computer Arithmetic (ARITH 10) |editor-first1=Peter |editor-last1=Komerup |editor-first2=David W. |editor-last2=Matula |date=June 1991 |publisher=IEEE Computer Society |location=Washington, DC, USA |pages=110-117}}
  • {{cite journal |title=The MasPar MP-1 As a Computer Arithmetic Laboratory |author-first1=Michael A. |author-last1=Anuta |author-first2=Daniel W. |author-last2=Lozier |author-first3=Peter R. |author-last3=Turner |journal=Journal of Research of the National Institute of Standards and Technology |volume=101 |number=2 |date=March–April 1996 |orig-year=1995-11-15 |doi=10.6028/jres.101.018 |url=https://www.researchgate.net/profile/Peter_Turner4/publication/268629107_The_MasPar_MP-1_As_a_Computer_Arithmetic_Laboratory/links/57fe2d1e08ae6b2da3c862d8/The-MasPar-MP-1-As-a-Computer-Arithmetic-Laboratory.pdf?origin=publication_detail |archive-url=https://web.archive.org/web/20180710152102/https://nvlpubs.nist.gov/nistpubs/jres/101/2/j2anut.pdf |archive-date=2018-07-10}}
  • {{cite web |title=Between Fixed and Floating Point |author-first=Gary |author-last=Ray |date=2010-02-04 |work=Electronic Systems Design Engineering incorporating Chip Design |url=http://chipdesignmag.com/display.php?articleId=3921 |access-date=2018-07-09 |dead-url=no |archive-url=https://web.archive.org/web/20180710035319/http://chipdesignmag.com/display.php?articleId=3921 |archive-date=2018-07-10}}
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1 : Computer arithmetic
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