tapered floating point#Leveling

{{Short description|Variant of floating-point numbers in computers}}

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.

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

{{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, and by Hozumi Hamada of Hitachi, Ltd.

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

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.

References

{{Reflist|refs=

{{cite journal |title=Tapered Floating Point: A New Floating-Point Representation |author-first=Robert H. |author-last=Morris, Sr. |author-link=Robert Morris (cryptographer) |journal=IEEE Transactions on Computers |publisher=IEEE |volume=C-20 |issue=12 |pages=1578–1579 |date=December 1971 |issn=0018-9340 |doi=10.1109/T-C.1971.223174 |s2cid=206618406 }}

{{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-date=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 |access-date=2018-07-09 }} [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–}}

{{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 |s2cid=12806462 |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.)

{{cite book |author-first=Hozumi |author-last=Hamada |title=1987 IEEE 8th Symposium on Computer Arithmetic (ARITH) |chapter=A new real number representation and its operation |editor-first1=Mary Jane |editor-last1=Irwin |editor-first2=Renato |editor-last2=Stefanelli |date=1987-05-18 |pages=153–157 |doi=10.1109/ARITH.1987.6158698 |location=Washington, D.C., USA |publisher=IEEE Computer Society Press |isbn=0-8186-0774-2 |s2cid=15189621 }} [http://www.acsel-lab.com/arithmetic/arith8/papers/ARITH8_Hamada.pdf]

{{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 |s2cid=121337883 }} [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 |pages=113–126 |url=https://books.google.com/books?id=E4c3DwAAQBAJ}}

{{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 |access-date=2018-07-09 }}

{{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 |url-status=live |archive-url=https://web.archive.org/web/20160606203112/http://www.johngustafson.net/presentations/Right-SizingPrecision1.pdf |archive-date=2016-06-06}}

{{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, Massachusetts, USA |date=2016-12-12 |isbn=978-1-4899-7981-0 }}

{{cite book |title=Accuracy and Stability of Numerical Algorithms |edition=2 |author-first=Nicholas John |author-link=Nicholas Higham |author-last=Higham |publisher=Society for Industrial and Applied Mathematics (SIAM) |date=2002 |isbn=978-0-89871-521-7 |id=0-89871-355-2 |pages=49 |url=https://books.google.com/books?id=epilvM5MMxwC}}

{{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 |url-status=live |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]

}}

tapered floating point#Leveling

History

See also

References

Further reading