View source for Tapered floating point

{{Short description|Variant of floating-point numbers in computers}}
{{Use list-defined references|date=January 2022}}
In computing, '''tapered floating point''' ('''TFP''') is a format similar to [[Floating-point arithmetic|floating point]], but with variable-sized entries for the [[Significand|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.<ref name="Zehendner_2008"/>

Thus numbers with a small exponent, i.e. whose [[Order of magnitude|order of magnitude]] is close to the one of 1, have a higher [[Precision (computer science)|relative precision]] than those with a large exponent.

== History ==

{{anchor|Leveling}}The tapered floating-point scheme was first proposed by [[Biography:Robert Morris (cryptographer)|Robert Morris]] of Bell Laboratories in 1971,<ref name="Morris_1971"/> and refined with ''leveling'' by Masao Iri and Shouichi Matsui of University of Tokyo in 1981,<ref name="Matsui_1981"/><ref name="Higham_2002"/><ref name="Zehendner_2008"/> and by Hozumi Hamada of Hitachi, Ltd.<ref name="Hamada_1983"/><ref name="Hamada_1987"/><ref name="Hayes_2009"/>

Alan Feldstein of [[Organization:Arizona State University|Arizona State University]] and Peter Turner<ref name="Feldstein_2006"/> of [[Organization:Clarkson University|Clarkson University]] described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions.<ref name="Hayes_2009"/>

In 2013, [[Biography:John Gustafson (scientist)|John Gustafson]] proposed the [[Unum (number format)|Unum]] number system, a variant of tapered floating-point arithmetic with an ''exact'' bit added to the representation and some [[Interval arithmetic|interval]] interpretation to the non-exact values.<ref name="Gustafson_2013"/><ref name="Muller_2016"/>

==See also==
* [[Logarithmic number system]] (LNS)
* [[Symmetric level-index arithmetic]] (SLI)

==References==
{{Reflist|refs=
<ref name="Morris_1971">{{cite journal |title=Tapered Floating Point: A New Floating-Point Representation |author-first=Robert H. |author-last=Morris, Sr. |journal=[[IEEE Transactions on Computers]] |publisher=IEEE |volume=C-20 <!-- |number=6? --> |issue=12 |pages=1578–1579 |date=December 1971 |issn=0018-9340 |doi=10.1109/T-C.1971.223174 |s2cid=206618406 }}</ref>
<ref name="Matsui_1981">{{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=[[Organization:Information Processing Society of Japan|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–}}</ref>
<ref name="Hamada_1983">{{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<!-- https://link.springer.com/article/10.1007/BF03037427 --><!-- https://www.semanticscholar.org/paper/URR%3A-Universal-representation-of-real-numbers-Hamada/ae987cfad0b240d49245995421d0ec147ac72ae1 --> |access-date=2018-07-09}} (NB. The URR representation coincides with Elias delta (δ) coding.)</ref>
<ref name="Hamada_1987">{{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<!-- /21 --> |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<!-- https://web.archive.org/web/20180709212237/http://www.acsel-lab.com/arithmetic/arith8/papers/ARITH8_Hamada.pdf -->]</ref>
<ref name="Hayes_2009">{{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<!-- https://web.archive.org/web/20180709200456/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}}</ref>
<ref name="Feldstein_2006">{{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 }}</ref>
<ref name="Gustafson_2013">{{cite web |author-first=John Leroy |author-last=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}}</ref>
<ref name="Muller_2016">{{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 }}</ref>
<ref name="Higham_2002">{{cite book |title=Accuracy and Stability of Numerical Algorithms |edition=2 |author-first=Nicholas John |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}}</ref>
<ref name="Zehendner_2008">{{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]</ref>
}}

==Further reading==
* {{cite book |author-first=Clement |author-last=Luk |title=Conference record of the 7th annual workshop on Microprogramming - MICRO 7 |chapter=Microprogrammed significance arithmetic with tapered floating point representation |pages=248–252 |location=Palo Alto, California, USA |orig-date=1974-09-30 |date=1974-10-02 |doi=10.1145/800118.803869 |isbn=9781450374217 |doi-access=free }}
* {{cite book |author-first1=Aquil M. |author-last1=Azmi |author-first2=Fabrizio |author-last2=Lombardi |title=Proceedings of 9th Symposium on Computer Arithmetic |chapter=On a tapered floating point system |date=1989-09-06<!-- /08 --> |isbn=0-8186-8963-3 |doi=10.1109/ARITH.1989.72803 |publisher=IEEE |location=Santa Monica, California, USA |pages=2–9 |s2cid=38180269 |chapter-url=http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Azmi.pdf |access-date=2018-07-13 |url-status=live |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= IEEE Transactions on Computers|volume=41 |issue=8 |date=August 1992 |issn=0018-9340 |pages=1033–1039 |publisher=[[Organization:IEEE Computer Society|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 |journal=Proceedings of the 10th IEEE Symposium on Computer Arithmetic (ARITH 10) |editor-first1=Peter |editor-last1=Komerup |editor-first2=David W. |editor-last2=Matula |editor2-link=David Matula|date=June 1991 |publisher=[[Organization:IEEE Computer Society|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 |pages=165–174 |number=2 |date=March–April 1996 |orig-date=1995-11-15 |doi=10.6028/jres.101.018 |pmid=27805123 |pmc=4907584}}
* {{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 |url-status=live |archive-url=https://web.archive.org/web/20180710035319/http://chipdesignmag.com/display.php?articleId=3921 |archive-date=2018-07-10}}
* {{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter H.8 - Unusual floating-point systems |date=2017-08-22 |location=Salt Lake City, Utah, USA |publisher=Springer International Publishing AG |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |page=966 |s2cid=30244721 |quote=[…] representation with a moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called ''tapered arithmetic'') […]}}

[[Category:Computer arithmetic]]


{{Sourceattribution|Tapered floating point}}