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Tapered floating point
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<h2 id="history">History</h2> <p>The tapered floating-point scheme was first proposed by <a href="/facts/Robert_Morris_(cryptographer)/3L0QovBu">Robert Morris</a> of <a href="/facts/Bell_Laboratories/fKP5Gk0H">Bell Laboratories</a> in 1971,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> and refined with <i>leveling</i> by Masao Iri and Shouichi Matsui of <a href="/facts/University_of_Tokyo/Td0xnQDF">University of Tokyo</a> in 1981,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and by Hozumi Hamada of <a href="/facts/Hitachi,_Ltd./KeK38e82">Hitachi, Ltd.</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>Alan Feldstein of <a href="/facts/Arizona_State_University/e4lLNfmP">Arizona State University</a> and Peter Turner<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> of <a href="/facts/Clarkson_University/DXf9Jf59">Clarkson University</a> described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>In 2013, <a href="/facts/John_Gustafson_(scientist)/h8qV1z0E">John Gustafson</a> proposed the <a href="/facts/Unum_(number_format)/Bo3XNr7C">Unum</a> number system, a variant of tapered floating-point arithmetic with an <i>exact</i> bit added to the representation and some <a href="/facts/Interval_arithmetic/1ylnKraf">interval</a> interpretation to the non-exact values.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Logarithmic_number_system/V8zygCjX">Logarithmic number system</a> (LNS)</li> <li><a href="/facts/Symmetric_level-index_arithmetic/2cl2lBwM">Symmetric level-index arithmetic</a> (SLI)</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Luk, Clement (1974-10-02) [1974-09-30]. "Microprogrammed significance arithmetic with tapered floating point representation". <i>Conference record of the 7th annual workshop on Microprogramming - MICRO 7</i>. Palo Alto, California, USA. pp. 248–252. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F800118.803869">10.1145/800118.803869</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781450374217.{{cite book}}: CS1 maint: location missing publisher (link)</li> <li>Azmi, Aquil M.; Lombardi, Fabrizio (1989-09-06). <a href="http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Azmi.pdf">"On a tapered floating point system"</a> (PDF). <i>Proceedings of 9th Symposium on Computer Arithmetic</i>. Santa Monica, California, USA: <a href="/facts/IEEE/UddC4pUW">IEEE</a>. pp. 2–9. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FARITH.1989.72803">10.1109/ARITH.1989.72803</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8186-8963-3. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:38180269">38180269</a>. <a href="https://web.archive.org/web/20180713190341/http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Azmi.pdf">Archived</a> (PDF) from the original on 2018-07-13. Retrieved 2018-07-13.</li> <li>Yokoo, Hidetoshi (August 1992). <a href="https://dl.acm.org/citation.cfm?id=141503">"Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field"</a>. <i>IEEE Transactions on Computers</i>. 41 (8). Washington, DC, USA: <a href="/facts/IEEE_Computer_Society/asUJfKHp">IEEE Computer Society</a>: 1033–1039. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F12.156546">10.1109/12.156546</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0018-9340">0018-9340</a>.. Previously published in: Yokoo, Hidetoshi (June 1991). Komerup, Peter; <a href="/facts/David_Matula/JYEFTzrF">Matula, David W.</a> (eds.). "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field". <i>Proceedings of the 10th IEEE Symposium on Computer Arithmetic (ARITH 10)</i>. Washington, DC, USA: <a href="/facts/IEEE_Computer_Society/asUJfKHp">IEEE Computer Society</a>: 110–117.</li> <li>Anuta, Michael A.; Lozier, Daniel W.; Turner, Peter R. (March–April 1996) [1995-11-15]. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4907584">"The MasPar MP-1 As a Computer Arithmetic Laboratory"</a>. <i><a href="/facts/Journal_of_Research_of_the_National_Institute_of_Standards_and_Technology/NQwepLqx">Journal of Research of the National Institute of Standards and Technology</a></i>. 101 (2): 165–174. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.6028%2Fjres.101.018">10.6028/jres.101.018</a>. <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4907584">4907584</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/27805123">27805123</a>.</li> <li>Ray, Gary (2010-02-04). <a href="http://chipdesignmag.com/display.php?articleId=3921">"Between Fixed and Floating Point"</a>. <i>Electronic Systems Design Engineering incorporating Chip Design</i>. <a href="https://web.archive.org/web/20180710035319/http://chipdesignmag.com/display.php?articleId=3921">Archived</a> from the original on 2018-07-10. Retrieved 2018-07-09.</li> <li>Beebe, Nelson H. F. (2017-08-22). "Chapter H.8 - Unusual floating-point systems". <i>The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library</i> (1 ed.). Salt Lake City, Utah, USA: <a href="/facts/Springer_International_Publishing_AG/TpTgApAh">Springer International Publishing AG</a>. p. 966. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-64110-2">10.1007/978-3-319-64110-2</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-64109-6. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/2017947446">2017947446</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:30244721">30244721</a>. […] representation with a moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called <i>tapered arithmetic</i>) […]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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] <a href="https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf" target="_blank">https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Morris, Sr., Robert H. (December 1971). "Tapered Floating Point: A New Floating-Point Representation". IEEE Transactions on Computers. C-20 (12). IEEE: 1578–1579. doi:10.1109/T-C.1971.223174. ISSN 0018-9340. S2CID 206618406. <a href="/wiki/Robert_Morris_(cryptographer)" target="_blank">/wiki/Robert_Morris_(cryptographer)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Matsui, Shourichi; Iri, Masao (1981-11-05) [January 1981]. "An Overflow/Underflow-Free Floating-Point Representation of Numbers". Journal of Information Processing. 4 (3). Information Processing Society of Japan (IPSJ): 123–133. ISSN 1882-6652. NAID 110002673298 NCID AA00700121. Retrieved 2018-07-09. [2]. Also reprinted in: Swartzlander, Jr., Earl E., ed. (1990). Computer Arithmetic. Vol. II. IEEE Computer Society Press. pp. 357–. <a href="https://www.researchgate.net/publication/243733790" target="_blank">https://www.researchgate.net/publication/243733790</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="978-0-89871-521-7" target="_blank">978-0-89871-521-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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] <a href="https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf" target="_blank">https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.09.handout.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Hamada, Hozumi (June 1983). "URR: Universal representation of real numbers". New Generation Computing. 1 (2): 205–209. doi:10.1007/BF03037427. ISSN 0288-3635. S2CID 12806462. Retrieved 2018-07-09. (NB. The URR representation coincides with Elias delta (δ) coding.) <a href="https://www.researchgate.net/publication/220619145_URR_Universal_Representation_of_Real_Numbers" target="_blank">https://www.researchgate.net/publication/220619145_URR_Universal_Representation_of_Real_Numbers</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Hamada, Hozumi (1987-05-18). "A new real number representation and its operation". In Irwin, Mary Jane; Stefanelli, Renato (eds.). 1987 IEEE 8th Symposium on Computer Arithmetic (ARITH). Washington, D.C., USA: IEEE Computer Society Press. pp. 153–157. doi:10.1109/ARITH.1987.6158698. ISBN 0-8186-0774-2. S2CID 15189621. [3] <a href="0-8186-0774-2" target="_blank">0-8186-0774-2</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>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. <a href="978-0-26203686-3" target="_blank">978-0-26203686-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>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. 56 (3–4). Amsterdam, Netherlands: International Association for Mathematics and Computers in Simulation (IMACS) / Elsevier Science Publishers B. V.: 517–532. doi:10.1016/j.apnum.2005.04.018. ISSN 0168-9274. Retrieved 2018-07-09. <a href="https://www.researchgate.net/publication/223110157" target="_blank">https://www.researchgate.net/publication/223110157</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="978-0-26203686-3" target="_blank">978-0-26203686-3</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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. <a href="/wiki/John_Leroy_Gustafson" target="_blank">/wiki/John_Leroy_Gustafson</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>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. <a href="978-1-4899-7981-0" target="_blank">978-1-4899-7981-0</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> </ol>