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<meta content="Frustrated by the limitations of integer arithmetic, scientists and engineers developed a floating-point number representation. The subsequent adoption of a binary floating-point standard greatly enhanced scientific computing. But floating-point is too specialized for spreadsheets and inadequate for computer graphics. It also complicates programming. Finally, it does not even yield exact results: One very popular calculator gives the answer -0.000000001 to the calculation ((1 \divide 3) x 3)-1! These circumstances, coupled with developments in circuit technology and computation, have encouraged and made possible a more general arithmetic. In this article I propose what I call composite arithmetic, which combines aspects of traditional integer and floating-point arithmetics with less familiar aspects of rational and logarithmic arithmetics. Composite arithmetic complements the binary floating-point standard and satisfies more diverse computational needs. Successful development of a composite arithmetic standard would be most timely, given the burgeoning ability to manufacture complex processors and the interest in extended forms of arithmetic being shown in the research literature. It would also be highly beneficial in support of better electronic calculator arithmetic and standard operation of generic software packages such as those including spreadsheet capabilities." name="eprints.abstract" />
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<meta content="[1] ANSI/IEEE Std. 754-1985, Binary Floating-Point Arithmetic, IEEE Press, Piscataway, N.J., 1985 (also called ISO/IEC 559).
[2] D.W. Lozier, "An Underflow-Induced Graphics Failure Solved by SLI Arithmetic," in Proc. 11th Symp. Computer Arithmetic, IEEE CS Press, Los Alamitos, Calif., 1993, pp. 10-17.
[3] L. Hatton and A. Roberts, "How Accurate Is Scientific Software?" IEEE Trans. Software Eng., Oct. 1994, pp. 785-797.
[4] H.W. Thimbleby, "A New Calculator and Why It Is Necessary," Computer J., Issue 6, 1995, pp. 418-433.
[5] D.W. Matula and P. Kornerup, "Foundations of Finite Precision Rational Arithmetic," Computing, Suppl. 2, 1980, pp. 85-111.
[6] J.K. Iliffe, Basic Machine Principles,Macdonald, London, 1968.
[7] D.M. Lewis, "Interleaved Memory Function Interpolators with Application to an Accurate LNS Arithmetic Unit," IEEE Trans. Computers, Aug. 1994, pp. 974-982.
[8] H. Yokoo, "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field," IEEE Trans. Computers, Aug. 1992, pp. 1,033-1,039.
[9] D.W. Lozier and F.W.J. Olver, "Closure and Precision in Level-Index Arithmetic," SIAM J. Numerical Analysis, Oct. 1990, pp. 1,295-1,304.
[10] M.G. Arnold et al., "Applying Features of IEEE 754 to Sign/Logarithm Arithmetic," IEEE Trans. Computers, Aug. 1992, pp. 1,040-1,050.
[11] R.L. Goodstein, Fundamental Concepts of Mathematics, Pergamon Press, Oxford, UK, second edition, 1979.
Additional References
[1] U.W. Kulisch and W.L. Miranker, "The Arithmetic of the Digital Computer: A New Approach," SIAM Review, Mar. 1986, pp. 1-40.
[2] E. Lange, "Implementation and Test of the ACRITH Facility in a System/370," IEEE Trans. Computers, Sept. 1987, pp. 1,088-1,096." name="eprints.referencetext" />
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<h1 class="ep_tm_pagetitle">Composite Arithmetic: Proposal for a New Standard</h1>
<p style="margin-bottom: 1em" class="not_ep_block"><span class="person_name">Holmes, Neville</span> (1997) <xhtml:em>Composite Arithmetic: Proposal for a New Standard.</xhtml:em> Computer, 30 (3). pp. 65-73. ISSN 0018-9162</p><p style="margin-bottom: 1em" class="not_ep_block"></p><table style="margin-bottom: 1em" class="not_ep_block"><tr><td valign="top" style="text-align:center"><a onmouseover="EPJS_ShowPreview( event, 'doc_preview_2479' );" href="http://eprints.utas.edu.au/2003/1/Cm97Mr.pdf" onmouseout="EPJS_HidePreview( event, 'doc_preview_2479' );"><img alt="[img]" src="http://eprints.utas.edu.au/style/images/fileicons/application_pdf.png" class="ep_doc_icon" border="0" /></a><div class="ep_preview" id="doc_preview_2479"><table><tr><td><img alt="" src="http://eprints.utas.edu.au/2003/thumbnails/1/preview.png" class="ep_preview_image" border="0" /><div class="ep_preview_title">Preview</div></td></tr></table></div></td><td valign="top"><a href="http://eprints.utas.edu.au/2003/1/Cm97Mr.pdf"><span class="ep_document_citation">PDF</span></a> - Requires a PDF viewer<br />139Kb</td></tr></table><p style="margin-bottom: 1em" class="not_ep_block">Official URL: <a href="http://dx.doi.org/10.1109/2.573666">http://dx.doi.org/10.1109/2.573666</a></p><div class="not_ep_block"><h2>Abstract</h2><p style="padding-bottom: 16px; text-align: left; margin: 1em auto 0em auto">Frustrated by the limitations of integer arithmetic, scientists and engineers developed a floating-point number representation. The subsequent adoption of a binary floating-point standard greatly enhanced scientific computing. But floating-point is too specialized for spreadsheets and inadequate for computer graphics. It also complicates programming. Finally, it does not even yield exact results: One very popular calculator gives the answer -0.000000001 to the calculation ((1 \divide 3) x 3)-1! These circumstances, coupled with developments in circuit technology and computation, have encouraged and made possible a more general arithmetic. In this article I propose what I call composite arithmetic, which combines aspects of traditional integer and floating-point arithmetics with less familiar aspects of rational and logarithmic arithmetics. Composite arithmetic complements the binary floating-point standard and satisfies more diverse computational needs. Successful development of a composite arithmetic standard would be most timely, given the burgeoning ability to manufacture complex processors and the interest in extended forms of arithmetic being shown in the research literature. It would also be highly beneficial in support of better electronic calculator arithmetic and standard operation of generic software packages such as those including spreadsheet capabilities.</p></div><table style="margin-bottom: 1em" cellpadding="3" class="not_ep_block" border="0"><tr><th valign="top" class="ep_row">Item Type:</th><td valign="top" class="ep_row">Article</td></tr><tr><th valign="top" class="ep_row">Additional Information:</th><td valign="top" class="ep_row">This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
Copyright 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.</td></tr><tr><th valign="top" class="ep_row">Keywords:</th><td valign="top" class="ep_row">arithmetic, computer arithmetic, standard, integer arithmetic, rational arithmetic, floating point arithmetic, symmetric level indexed arithmetic, number representation</td></tr><tr><th valign="top" class="ep_row">Subjects:</th><td valign="top" class="ep_row"><a href="http://eprints.utas.edu.au/view/subjects/291601.html">290000 Engineering and Technology > 291600 Computer Hardware > 291601 Arithmetic and Logic Structures</a><br /><a href="http://eprints.utas.edu.au/view/subjects/230116.html">230000 Mathematical Sciences > 230100 Mathematics > 230116 Numerical Analysis</a><br /><a href="http://eprints.utas.edu.au/view/subjects/280502.html">280000 Information, Computing and Communication Sciences > 280500 Data Format > 280502 Data Storage Representations</a></td></tr><tr><th valign="top" class="ep_row">ID Code:</th><td valign="top" class="ep_row">2003</td></tr><tr><th valign="top" class="ep_row">Deposited By:</th><td valign="top" class="ep_row"><span class="ep_name_citation"><span class="person_name">Mr Neville Holmes</span></span></td></tr><tr><th valign="top" class="ep_row">Deposited On:</th><td valign="top" class="ep_row">26 Sep 2007</td></tr><tr><th valign="top" class="ep_row">Last Modified:</th><td valign="top" class="ep_row">09 Jan 2008 02:30</td></tr><tr><th valign="top" class="ep_row">ePrint Statistics:</th><td valign="top" class="ep_row"><a target="ePrintStats" href="/es/index.php?action=show_detail_eprint;id=2003;">View statistics for this ePrint</a></td></tr></table><p align="right">Repository Staff Only: <a href="http://eprints.utas.edu.au/cgi/users/home?screen=EPrint::View&eprintid=2003">item control page</a></p>
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