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Basic properties of floating-point and interval arithmetic can directly be derived from this abstract mathematical model. Interval operations are defined as set operations for elements of the set \u00afIR of closed and connected sets of real numbers. As such, they form an algebraically closed subset of the powerset of the real numbers. This property leads to explicit formulas for the arithmetic operations of floating-point intervals of \u00afIF, which are executable on the computer. Arithmetic for intervals of \u00afIF forms an exception free calculus, i.e., arithmetic operations for intervals of \u00afIF always lead to intervals of \u00afIF again.<\/jats:p>\n          <jats:p>Later sections are concerned with programming support and hardware for interval arithmetic. Both are a must and absolutely necessary to move interval arithmetic more into the center of scientific computing. With some minor hardware additions, interval operations can be made as fast as simple floating-point operations.<\/jats:p>\n          <jats:p>\n            In vector and matrix spaces for real, complex, and interval data, the dot product is a fundamental arithmetic operation. Computing the dot product of two vectors with floating-point components\n            <jats:bold>exactly<\/jats:bold>\n            substantially speeds up floating-point and interval arithmetic as well as the accuracy of the computed result. Hardware needed for the exact dot product is very modest. The exact dot product is essential for long real and long interval arithmetic.\n          <\/jats:p>\n          <jats:p>Section 9 illustrates that interval arithmetic as developed in this article already has a long tradition. Products based on these ideas have been available since 1980. Implementing what the article advocates would have a profound effect on mathematical software. Modern processor architecture from Intel, for example, comes quite close to what is requested in this article.<\/jats:p>","DOI":"10.1145\/3264448","type":"journal-article","created":{"date-parts":[[2019,3,14]],"date-time":"2019-03-14T17:11:58Z","timestamp":1552583518000},"page":"1-22","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Mathematics and Speed for Interval Arithmetic"],"prefix":"10.1145","volume":"45","author":[{"given":"Ulrich","family":"Kulisch","sequence":"first","affiliation":[{"name":"KIT, Institut f\u00fcr Angewandte Mathematik, Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2019,3,14]]},"reference":[{"key":"e_1_2_1_1_1","first-page":"297","article-title":"IMACS-GAMM resolution on computer arithmetic","volume":"31","author":"MACS","year":"1989","unstructured":"I MACS and GAMM. 1989 . IMACS-GAMM resolution on computer arithmetic . Math. Comput. Simul. 31 (1989), 297 -- 298 , or in Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik 70:4 (1990). IMACS and GAMM. 1989. IMACS-GAMM resolution on computer arithmetic. Math. Comput. Simul. 31 (1989), 297--298, or in Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik 70:4 (1990).","journal-title":"Math. Comput. Simul."},{"key":"e_1_2_1_2_1","volume-title":"9--16, and Math. Comput. Simul","author":"MACS","year":"1993","unstructured":"GAMM-I MACS proposal for accurate floating-point vector arithmetic. GAMM Rundbrief 2 ( 1993 ), 9--16, and Math. Comput. Simul ., Vol. 35 , IMACS, North Holland , 1993. News of IMACS , Vol. 35, No. 4 (1993), 375--382. GAMM-IMACS proposal for accurate floating-point vector arithmetic. GAMM Rundbrief 2 (1993), 9--16, and Math. Comput. Simul., Vol. 35, IMACS, North Holland, 1993. News of IMACS, Vol. 35, No. 4 (1993), 375--382."},{"key":"e_1_2_1_3_1","first-page":"9","article-title":"New York, 1985","volume":"22","author":"American National Standards Institute\/Institute of Electrical and Electronics Engineers: A Standard for Binary Floating-Point Arithmetic. ANSI\/IEEE Std. 754--","year":"1987","unstructured":"American National Standards Institute\/Institute of Electrical and Electronics Engineers: A Standard for Binary Floating-Point Arithmetic. ANSI\/IEEE Std. 754-- 1987 , New York, 1985 . Reprinted in SIGPLAN 22 , 2 (1987), 9 -- 25 . Also adopted as IEC Standard 559:1989, Revised version 2008, and in ISO\/IEC\/IEEE 60559:2011. American National Standards Institute\/Institute of Electrical and Electronics Engineers: A Standard for Binary Floating-Point Arithmetic. ANSI\/IEEE Std. 754--1987, New York, 1985. Reprinted in SIGPLAN 22, 2 (1987), 9--25. Also adopted as IEC Standard 559:1989, Revised version 2008, and in ISO\/IEC\/IEEE 60559:2011.","journal-title":"Reprinted in SIGPLAN"},{"key":"e_1_2_1_4_1","volume-title":"A Standard for Radix-Independent Floating-Point Arithmetic","author":"American National Standards Institute\/Institute of Electrical and Electronics Engineers","year":"1987","unstructured":"American National Standards Institute\/Institute of Electrical and Electronics Engineers : A Standard for Radix-Independent Floating-Point Arithmetic . ANSI\/IEEE Std. 854- 1987 , New York , 1987. American National Standards Institute\/Institute of Electrical and Electronics Engineers: A Standard for Radix-Independent Floating-Point Arithmetic. ANSI\/IEEE Std. 854-1987, New York, 1987."},{"key":"e_1_2_1_5_1","volume-title":"dated","author":"IFIP","year":"2007","unstructured":"The IFIP WG 2.5 \u2014IEEE 754R letter , dated September 4, 2007 . The IFIP WG 2.5 \u2014IEEE 754R letter, dated September 4, 2007."},{"key":"e_1_2_1_6_1","volume-title":"dated","author":"IFIP","year":"2009","unstructured":"The IFIP WG 2.5 \u2014IEEE P1788 letter , dated September 9, 2009 . The IFIP WG 2.5 \u2014IEEE P1788 letter, dated September 9, 2009."},{"key":"e_1_2_1_7_1","unstructured":"G. Alefeld. 1968. Intervallrechnung \u00fcber den komplexen Zahlen und einige Anwendungen. Dissertation. Universit\u00e4t Karlsruhe.  G. Alefeld. 1968. Intervallrechnung \u00fcber den komplexen Zahlen und einige Anwendungen. Dissertation. Universit\u00e4t Karlsruhe."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.5555\/786447.786506"},{"key":"e_1_2_1_9_1","unstructured":"Ch. Baumhof. 1996. Ein Vektorarithmetik-Koprozessor in VLSI-Technik zur Unterst\u00fctzung des Wissenschaft-lichen Rechnens. Dissertation. Universit\u00e4t Karlsruhe.  Ch. Baumhof. 1996. Ein Vektorarithmetik-Koprozessor in VLSI-Technik zur Unterst\u00fctzung des Wissenschaft-lichen Rechnens. Dissertation. Universit\u00e4t Karlsruhe."},{"key":"e_1_2_1_10_1","unstructured":"D. Biancolin and J. Koenig. 2015. Hardware Accelerator for Exact Dot Product ASPIRE Laboratory University of California Berkeley.  D. Biancolin and J. Koenig. 2015. Hardware Accelerator for Exact Dot Product ASPIRE Laboratory University of California Berkeley."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1977.1674894"},{"key":"e_1_2_1_12_1","doi-asserted-by":"crossref","unstructured":"G. Bohlender. 1978. Genaue Berechnung mehrfacher Summen Produkte und Wurzeln von Gleitkommazahlen und allgemeine Arithmetik in h\u00f6heren Programmiersprachen. Dissertation. Universit\u00e4t Karlsruhe.  G. Bohlender. 1978. Genaue Berechnung mehrfacher Summen Produkte und Wurzeln von Gleitkommazahlen und allgemeine Arithmetik in h\u00f6heren Programmiersprachen. Dissertation. Universit\u00e4t Karlsruhe.","DOI":"10.1007\/978-3-7091-8471-4_4"},{"key":"e_1_2_1_13_1","unstructured":"T. Teufel. 1984. Ein optimaler Gleitkommaprozessor. Dissertation. Universit\u00e4t Karlsruhe.  T. Teufel. 1984. Ein optimaler Gleitkommaprozessor. Dissertation. Universit\u00e4t Karlsruhe."},{"volume-title":"PASCAL-XSC \u2014Sprachbeschreibung mit Beispielen","author":"Klatte R.","key":"e_1_2_1_14_1","unstructured":"R. Klatte , U. Kulisch , M. Neaga , D. Ratz , and Ch. Ullrich . 1991. PASCAL-XSC \u2014Sprachbeschreibung mit Beispielen , Springer , Berlin . Retrieved from http:\/\/www2.math.uni-wuppertal.de\/&sim;xsc\/ or http:\/\/www.xsc.de\/. R. Klatte, U. Kulisch, M. Neaga, D. Ratz, and Ch. Ullrich. 1991. PASCAL-XSC \u2014Sprachbeschreibung mit Beispielen, Springer, Berlin. 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See &lsqb;29&rsqb;."},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-010-0127-7"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-31500-8_50"},{"volume-title":"P1788","author":"Pryce J. D.","key":"e_1_2_1_29_1","unstructured":"J. D. Pryce (Ed.), P1788 , IEEE Standard for Interval Arithmetic , Retrieved from http:\/\/grouper.ieee.org\/groups\/1788\/email\/pdfOWdtH2mOd9.pdf. J. D. Pryce (Ed.), P1788, IEEE Standard for Interval Arithmetic, Retrieved from http:\/\/grouper.ieee.org\/groups\/1788\/email\/pdfOWdtH2mOd9.pdf."},{"key":"e_1_2_1_30_1","unstructured":"IEEE 1788-2015\u2014Standard for Interval Arithmetic. Retrieved from http:\/\/standards.ieee.org\/findstds\/standard\/1788-2015.html.  IEEE 1788-2015\u2014Standard for Interval Arithmetic. 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