270:
746:
The simplification of multiplication, division, roots, and powers is counterbalanced by the cost of evaluating these functions for addition and subtraction. This added cost of evaluation may not be critical when using an LNS primarily for increasing the precision of floating-point math operations.
893:(ALU), demonstrated LNSs as a "more accurate alternative to floating-point", with improved speed. Further improvement of the LNS design based on the ELM architecture has shown its capability to offer significantly higher speed and accuracy than floating-point as well.
1279:
LEONELLIs logarithmische
Supplemente, als ein Beitrag, Mängel der gewöhnlichen Logarithmentafeln zu ersetzen. Aus dem Französischen nebst einigen Zusätzen von GOTTFRIED WILHELM LEONHARDI, Souslieutenant beim kurfürstlichen sächsischen
126:
815:
and developed one that could solve algebraic equations with eight terms, finding the roots, including the complex ones. One part of this machine called an "endless spindle" allowed the mechanical expression of the relation
1733:
1468:
Coleman, John
Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Licko, Miroslav; Pohl, Zdenek; Hermanek, Antonin (2002-08-07) . "The European Logarithmic Microprocessor – a QR RLS application".
506:
1514:
Coleman, John
Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Tichy, Milan; Pohl, Zdenek; Hermanek, Antonin; Benschop, Nico F. (April 2008) . "The European Logarithmic Microprocessor".
665:
885:
A substantial effort to explore the applicability of LNSs as a viable alternative to floating point for general-purpose processing of single-precision real numbers is described in the context of the
394:
287:. This formulation simplifies the operations of multiplication, division, powers and roots, since they are reduced down to addition, subtraction, multiplication, and division, respectively.
583:
1795:
1810:
1710:
865:
1695:
737:
701:
265:{\displaystyle X\rightarrow {\begin{cases}x=\log _{b}{\big |}X{\big |}\,,\\s={\begin{cases}0{\text{ if }}X>0\,,\\1{\text{ if }}X<0\,.\end{cases}}\end{cases}}}
118:
86:
62:
1430:
1768:
1601:
Fu, Haohuan; Mencer, Oskar; Luk, Wayne (2007-01-02) . "Comparing floating-point and logarithmic number representations for reconfigurable acceleration".
792:
1838:
1318:
1676:
1021:
1850:
1322:
400:
1784:
1950:
1035:
812:
588:
1791:
1910:
1622:
1577:
1452:
1354:
290:
On the other hand, the operations of addition and subtraction are more complicated and they are calculated by the formulae:
1894:
1869:
1346:
296:
1970:
1374:
1727:
1490:
1200:
1879:
929:
514:
1651:
1517:
1136:
1083:
1078:
980:
1941:
esprit – European
Logarithmic Microprocessor (formerly the 'High Speed Logarithmic Arithmetic' (HSLA) project)
788:
Samuel Lee and Albert Edgar described a similar system, which they called the "Focus" number system, in 1977.
1760:
1693:
Muller, Jean-Michel; Scherbyna, Alexandre; Tisserand, Arnaud (July 1995). "Semi-Logarithmic Number
Systems".
897:
777:
A similar LNS named "signed logarithmic number system" (SLNS) was described in 1975 by Earl
Swartzlander and
1300:
1965:
1471:
Conference Record of Thirty-Fifth
Asilomar Conference on Signals, Systems and Computers (Cat.No.01CH37256)
1207:
1120:
1063:
1719:
1264:
819:
1224:
771:
33:
199:
141:
1816:
1415:
1370:
808:
1643:
1383:
Modern instruments and methods of calculation: a handbook of the Napier
Tercentenary Exhibition
925:
919:
760:
1340:
889:(ELM). A fabricated prototype of the processor, which has a 32-bit cotransformation-based LNS
706:
670:
1176:
890:
778:
978:
Lee, Samuel C.; Edgar, Albert D. (September 1979). "Addendum to "The Focus Number System"".
1888:
1440:
1436:
1336:
1184:
796:
8:
1975:
1887:
Amir
Sabbagh, Molahosseini; de Sousa, Leonel Seabra; Chip-Hong Chang, eds. (2017-03-21).
1026:
940:
93:
29:
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1821:
1749:
1192:
935:
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103:
71:
47:
1945:
955:
1916:
1906:
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1618:
1583:
1573:
1538:
1496:
1486:
1448:
1350:
1196:
1157:
1104:
1047:
1001:
900:-based applications where most arithmetic operations are multiplication or division.
879:
1898:
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1660:
1610:
1565:
1530:
1478:
1396:
1237:
1188:
1149:
1096:
1039:
993:
914:
791:
The mathematical foundations for addition and subtraction in an LNS trace back to
1855:
950:
764:
284:
276:
89:
1902:
1642:
Muller, Jean-Michel; Scherbyna, Alexandre; Tisserand, Arnaud (February 1998).
1482:
1959:
1861:
1672:
1587:
1542:
1500:
1161:
1134:
Lee, Samuel C.; Edgar, Albert D. (November 1977). "The Focus Number System".
1108:
1051:
1005:
782:
1614:
1153:
1100:
997:
867:, with the aim of extracting the logarithm of a sum as a sum of logarithms.
770:
Nicholas
Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for
871:
1569:
1534:
1287:
Supplément logarithmique. Théorie des logarithmes additionels et diductifs
1259:
Supplément logarithmique. Théorie des logarithmes additionels et diductifs
1242:
1219:
1043:
945:
280:
25:
1834:
1432:
Scientific Simulations with Special Purpose Computers: The GRAPE Systems
1177:"Chapter I.1.: Microcomputer Design – Focus Microcomputer Number System"
1010:(NB. Nicholas Kingsbury's name is incorrectly spelled in this citation.)
1664:
1556:
Ismail, R. Che; Coleman, John Nicholas (2011-08-18) . "ROM-less LNS".
1385:. Gerstein – University of Toronto. London, UK: G. Bell. p. 263.
1233:
781:; rather than use two's complement notation for the logarithms, they
65:
1419:, Mechanism and Machine Theory, Vol. 43, No. 8, pp. 1055-1063, 2008.
1886:
1699:
1603:
2006 IEEE International Conference on Field Programmable Technology
785:
them (scale the numbers being represented) to avoid negative logs.
97:
1940:
1920:
1890:
Embedded Systems Design with Special Arithmetic and Number Systems
501:{\displaystyle \log _{b}{\bigg |}|X|-|Y|{\bigg |}=x+d_{b}(y-x)\,,}
909:
875:
1712:
Applications of Digital Signal Processing to Audio and Acoustics
1935:
1696:
Proceedings of the 12th IEEE Symposium on Computer Arithmetic
1319:"Logarithm: Addition and Subtraction, or Gaussian Logarithms"
1513:
1467:
1076:
1606:
1561:
1526:
1474:
1401:
Encyclopédie des sciences mathematiques pures et appliquées
1145:
1092:
989:
258:
251:
1020:
Kingsbury, Nicholas G.; Rayner, Peter J. W. (1971-01-28).
660:{\displaystyle d_{b}(z)=\log _{b}{\big |}1-b^{z}{\big |}}
1761:"Gauss'sche Additionslogarithmen feiern 200. Geburtstag"
1692:
1641:
759:
and published at least three times as an alternative to
1303:(1808-02-12). "LEONELLI, Logarithmische Supplemente".
1854:
Hayes, Brian (2017). "Chapter 8: Higher Arithmetic".
1081:(December 1975). "The Sign/Logarithm Number System".
822:
709:
673:
591:
517:
403:
299:
129:
106:
74:
50:
1363:
1284:(NB. An expanded translation of Zecchini Leonelli's
1951:
A Short Account on Leonardo Torres’ Endless Spindle
1709:Kahrs, Mark; Brandenburg, Karlheinz, eds. (2002) .
1416:
A Short Account on Leonardo Torres' Endless Spindle
1339:(2004) . Gray, Jeremy; Dohse, Fritz-Egbert (eds.).
1282:(in German). Dresden: Walther'sche Hofbuchhandlung.
859:
731:
695:
659:
577:
500:
389:{\displaystyle \log _{b}(|X|+|Y|)=x+s_{b}(y-x)\,,}
388:
264:
112:
80:
64:, is represented in an LNS by two components: the
56:
1785:"Rechnerarithmetik: Logarithmische Zahlensysteme"
1114:
1057:
455:
419:
1957:
1708:
1377:The Instrumental Solution of Numerical Equations
1070:
1022:"Digital filtering using logarithmic arithmetic"
24:) is an arithmetic system used for representing
1558:2011 IEEE 20th Symposium on Computer Arithmetic
1403:. Paris, France: Gauthier-Villars. p. 351.
1329:
1218:Edgar, Albert D.; Lee, Samuel C. (March 1979).
1019:
1381:. Written at Napier Tercentenary Exhibition.
1857:Foolproof, and Other Mathematical Meditations
1555:
1270:
1013:
878:) special-purpose supercomputer that won the
652:
629:
175:
165:
1507:
1461:
1250:
973:
971:
1600:
1549:
1428:
1422:
1389:
578:{\displaystyle s_{b}(z)=\log _{b}(1+b^{z})}
1946:A VHDL library for LNS hardware generation
1407:
1335:
283:being always equal to 1 and a non-integer
1782:
1429:Makino, Junichiro; Taiji, Makoto (1998).
1307:(in German) (45). Halle-Leipzig: 353–356.
1293:
1276:
1241:
1211:
1175:Lee, Samuel C.; Edgar, Albert D. (1977).
1168:
1127:
1036:Institution of Engineering and Technology
968:
494:
382:
244:
219:
180:
1369:
1323:Encyclopædia Britannica Eleventh Edition
1311:
1256:
1217:
1174:
1133:
1115:Swartzlander, Jr., Earl E., ed. (1990).
1058:Swartzlander, Jr., Earl E., ed. (1990).
977:
813:analogue calculating mechanical machines
807:In the late 1800s, the Spanish engineer
1815:Hayes, Brian (September–October 2009).
1594:
1342:Carl Friedrich Gauss – Titan of Science
511:where the "sum" function is defined by
1958:
1758:
1395:
1345:. Spectrum series (revised ed.).
1263:(NB. 1802/1803 is the year XI. in the
1119:. Vol. I. Los Alamitos, CA, USA:
1062:. Vol. I. Los Alamitos, CA, USA:
1853:
1814:
1277:Leonhardi, Gottfried Wilhelm (1806).
1181:Microcomputer Design and Applications
755:Logarithmic number systems have been
1895:Springer International Publishing AG
1748:(NB. Describes a 13-bit LNS used in
1299:
1792:Friedrich-Schiller-Universität Jena
1783:Zehendner, Eberhard (Summer 2008).
1347:Mathematical Association of America
1220:"FOCUS Microcomputer Number System"
887:European Logarithmic Microprocessor
585:, and the "difference" function by
13:
1635:
1473:. Vol. 1. Monterey, CA, USA:
1286:
1193:10.1016/B978-0-12-442350-3.50005-5
14:
1987:
1929:
1644:"Semi-Logarithmic Number Systems"
1261:(in French). Bordeaux: Brossier.
930:symmetric level-index arithmetic
860:{\displaystyle y=\log(1+10^{x})}
1841:from the original on 2018-07-09
1801:from the original on 2018-07-09
1771:from the original on 2018-07-07
1739:from the original on 2018-07-07
1682:from the original on 2018-07-13
1079:Alexopoulos, Aristides Georgiou
802:
1790:(Lecture script) (in German).
1759:Kremer, Hermann (2002-08-29).
1652:IEEE Transactions on Computers
1518:IEEE Transactions on Computers
1137:IEEE Transactions on Computers
1084:IEEE Transactions on Computers
981:IEEE Transactions on Computers
854:
835:
726:
720:
690:
684:
608:
602:
572:
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528:
491:
479:
449:
441:
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379:
367:
345:
341:
333:
325:
317:
313:
275:An LNS can be considered as a
133:
1:
961:
92:(as a binary word usually in
1936:A site that lists LNS papers
1257:Leonelli, Zecchini (1803) .
1077:Swartzlander, Jr., Earl E.;
7:
1305:Allgemeine Literaturzeitung
1301:GauĂź, Johann Carl Friedrich
1179:. In Lee, Samuel C. (ed.).
1121:IEEE Computer Society Press
1064:IEEE Computer Society Press
903:
896:LNSs are sometimes used in
870:A LNS has been used in the
39:
10:
1992:
1720:Kluwer Academic Publishing
1265:French Republican Calendar
750:
1971:Digital signal processing
1903:10.1007/978-3-319-49742-6
1691:Previously published in:
1483:10.1109/ACSSC.2001.986897
1225:Communications of the ACM
772:digital signal processing
34:digital signal processing
18:logarithmic number system
732:{\displaystyle d_{b}(z)}
696:{\displaystyle s_{b}(z)}
1817:"The Higher Arithmetic"
1615:10.1109/FPT.2006.270342
1371:Horsburg, Ellice Martin
1154:10.1109/TC.1977.1674770
1101:10.1109/T-C.1975.224172
998:10.1109/TC.1979.1675442
809:Leonardo Torres Quevedo
926:Level-index arithmetic
920:Tapered floating point
861:
811:conceived a series of
757:independently invented
733:
697:
661:
579:
502:
390:
266:
114:
82:
58:
1852:. Also reprinted in:
1570:10.1109/ARITH.2011.15
1535:10.1109/TC.2007.70791
1437:John Wiley & Sons
1337:Dunnington, Guy Waldo
1243:10.1145/359080.359085
891:arithmetic logic unit
862:
779:Aristides Alexopoulos
734:
698:
662:
580:
503:
391:
267:
115:
83:
59:
1864:. pp. 113–126.
1609:. pp. 337–340.
1477:. pp. 155–159.
1185:Academic Press, Inc.
820:
799:in the early 1800s.
797:Carl Friedrich Gauss
707:
671:
589:
515:
401:
297:
127:
104:
72:
48:
1966:Computer arithmetic
1835:10.1511/2009.80.364
1445:1998sssc.book.....M
1280:Feldartilleriecorps
1117:Computer Arithmetic
1113:Also reprinted in:
1060:Computer Arithmetic
1056:Also reprinted in:
1044:10.1049/el:19710039
1027:Electronics Letters
741:Gaussian logarithms
1822:American Scientist
1755:during the 1980s.)
1753:music synthesizers
1564:. pp. 43–51.
936:Gaussian logarithm
857:
739:are also known as
729:
693:
667:. These functions
657:
575:
498:
386:
262:
257:
250:
110:
78:
54:
1912:978-3-319-49741-9
1765:de.sci.mathematik
1665:10.1109/12.663760
1624:978-0-7803-9728-6
1579:978-1-4244-9457-6
1454:978-0-471-96946-4
1379:by D. Gibb, M.A."
1356:978-0-88385-547-8
880:Gordon Bell Prize
793:Zecchini Leonelli
233:
208:
113:{\displaystyle s}
81:{\displaystyle x}
57:{\displaystyle X}
32:, especially for
1983:
1924:
1883:
1871:978-0-26203686-3
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975:
941:Zech's logarithm
915:Subnormal number
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767:number systems.
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279:number with the
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30:digital hardware
28:in computer and
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1636:Further reading
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1399:(1908). "I23".
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1187:pp. 1–40.
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956:ÎĽ-law algorithm
951:A-law algorithm
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774:(DSP) in 1971.
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1931:
1930:External links
1928:
1927:
1926:
1911:
1893:(1 ed.).
1884:
1870:
1860:(1 ed.).
1829:(5): 364–368.
1812:
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1659:(2): 145–151.
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1397:Mehmke, Rudolf
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1038:(IET): 56–58.
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90:absolute value
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1862:The MIT Press
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1770:
1767:(in German).
1766:
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1729:0-7923-8130-0
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1492:0-7803-7147-X
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1198:
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1148:: 1167–1170.
1147:
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1095:: 1238–1242.
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541:
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107:
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75:
67:
51:
37:
35:
31:
27:
23:
19:
1889:
1880:0-26203686-X
1856:
1843:. Retrieved
1826:
1820:
1803:. Retrieved
1773:. Retrieved
1764:
1741:. Retrieved
1711:
1694:
1684:. Retrieved
1656:
1650:
1602:
1596:
1557:
1551:
1522:
1516:
1509:
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1463:
1431:
1424:
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1088:
1082:
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1059:
1031:
1025:
1015:
985:
979:
895:
886:
884:
872:Gravity Pipe
869:
806:
803:Applications
790:
787:
776:
769:
754:
745:
510:
289:
274:
43:
26:real numbers
21:
17:
15:
1925:(389 pages)
1704:. Bath, UK.
1529:: 532–546.
1413:F. Thomas.
1236:: 166–177.
946:ITU-T G.711
761:fixed-point
281:significand
96:), and its
1976:Logarithms
1960:Categories
1921:2017934074
1845:2018-07-09
1805:2018-07-09
1775:2018-07-07
1743:2018-07-07
1686:2018-07-11
962:References
44:A number,
1673:0018-9340
1588:1063-6889
1543:0018-9340
1501:1058-6393
1234:ACM Press
1162:0018-9340
1109:0018-9340
1052:0013-5194
1006:0018-9340
928:(LI) and
882:in 1999.
833:
638:−
625:
551:
486:−
438:−
415:
374:−
311:
161:
134:→
88:) of its
66:logarithm
1839:Archived
1796:Archived
1769:Archived
1734:Archived
1700:ARITH 12
1677:Archived
1373:(1914).
904:See also
285:exponent
98:sign bit
40:Overview
1441:Bibcode
1349:(MAA).
992:: 693.
910:Decibel
876:GRAPE-5
751:History
1919:
1909:
1878:
1868:
1750:Yamaha
1726:
1671:
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1586:
1576:
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1499:
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1160:
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1107:
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1050:
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1799:(PDF)
1788:(PDF)
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1034:(2).
988:(9).
932:(SLI)
922:(TFP)
1917:LCCN
1907:ISBN
1876:ISBN
1866:ISBN
1724:ISBN
1669:ISSN
1619:ISBN
1607:IEEE
1584:ISSN
1574:ISBN
1562:IEEE
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1527:IEEE
1497:ISSN
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1475:IEEE
1449:ISBN
1351:ISBN
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1002:ISSN
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898:FPGA
795:and
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1899:doi
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.