Logarithmic number system

A logarithmic number system (LNS) is an arithmetic system used for representing real numbers in computer and digital hardware, especially for digital signal processing.

Overview

A number, $X$ , is represented in an LNS by two components: the logarithm ( $x$ ) of its absolute value (as a binary word usually in two's complement), and its sign bit ( $s$ ):

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}}

An LNS can be considered as a floating-point number with the significand being always equal to 1 and a non-integer exponent. This formulation simplifies the operations of multiplication, division, powers and roots, since they are reduced down to addition, subtraction, multiplication, and division, respectively.

On the other hand, the operations of addition and subtraction are more complicated and they are calculated by the formulae:

\log _{b}(|X|+|Y|)=x+s_{b}(y-x)\,,

\log _{b}{\bigg |}|X|-|Y|{\bigg |}=x+d_{b}(y-x)\,,

where the "sum" function is defined by $s_{b}(z)=\log _{b}(1+b^{z})$ , and the "difference" function by $d_{b}(z)=\log _{b}{\big |}1-b^{z}{\big |}$ . These functions $s_{b}(z)$ and $d_{b}(z)$ are also known as Gaussian logarithms.

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.

History

Logarithmic number systems have been independently invented and published at least three times as an alternative to fixed-point and floating-point number systems.^[1]

Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal processing (DSP) in 1971.^[2]

A similar LNS named "signed logarithmic number system" (SLNS) was described in 1975 by Earl Swartzlander and Aristides Alexopoulos; rather than use two's complement notation for the logarithms, they offset them (scale the numbers being represented) to avoid negative logs.^[3]

Samuel Lee and Albert Edgar described a similar system, which they called the "Focus" number system, in 1977.^[4]^[1]^[5]^[6]

The mathematical foundations for addition and subtraction in an LNS trace back to Zecchini Leonelli and Carl Friedrich Gauss in the early 1800s.^[7]^[8]^[9]^[10]^[11]

Applications

In the late 1800s, the Spanish engineer Leonardo Torres Quevedo conceived a series of analogue calculating mechanical machines^[12]^[13] 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 $y=\log(1+10^{x})$ ,^[14] with the aim of extracting the logarithm of a sum as a sum of logarithms.

A LNS has been used in the Gravity Pipe (GRAPE-5) special-purpose supercomputer^[15] that won the Gordon Bell Prize in 1999.

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 European Logarithmic Microprocessor (ELM).^[16]^[17] A fabricated prototype of the processor, which has a 32-bit cotransformation-based LNS arithmetic logic unit (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.^[18]

LNSs are sometimes used in FPGA-based applications where most arithmetic operations are multiplication or division.^[19]

References

^ ^a ^b Lee, Samuel C.; Edgar, Albert D. (September 1979). "Addendum to "The Focus Number System"". IEEE Transactions on Computers. C-28 (9). IEEE: 693. doi:10.1109/TC.1979.1675442. ISSN 0018-9340. (NB. Nicholas Kingsbury's name is incorrectly spelled in this citation.)
^ Kingsbury, Nicholas G.; Rayner, Peter J. W. (1971-01-28). "Digital filtering using logarithmic arithmetic". Electronics Letters. 7 (2). Institution of Engineering and Technology (IET): 56–58. doi:10.1049/el:19710039. ISSN 0013-5194. Also reprinted in: Swartzlander, Jr., Earl E., ed. (1990). Computer Arithmetic. Vol. I. Los Alamitos, CA, USA: IEEE Computer Society Press.
^ Swartzlander, Jr., Earl E.; Alexopoulos, Aristides Georgiou (December 1975). "The Sign/Logarithm Number System". IEEE Transactions on Computers. C-24 (12). IEEE: 1238–1242. doi:10.1109/T-C.1975.224172. ISSN 0018-9340. Also reprinted in: Swartzlander, Jr., Earl E., ed. (1990). Computer Arithmetic. Vol. I. Los Alamitos, CA, USA: IEEE Computer Society Press.
^ Lee, Samuel C.; Edgar, Albert D. (November 1977). "The Focus Number System". IEEE Transactions on Computers. C-26 (11). IEEE: 1167–1170. doi:10.1109/TC.1977.1674770. ISSN 0018-9340.
^ Lee, Samuel C.; Edgar, Albert D. (1977). "Chapter I.1.: Microcomputer Design – Focus Microcomputer Number System". In Lee, Samuel C. (ed.). Microcomputer Design and Applications. Academic Press, Inc. pp. 1–40. doi:10.1016/B978-0-12-442350-3.50005-5. ISBN 0-12-442350-7. [1]
^ Edgar, Albert D.; Lee, Samuel C. (March 1979). "FOCUS Microcomputer Number System". Communications of the ACM. 22 (3). ACM Press: 166–177. doi:10.1145/359080.359085.
^ Leonelli, Zecchini (1803) [1802]. Supplément logarithmique. Théorie des logarithmes additionels et diductifs (in French). Bordeaux: Brossier. (NB. 1802/1803 is the year XI. in the French Republican Calendar.)
^ Leonhardi, Gottfried Wilhelm (1806). 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 Feldartilleriecorps (in German). Dresden: Walther'sche Hofbuchhandlung. (NB. An expanded translation of Zecchini Leonelli's Supplément logarithmique. Théorie des logarithmes additionels et diductifs.)
^ Gauß, Johann Carl Friedrich (1808-02-12). "LEONELLI, Logarithmische Supplemente". Allgemeine Literaturzeitung (in German) (45). Halle-Leipzig: 353–356.
^ "Logarithm: Addition and Subtraction, or Gaussian Logarithms". Encyclopædia Britannica Eleventh Edition.
^ Dunnington, Guy Waldo (2004) [1955]. Gray, Jeremy; Dohse, Fritz-Egbert (eds.). Carl Friedrich Gauss – Titan of Science. Spectrum series (revised ed.). Mathematical Association of America (MAA). ISBN 978-0-88385-547-8.
^ Horsburg, Ellice Martin (1914). "The Instrumental Solution of Numerical Equations by D. Gibb, M.A.". Written at Napier Tercentenary Exhibition. Modern instruments and methods of calculation: a handbook of the Napier Tercentenary Exhibition. Gerstein – University of Toronto. London, UK: G. Bell. p. 263.
^ Mehmke, Rudolf [in German] (1908). "I23". Encyclopédie des sciences mathematiques pures et appliquées. Paris, France: Gauthier-Villars. p. 351.
^ F. Thomas. A Short Account on Leonardo Torres' Endless Spindle, Mechanism and Machine Theory, Vol. 43, No. 8, pp. 1055-1063, 2008.
^ Makino, Junichiro; Taiji, Makoto (1998). Scientific Simulations with Special Purpose Computers: The GRAPE Systems. John Wiley & Sons. Bibcode:1998sssc.book.....M. ISBN 978-0-471-96946-4.
^ Coleman, John Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Licko, Miroslav; Pohl, Zdenek; Hermanek, Antonin (2002-08-07) [2001-11-04]. "The European Logarithmic Microprocessor – a QR RLS application". Conference Record of Thirty-Fifth Asilomar Conference on Signals, Systems and Computers (Cat.No.01CH37256). Vol. 1. Monterey, CA, USA: IEEE. pp. 155–159. doi:10.1109/ACSSC.2001.986897. ISBN 0-7803-7147-X. ISSN 1058-6393.
^ Coleman, John Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Tichy, Milan; Pohl, Zdenek; Hermanek, Antonin; Benschop, Nico F. (April 2008) [2008-02-26]. "The European Logarithmic Microprocessor". IEEE Transactions on Computers. 57 (4). IEEE: 532–546. doi:10.1109/TC.2007.70791. ISSN 0018-9340.
^ Ismail, R. Che; Coleman, John Nicholas (2011-08-18) [2011-07-25]. "ROM-less LNS". 2011 IEEE 20th Symposium on Computer Arithmetic. IEEE. pp. 43–51. doi:10.1109/ARITH.2011.15. ISBN 978-1-4244-9457-6. ISSN 1063-6889.
^ Fu, Haohuan; Mencer, Oskar; Luk, Wayne (2007-01-02) [2006-12-13]. "Comparing floating-point and logarithmic number representations for reconfigurable acceleration". 2006 IEEE International Conference on Field Programmable Technology. IEEE. pp. 337–340. doi:10.1109/FPT.2006.270342. ISBN 978-0-7803-9728-6.

External links

[Lee_1979_1-1] Lee, Samuel C.; Edgar, Albert D. (September 1979). "Addendum to "The Focus Number System"". IEEE Transactions on Computers. C-28 (9). IEEE: 693. doi:10.1109/TC.1979.1675442. ISSN 0018-9340. (NB. Nicholas Kingsbury's name is incorrectly spelled in this citation.)

[Kingsbury_1971-2] Kingsbury, Nicholas G.; Rayner, Peter J. W. (1971-01-28). "Digital filtering using logarithmic arithmetic". Electronics Letters. 7 (2). Institution of Engineering and Technology (IET): 56–58. doi:10.1049/el:19710039. ISSN 0013-5194. Also reprinted in: Swartzlander, Jr., Earl E., ed. (1990). Computer Arithmetic. Vol. I. Los Alamitos, CA, USA: IEEE Computer Society Press.

[Swartzlander_1975-3] Swartzlander, Jr., Earl E.; Alexopoulos, Aristides Georgiou (December 1975). "The Sign/Logarithm Number System". IEEE Transactions on Computers. C-24 (12). IEEE: 1238–1242. doi:10.1109/T-C.1975.224172. ISSN 0018-9340. Also reprinted in: Swartzlander, Jr., Earl E., ed. (1990). Computer Arithmetic. Vol. I. Los Alamitos, CA, USA: IEEE Computer Society Press.

[Lee_1977_1-4] Lee, Samuel C.; Edgar, Albert D. (November 1977). "The Focus Number System". IEEE Transactions on Computers. C-26 (11). IEEE: 1167–1170. doi:10.1109/TC.1977.1674770. ISSN 0018-9340.

[Lee_1977_2-5] Lee, Samuel C.; Edgar, Albert D. (1977). "Chapter I.1.: Microcomputer Design – Focus Microcomputer Number System". In Lee, Samuel C. (ed.). Microcomputer Design and Applications. Academic Press, Inc. pp. 1–40. doi:10.1016/B978-0-12-442350-3.50005-5. ISBN 0-12-442350-7. [1]

[Lee_1979_2-6] Edgar, Albert D.; Lee, Samuel C. (March 1979). "FOCUS Microcomputer Number System". Communications of the ACM. 22 (3). ACM Press: 166–177. doi:10.1145/359080.359085.

[Leonelli_1802-7] Leonelli, Zecchini (1803) [1802]. Supplément logarithmique. Théorie des logarithmes additionels et diductifs (in French). Bordeaux: Brossier. (NB. 1802/1803 is the year XI. in the French Republican Calendar.)

[Leonhardi_1806-8] Leonhardi, Gottfried Wilhelm (1806). 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 Feldartilleriecorps (in German). Dresden: Walther'sche Hofbuchhandlung. (NB. An expanded translation of Zecchini Leonelli's Supplément logarithmique. Théorie des logarithmes additionels et diductifs.)

[Gauss_1808-9] Gauß, Johann Carl Friedrich (1808-02-12). "LEONELLI, Logarithmische Supplemente". Allgemeine Literaturzeitung (in German) (45). Halle-Leipzig: 353–356.

[Theodora-10] "Logarithm: Addition and Subtraction, or Gaussian Logarithms". Encyclopædia Britannica Eleventh Edition.

[Dunnington_2004-11] Dunnington, Guy Waldo (2004) [1955]. Gray, Jeremy; Dohse, Fritz-Egbert (eds.). Carl Friedrich Gauss – Titan of Science. Spectrum series (revised ed.). Mathematical Association of America (MAA). ISBN 978-0-88385-547-8.

[Horsburg-Napier_1914-12] Horsburg, Ellice Martin (1914). "The Instrumental Solution of Numerical Equations by D. Gibb, M.A.". Written at Napier Tercentenary Exhibition. Modern instruments and methods of calculation: a handbook of the Napier Tercentenary Exhibition. Gerstein – University of Toronto. London, UK: G. Bell. p. 263.

[Mehmke_1908-13] Mehmke, Rudolf [in German] (1908). "I23". Encyclopédie des sciences mathematiques pures et appliquées. Paris, France: Gauthier-Villars. p. 351.

[Thomas2008-14] F. Thomas. A Short Account on Leonardo Torres' Endless Spindle, Mechanism and Machine Theory, Vol. 43, No. 8, pp. 1055-1063, 2008.

[Makino_1998-15] Makino, Junichiro; Taiji, Makoto (1998). Scientific Simulations with Special Purpose Computers: The GRAPE Systems. John Wiley & Sons. Bibcode:1998sssc.book.....M. ISBN 978-0-471-96946-4.

[Coleman_2001-16] Coleman, John Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Licko, Miroslav; Pohl, Zdenek; Hermanek, Antonin (2002-08-07) [2001-11-04]. "The European Logarithmic Microprocessor – a QR RLS application". Conference Record of Thirty-Fifth Asilomar Conference on Signals, Systems and Computers (Cat.No.01CH37256). Vol. 1. Monterey, CA, USA: IEEE. pp. 155–159. doi:10.1109/ACSSC.2001.986897. ISBN 0-7803-7147-X. ISSN 1058-6393.

[Coleman_2008-17] Coleman, John Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Tichy, Milan; Pohl, Zdenek; Hermanek, Antonin; Benschop, Nico F. (April 2008) [2008-02-26]. "The European Logarithmic Microprocessor". IEEE Transactions on Computers. 57 (4). IEEE: 532–546. doi:10.1109/TC.2007.70791. ISSN 0018-9340.

[Ismail_2011-18] Ismail, R. Che; Coleman, John Nicholas (2011-08-18) [2011-07-25]. "ROM-less LNS". 2011 IEEE 20th Symposium on Computer Arithmetic. IEEE. pp. 43–51. doi:10.1109/ARITH.2011.15. ISBN 978-1-4244-9457-6. ISSN 1063-6889.

[Fu_2006-19] Fu, Haohuan; Mencer, Oskar; Luk, Wayne (2007-01-02) [2006-12-13]. "Comparing floating-point and logarithmic number representations for reconfigurable acceleration". 2006 IEEE International Conference on Field Programmable Technology. IEEE. pp. 337–340. doi:10.1109/FPT.2006.270342. ISBN 978-0-7803-9728-6.

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