Semi-logarithmic number systems

1998 ◽  
Vol 47 (2) ◽  
pp. 145-151 ◽  
Author(s):  
J.-M. Muller ◽  
A. Scherbyna ◽  
A. Tisserand
Keyword(s):  
Number Systems ◽  
Logarithmic Number

Implementation of elementary functions for logarithmic number systems

2008 ◽  
Vol 2 (4) ◽  
pp. 295 ◽  
Author(s):  
K. Johansson ◽  
O. Gustafsson ◽  
L. Wanhammar
Keyword(s):  
Elementary Functions ◽  
Number Systems ◽  
Logarithmic Number

scholarly journals Low-precision Logarithmic Number Systems

2021 ◽  
Vol 18 (4) ◽  
pp. 1-25
Author(s):  
Syed Asad Alam ◽  
James Garland ◽  
David Gregg
Keyword(s):  
Input Data ◽  
Global Optimum ◽  
Average Error ◽  
Conversion Error ◽  
Real Numbers ◽  
Number Systems ◽  
External Sources ◽  
Addition And Subtraction ◽  
Base Selection ◽  
Logarithmic Number

Logarithmic number systems (LNS) are used to represent real numbers in many applications using a constant base raised to a fixed-point exponent making its distribution exponential. This greatly simplifies hardware multiply, divide, and square root. LNS with base-2 is most common, but in this article, we show that for low-precision LNS the choice of base has a significant impact. We make four main contributions. First, LNS is not closed under addition and subtraction, so the result is approximate. We show that choosing a suitable base can manipulate the distribution to reduce the average error. Second, we show that low-precision LNS addition and subtraction can be implemented efficiently in logic rather than commonly used ROM lookup tables, the complexity of which can be reduced by an appropriate choice of base. A similar effect is shown where the result of arithmetic has greater precision than the input. Third, where input data from external sources is not expected to be in LNS, we can reduce the conversion error by selecting a LNS base to match the expected distribution of the input. Thus, there is no one base that gives the global optimum, and base selection is a trade-off between different factors. Fourth, we show that circuits realized in LNS require lower area and power consumption for short word lengths.

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