Thelevel-index (LI) representation of numbers, and itsalgorithms forarithmetic operations, were introduced byCharles Clenshaw andFrank Olver in 1984.^[1]

The symmetric form of the LI system and its arithmetic operations were presented by Clenshaw and Peter Turner in 1987.^[2]

Michael Anuta, Daniel Lozier, Nicolas Schabanel and Turner developed the algorithm forsymmetric level-index (SLI) arithmetic, and a parallel implementation of it. There has been extensive work on developing the SLI arithmetic algorithms and extending them tocomplex andvector arithmetic operations.

The idea of the level-index system is to represent a non-negativereal numberX as

${\displaystyle X=e^{e^{e^{{\cdots }^{e^f}}}},}$

where, and the process of exponentiation is performedℓ times, with${\displaystyle \ell \ge 0}$.ℓ andf are thelevel andindex ofX respectively.x =ℓ +f is the LI image ofX. For example,

${\displaystyle X=1234567=e^{e^{e^{0.9711308}}},}$

so its LI image is

${\displaystyle x=\ell +f=3+0.9711308=\mathrm{3.9711308.}}$

The symmetric form is used to allow negative exponents, if the magnitude ofX is less than 1. One takessgn(log(X)) orsgn(|X| − |X|⁻¹) and stores it (after substituting +1 for 0 for the reciprocal sign; since forX = 1 = e⁰ the LI image isx = 1.0 and uniquely definesX = 1, we can do away without a third state and use only one bit for the two states −1 and +1^{[clarification needed]}) as the reciprocal signr_X. Mathematically, this is equivalent to taking thereciprocal (multiplicative inverse) of a small-magnitude number, and then finding the SLI image for the reciprocal. Using one bit for the reciprocal sign enables the representation of extremely small numbers.

Asign bit may also be used to allow negative numbers. One takessgn(X) and stores it (after substituting +1 for 0 for the sign; since forX = 0 the LI image isx = 0.0 and uniquely definesX = 0, we can do away without a third state and use only one bit for the two states −1 and +1^{[clarification needed]}) as the signs_X. Mathematically, this is equivalent to taking the inverse (additive inverse) of a negative number, and then finding the SLI image for the inverse. Using one bit for the sign enables the representation of negative numbers.

The mapping function is called thegeneralized logarithm function. It is defined as

and it maps${\displaystyle [0,\mathrm{\infty })}$ onto itself monotonically, thus being invertible on this interval. The inverse, thegeneralized exponential function, is defined by

The density of valuesX represented byx has no discontinuities as we go from levelℓ toℓ + 1 (a very desirable property) since

${\displaystyle {\frac{d\phi (x)}{dx}|}_{x=1}={\frac{d\phi (e^x)}{dx}|}_{x=0}.}$

The generalized logarithm function is closely related to theiterated logarithm used in computer science analysis of algorithms.

Formally, we can define the SLI representation for an arbitrary realX (not 0 or 1) as

${\displaystyle X=s_X\phi (x{)}^{r_X},}$

wheres_X is the sign (additive inversion or not) ofX, andr_X is the reciprocal sign (multiplicative inversion or not) as in the following equations:

${\displaystyle s_X=\mathrm{sgn}(X),r_X=\mathrm{sgn}(|X|-|X{|}^{-1}),x=\psi (max(|X|,|X{|}^{-1}))=\psi (|X{|}^{r_X}),}$

whereas forX = 0 or 1, we have

${\displaystyle s_0=+1,r_0=+1,x=0.0,}$

${\displaystyle s_1=+1,r_1=+1,x=\mathrm{1.0.}}$

For example,

${\displaystyle X=-{\displaystyle \frac{1}{1234567}}=-e^{-e^{e^{0.9711308}}},}$

and its SLI representation is

${\displaystyle x=-\phi (3.9711308{)}^{-1}.}$

• Tetration
• Floating point (FP)
• Tapered floating point (TFP)
• Logarithmic number system (LNS)
• Level (logarithmic quantity)

• Clenshaw, Charles William;Olver, Frank William John; Turner, Peter R. (1989). "Level-index arithmetic: An introductory survey".Numerical Analysis and Parallel Processing (Conference proceedings / The Lancaster Numerical Analysis Summer School 1987). Lecture Notes in Mathematics (LNM).1397:95–168.doi:10.1007/BFb0085718.ISBN 978-3-540-51645-3.
• Clenshaw, Charles William; Turner, Peter R. (1989-06-23) [1988-10-04]. "Root Squaring Using Level-Index Arithmetic".Computing.43 (2).Springer-Verlag:171–185.doi:10.1007/BF02241860.ISSN 0010-485X.
• Zehendner, Eberhard (Summer 2008)."Rechnerarithmetik: Logarithmische Zahlensysteme"(PDF) (Lecture script) (in German).Friedrich-Schiller-Universität Jena. pp. 21–22.Archived(PDF) from the original on 2018-07-09. Retrieved2018-07-09.[1]
• Hayes, Brian (September–October 2009)."The Higher Arithmetic".American Scientist.97 (5):364–368.doi:10.1511/2009.80.364.Archived from the original on 2018-07-09. Retrieved2018-07-09.[2]. Also reprinted in:Hayes, Brian (2017). "Chapter 8: Higher Arithmetic".Foolproof, and Other Mathematical Meditations (1 ed.).The MIT Press. pp. 113–126.ISBN 978-0-26203686-3.ISBN 0-26203686-X.

• sli-c-library (hosted by Google Code),"C++ Implementation of Symmetric Level-Index Arithmetic".

• Computer arithmetic

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