Symmetric level-index arithmetic

The level-index (LI) representation of numbers, and its algorithms for arithmetic operations, were introduced by Charles Clenshaw and Frank 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 for symmetric level-index (SLI) arithmetic, and a parallel implementation of it. There has been extensive work on developing the SLI arithmetic algorithms and extending them to complex and vector arithmetic operations.

The idea of the level-index system is to represent a non-negative real number X as

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

where ${\displaystyle 0\leq f<1}$, and the process of exponentiation is performed ℓ times, with ${\displaystyle \ell \geq 0}$. ℓ and f are the level and index of X respectively. x = ℓ + f is the LI image of X. For example,

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

so its LI image is

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

The symmetric form is used to allow negative exponents, if the magnitude of X is less than 1. One takes sgn(log(X)) or sgn(|X| − |X|⁻¹) and stores it (after substituting +1 for 0 for the reciprocal sign; since for X = 1 = e⁰ the LI image is x = 1.0 and uniquely defines X = 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 sign r_X. Mathematically, this is equivalent to taking the reciprocal (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.

A sign bit may also be used to allow negative numbers. One takes sgn(X) and stores it (after substituting +1 for 0 for the sign; since for X = 0 the LI image is x = 0.0 and uniquely defines X = 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 sign s_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 the generalized logarithm function. It is defined as

${\displaystyle \psi (X)={\begin{cases}X&{\text{if }}0\leq X<1,\\1+\psi (\ln X)&{\text{if }}X\geq 1,\end{cases}}}$

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

${\displaystyle \varphi (x)={\begin{cases}x&{\text{if }}0\leq x<1,\\e^{\varphi (x-1)}&{\text{if }}x\geq 1.\end{cases}}}$

The density of values X represented by x has no discontinuities as we go from level ℓ to ℓ + 1 (a very desirable property) since

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

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

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

${\displaystyle X=s_{X}\varphi (x)^{r_{X}},}$

where s_X is the sign (additive inversion or not) of X, and r_X is the reciprocal sign (multiplicative inversion or not) as in the following equations:

${\displaystyle s_{X}=\operatorname {sgn}(X),\quad r_{X}=\operatorname {sgn} {\big (}|X|-|X|^{-1}{\big )},\quad x=\psi {\big (}\max {\big (}|X|,|X|^{-1}{\big )}{\big )}=\psi {\big (}|X|^{r_{X}}{\big )},}$

whereas for X = 0 or 1, we have

${\displaystyle s_{0}=+1,\quad r_{0}=+1,\quad x=0.0,}$

${\displaystyle s_{1}=+1,\quad r_{1}=+1,\quad x=1.0.}$

For example,

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

and its SLI representation is

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

• 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. Retrieved 2018-07-09. [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. Retrieved 2018-07-09. [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".