Symmetric level-index arithmetic

The LI (level-index) representation of numbers, and its algorithms for arithmetic operations, were introduced by Clenshaw & Olver. The symmetric form of the LI system and its arithmetic operations were presented by Clenshaw & Turner. Anuta, Lozier, Schabanel and Turner developed the algorithm for SLI (symmetric level-index) 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 positive real number X as

           $X=e^{e^{e^{...^{e^{f}}}}}$

where $0\leq f<1$ and the process of exponentiation is performed l times. l and f are the level and index of X respectively. X = l + f is the LI image of X. For an instance,

           $X=1234567=e^{e^{e^{0.9711308}}}$

so its LI image is

           $x=l+f=3+0.9711308=3.9711308$

The symmetric form is used to allow negative exponents, if the magnitude of X is less than 1. We take the logarithm of X and store its sign as the reciprocal sign. Mathematically, this is equivalent to taking the reciprocal 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, while a sign bit allows negative numbers.

The mapping function is called the generalized logarithm function. It is defined as

           Failed to parse (unknown function "\begin{array}"): {\displaystyle \psi (X)= \begin{array} X, 0\leq X<1 \\ 1+\psi (\ln X), X\geq 1 \end{array}}

and it maps $(0,\infty )$ onto itself monotonically and so it is invertible on this interval. The inverse, the generalized exponential function, is defined by

           Failed to parse (unknown function "\begin{array}"): {\displaystyle phi (x)= \begin{array} x, 0\leq x<1 \\ e^{\phi (x-1)}, x\geq 1 \end{array}}

Formally, we can define the SLI representation for an arbitrary nonzero X as

           $X=s_{X}\phi (x)^{r_{X}}$

where s_X is the sign and r_X is the reciprocal sign as in the following equations.

           $x=\psi (\max(|X|,|X|^{-1}))=\psi (|X|^{r_{X}}),s_{X}={\text{sign}}(X)$

For example,

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

and its SLI representation is

           $X=-\phi (3.9711308)^{-1}$