TÄNA AJALOOS, 11. juuni ⟩ Hävitati Trooja linn

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.

Definition

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

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

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

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

so its LI image is

x=\ell +f=3+0.9711308=3.9711308.

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

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

\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

\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.

A sign bit may also be used to allow negative numbers. One takes sgn(X) and stores it as a boolean $s X$ (0 only occurs when $X = 0$ so could be stored either way, +1 chosen here). Mathematically, this is equivalent to taking the negation (additive inverse) of a number, and finding the SLI image for that. Using one bit for the sign enables the representation of negative numbers.

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 as a boolean $r X$ (0 only occurs when $X = 1$ so could be stored either way, +1 chosen here). Mathematically, this is equivalent to taking the reciprocal (multiplicative inverse) of a small-magnitude number, and finding the SLI image for that. Using one bit for the reciprocal sign enables the representation of extremely small numbers.

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

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: