Symmetric level-index arithmetic
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The level-index (LI) 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 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.
[edit] Definition
The idea of the level-index system is to represent a positive real number X as
where 
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,
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
and it maps 
onto itself monotonically and so it is invertible on this interval. The inverse, the generalized exponential function, is defined by
Formally, we can define the SLI representation for an arbitrary nonzero X as
where sX is the sign and rX is the reciprocal sign as in the following equations.
For example,
and its SLI representation is
X = − φ(3.9711308) − 1
