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.
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 rX.
The inverse, the generalized exponential function, is defined by 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 The generalized logarithm function is closely related to the iterated logarithm used in computer science analysis of algorithms.