The idea of the level-index system is to represent a non-negative
real number as : X = e^{e^{e^{\cdots^{e^f}}}}, where 0 \leq f , and the process of exponentiation is performed times, with \ell \geq 0. and are the
level and
index of respectively. is the LI image of . For example, : X = 1234567 = e^{e^{e^{0.9711308}}}, so its LI image is : x = \ell + f = 3 + 0.9711308 = 3.9711308. The symmetric form is used to allow negative exponents, if the magnitude of is less than 1. One takes or and stores it (after substituting +1 for 0 for the reciprocal sign; since for the LI image is and uniquely defines , we can do away without a third state and use only one bit for the two states −1 and +1) as the reciprocal sign . 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 the LI image is and uniquely defines , we can do away without a third state and use only one bit for the two states −1 and +1) as the sign . 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 : \psi(X) = \begin{cases} X & \text{if } 0 \leq X 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 The density of values represented by has no discontinuities as we go from level to (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. Formally, we can define the SLI representation for an arbitrary real (not 0 or 1) as : X = s_X\varphi(x)^{r_X}, where is the sign (additive inversion or not) of , and is the reciprocal sign (multiplicative inversion or not) as in the following equations: : s_X = \sgn(X),\quad r_X = \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 = 0 or 1, we have : s_0 = +1,\quad r_0 = +1,\quad x = 0.0, : s_1 = +1,\quad r_1 = +1,\quad x = 1.0. For example, : X = -\dfrac{1}{1234567} = -e^{-e^{e^{0.9711308}}}, and its SLI representation is : x = -\varphi(3.9711308)^{-1}. ==See also==