can use their own set of constants, functions and binary operations, but they must be merged with irreducible set or .

It is illustrative to compare and with *Mathematica* core language [@Mathematica], composed of:

\[mma\] $$\begin{aligned}
\text{addition (Plus)} & \qquad x+y,\end{aligned}$$ $$\begin{aligned}
\text{multiplication (Times)} & \qquad x \times y,\end{aligned}$$ $$\begin{aligned}
\text{exponentiation (Power)} & \qquad x^y,\end{aligned}$$ $$\begin{aligned}
\text{natural logarithm (Log)} &\qquad  \ln{x},\end{aligned}$$ $$\begin{aligned}
\text{mathematical constants (E, Pi, I)} \qquad e, \pi, i, \ldots\end{aligned}$$

augmented with arbitrarily large integers, including their complex combinations (Complex) and rationals (Rational). Visual comparison of , and is presented in Fig. \[BaseSet\].

![\[BaseSet\] Illustration of base constants, functions and binary operations used to define $\exp-\log$ number class. Irreducible set is labeled ”up-botton”, because it is based on highest rank complex (hyper-)operation currently implemented (exponentiation), while (bottom-up) starts with low-rank arithmetic (subtraction). *Mathematica* core language is included for comparison. All three sets allow generation of any explicit elementary transcendental constant in the EL class. Question mark symbolize unknown ”holy-grail” complex function, which could, if exists, by repeated composition, generate all $\exp-\log$ numbers in unique order. ](figs/BaseSet){width="50.00000%"}

Accounting for the above considerations, for further analysis we have selected four base sets (buttons) to attack the inverse calculator problem:

\[CALC\]

