1.  `E`, `POW`, `LOG` \[$n=3$ buttons, eq.  \] $$\label{CALC1}$$

2.  `X`, `EXP`, `LN`, `-` \[$n=4$ buttons, eq. \] $$\label{CALC2}$$

3.  `Pi`, `E`, `I`, `-1`, `2`, `1/2`, `Log`, `Plus`, `Times`, `Power` \[$n=10$ buttons, eq.  \] $$\label{CALC3}$$

4.  full scale scientific calc \[$n=36$ buttons, see below\] $$\label{CALC4}$$

Calculator 1 (up-bottom), based on is the simplest found. It use one constant ($e\simeq 2.71828\ldots$, PUSH on stack operation), no functions, and only two non-commutative binary operations: arbitrary base logarithm $\log_x{y}$ and exponentiation $y^x$. Noteworthy, AST is a binary tree in case of .

Calculator 2 (bottom-up), based on is the second shortest, and have remarkable property: it can use any numerical constant/symbol $x$ to generate all other numbers. For example, $0 = x - x$, $1 = \exp{(0)}$, $e=\exp{(1)}$ and so on. We can use any of $x=0$, $x=-1$, $x=e$, $x=i$, $x=\phi$, …This property can be exploited in two applications: (1) use our calculator to operate on vectors, (2) use $x$ with large Kolmogorov complexity to ”shift” formula generator. The former can be used e.g. to take advantage of modern CPU AVX extensions, the latter to use $x$ as some form of random seed for enumeration procedure. This can be done extending any of 