Can you encode if-then-else in arithmetic Is there a general way to encode if-then-else/ITE in arithmetic, i.e. using the usual mathematical operators? Example: let $f(x) = x^2$ if $x < 10^{30}$ and $log x$ otherwise...--prophetes.ai

Can you encode if-then-else in arithmetic Is there a general way to encode if-then-else/ITE in arithmetic, i.e. using the usual mathematical operators? Example: let $f(x) = x^2$ if $x < 10^{30}$ and $log x$ otherwise, shortly written as $f(x) = ITE(x<10^{30},x^2,log x)$.

If you allow the use of indicator or characteristic functions, then the answer is yes. For your example, if you let $A = \\{x \in \mathbb{R}| x < 10^{30}\\}$, then $f(x) = x^2 \cdot \chi_{A}(x) + \log x \cdot \chi_{A^{C}}(x)$ where $\chi_A$ and $\chi_A^{C}$ are the characteristic functions of $A$ and $A$ complement, respectively.