Can we recover integers $a_i$ from the sum $a_0 + a_1e+a_2e^2+\cdots+a_ne^n$?
This is impossible. No finite number of bits of $f(a_0,\dots,a_n)$ suffices to determine any of $a_0,\dots,a_n$; in fact, any nondegenerate real interval contains the values $f(a_0,\dots,a_n)$ for infinitely many vectors $(a_0,\dots,a_n)\in\mathbb Z^{n+1}$, and this holds even you fix all but two of the $a_0,\dots,a_n$ in advance.