I do not think there is an algorithm (or you have to change somethingto allow algorithms that never end with some data). Consider the two numbers $$a=(3/2)^\sqrt{2}=1.77431468418218794421950\dots\quad b=\frac{1}{\sqrt{2}}=0.70710678118654752440084436210\dots$$We have $$a^b=\frac{3}{2}=1.5$$Then slightly changing $a$ or $b$ will make the second digit after the decimalpoint equal to $9$ or to $0$. Therefore we need to know all the digits of $a$ or $b$ to determine this second digit.
If the given numbers are $a$ and $b$ you will need to test all its digit to determinethis second digit.
Of course it is an issue. Since your data $f_a$ and $f_b$ are infinite, I assume that they act as oracles, i.e. your algorithm at any moment can ask for the value of $f_a(i)$ or $f_b(i)$. Assume that you give the above numbers and your program stop giving a value $0$ or $9$ for the second digit. Since your algorithm has stoppedat a finite time he would have ask only for a finite number of values of $f_a(i)$ and $f_b(i)$. Meaning that this algorithm will give the same answer for any irrational with these values in common. But it is clear that there are other irrational numbers sharing these digits and for which the correct value is just the opposite your program gives. And also this value will not be equivalent because it will be a numbernot ending in all "0" or all "9".
So any algorithm will give wrong answers for some numbers.