If I multiply $N$ irrational numbers together to generate a product $P$, where the irrational number are specified to a working precision of $m$ decimal digits, how many decimal digits, $k$, should I have confidence in for $P$?
For example, how large of an $N$ can I get away with standard double floating point precision to have a confidence of $k$ decimal digits in the product (about m ~ 15 stable decimal places)?