Confidence in the first $k$ decimal places of a product after multiplying $N$ irrational numbers together

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)?

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Roughly speaking, if you have an error of a factor of $1 \pm 10^{-15}$ in each multiply, the standard deviation is a factor of $1 \pm \frac 1{\sqrt 3}10^{-15}$ (assuming a uniform distribution). You need about 3 of these to lose the first digit, 300 to lose the second.