How to compare products of prime factors efficiently? Let's say that $n$ and $m$ are two very large natural numbers, both expressed as product of prime factors, for example:
$n = 3×5×43×367×4931×629281$
$m = 8219×138107×647099$
Now I'd like to know which is smaller. Unfortunately, all I have is an old pocket calculator that can show at most (say) ten digits. So while there are enough digits to enter each factor individually, $n$ and $m$ are both too large to fit in the calculator. To my disappoint, they are also so close that even their logarithms are indistinguishable looking at the first 10 digits.
Question: how would one go to determine which one of two integers is smaller in a case like this? Any easier alternative than calculating the full decimal expansion of both products with pen and paper?

As a practical example, in a programming language like C or C++, you can easily calculate both products using floating-point arithmetic, and you can easily calculate both products modulo 2^64 using unsigned integer arithmetic. 
If you can estimate the rounding error in the floating-point arithmetic products, and prove that the total rounding error is less than 2^63, then it's easy: Calculate the difference d between the products in floating-point arithmetic. If the difference is >= 2^63 or <= -2^63 then that decides. Otherwise, you know that the exact difference is greater than $d - 2^63$ and less than $d + 2^63$. We also know the exact difference modulo 2^64; this is enough to determine the exact difference. This should work for numbers up to 33 or 34 digits. 
