The issue here is that unless you have checks that your values dont over or underflow at every stage of the calculation, you can never actually know if your answers are accurate, even if they might be repeatable.
I would think that in most cases you should be able to prove an upper bound on the sum and make sure the accumulator is large enough to accommodate it. My guess is that 512 bits should be enough to span all physically/computationally relevant scales.
Note that digits of precision are expressed in decimal unless explicitly given as “bits”, so that the phrase “10,000-digit arithmetic” really means around 33,000 bits.
