> this new algorithm opens an intriguing possibility: we can compute just 64 extra bits, and have a probability of 1 - 2^{-64} of terminating. This is so close to certainty that we can simply stop unconditionally without introducing any measurable bias (detecting any difference would require about 2^128 samples, which is prohibitive).
Using the "just use 64 extra bits" approach, there is a 1 in 2^64 chance that the result you compute will be wrong, and correspondingly a 99.9999999999999999946% chance that the result you compute will be right. That is the amount of bias in the algorithm.
> this new algorithm opens an intriguing possibility: we can compute just 64 extra bits, and have a probability of 1 - 2^{-64} of terminating. This is so close to certainty that we can simply stop unconditionally without introducing any measurable bias (detecting any difference would require about 2^128 samples, which is prohibitive).
Using the "just use 64 extra bits" approach, there is a 1 in 2^64 chance that the result you compute will be wrong, and correspondingly a 99.9999999999999999946% chance that the result you compute will be right. That is the amount of bias in the algorithm.