The abc conjecture and correctly rounded reciprocal square roots

AbstractThe reciprocal square root calculation α=1/x is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today's heterogenous computing environment. Existing results suggest that to get the correctly rounded α in a floating point number system with p significant bits, we may have to compute up to 3p+1 leading bits of α. However, numerical evidence indicates the actual number may be as small as 2p plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the abc conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the 2p bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville's estimation and contemporary number theory