TITLE: Quanitative de Moivre–Laplace theorem (reference request)
QUESTION [6 upvotes]: The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution: 
$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / (2npq)}.$$
My question is: are there more precise, quantitative versions of this theorem in the literature? Are there good estimates how to measure the error? I am unfortunately not familiar with the subject but need a result of this type. 
Of course there is always the option of going through existing proofs and checking the details, and turning them from "soft" to "hard", but I suspect this has to be already done. And maybe this is not optimal, maybe there are good accessible ways. 
Can someone point me a good reference in this direction?

REPLY [2 votes]: You might be interested in this paper (a very precise estimate, apparently overlooked by most people!)

J. E. Littlewood, On the probability in the tail of a binomial
  distribution,  Adv. Appl. Prob. 1 (1969) 43–72.

revisited and corrected by McKay

Brendan D. McKay, On Littlewood's Estimate for the Binomial
  Distribution,  Advances in Applied Probability, Vol. 21, No. 2 (Jun.,
  1989), pp. 475-478