Central limit estimate of accumulated round-off error

Consider the sum \(a\) of \(10^N\) real numbers \(a_k\), \(a = \sum_{k=1}^{10^N} a_k\). Let \(\tilde{a}_k\) be an approximation of \(a_k\) with precision \(10^{-m}\). Assume that the round-off errors \(\delta_k = a_k - \tilde{a}_k\) are distributed uniformly within the interval \((-0.5 \cdot 10^{-m},\ 0.5 \cdot 10^{-m})\). Compose the sum \(\tilde{a} = \sum_{k=1}^{10^N} \tilde{a}_k\) and let \(\delta = a - \tilde{a}\) be the total error of the approximation. For the given values of \(N\) and \(m\) find \(\epsilon\) such that \[ P(|\delta| < \epsilon) > 0.99. \tag{52} \]