Before revision:  The complexity of our algorithms is 2^{( d /\varepsilon) ^{O(d) } n \log^4 n + 2^{O(d)} n \log^9 n, making a significant improvement over the state-of-the-art algorithm which runs in time n^{(d/\varepsilon)^{O(d)}}.
After revision:  The complexity of our algorithms is 2^{( \log(1 /\varepsilon) ^{O(d) } n \log^4 n + 2^{O(d)} n \log^9 n, making a significant improvement over the state-of-the-art algorithm which runs in time n^{(d/\varepsilon)^{O(d)}}.