We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a $(2Δ-1)$-edge coloring can be computed in time $\mathrm{poly}\logΔ+ O(\log^* n)$ in the LOCAL model. This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on $Δ$. We further show that in the CONGEST model, an $(8+\varepsilon)Δ$-edge coloring can be computed in $\mathrm{poly}\logΔ+ O(\log^* n)$ rounds. The best previous $O(Δ)$-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it computes a $2^{O(1/\varepsilon)}Δ$-edge coloring in time $O(Δ^\varepsilon + \log^* n)$ for any $\varepsilon\in(0,1]$.