We present a deterministic $(1+\varepsilon)$-approximate maximum matching algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in $1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n / \log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18], that is allowed a dependence on the number of nodes $n$. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing an $O(1)$-approximate maximum matching and an algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing $(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on $1/\varepsilon$, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.