The constant-time programming discipline is an effective countermeasure against timing attacks, which can lead to complete breaks of otherwise secure systems. However, adhering to constant-time programming is hard on its own, and extremely hard under additional efficiency and legacy constraints. This makes automated verification of constant-time code an essential component for building secure software. We propose a novel approach for verifying constant- time security of real-world code. Our approach is able to validate implementations that locally and intentionally violate the constant-time policy, when such violations are benign and leak no more information than the pub- lic outputs of the computation. Such implementations, which are used in cryptographic libraries to obtain impor- tant speedups or to comply with legacy APIs, would be declared insecure by all prior solutions. We implement our approach in a publicly available, cross-platform, and fully automated prototype, ct-verif, that leverages the SMACK and Boogie tools and verifies optimized LLVM implementations. We present verifica- tion results obtained over a wide range of constant-time components from the NaCl, OpenSSL, FourQ and other off-the-shelf libraries. The diversity and scale of our ex- amples, as well as the fact that we deal with top-level APIs rather than being limited to low-level leaf functions, distinguishes ct-verif from prior tools. Our approach is based on a simple reduction of constant-time security of a program P to safety of a prod- uct program Q that simulates two executions of P. We formalize and verify the reduction for a core high-level language using the Coq proof assistant.