Floating-point numbers are finite-precision approximations to real numbers and are ubiquitous in computer applications in nearly every field. Selecting the right floating-point representation that balances performance and numerical accuracy is a difficult task, one that has become even more critical as hardware trends toward high-performance, low-precision operations. Although a plethora of numerical techniques, including algebraic rewriting and mixed-precision tuning, can help navigate the tradeoff between performance and accuracy, applying complex numerical optimizations to real-world code is still a challenging engineering task that requires mathematical expertise and application-specific numerical context. We present Poseidon, a modular and extensible framework that fully automates floating-point optimizations for real-world applications within a production compiler. Within a production compiler, Poseidon auto-extracts symbolic expressions from pre-optimized IR, derives their numerical context from a small surrogate profile, and composes candidates from diverse techniques (e.g., Herbie’s e-graph-based algebraic rewrites and ADAPT-style precision tuning) via a shared dynamic-programming solver. On a quaternion differentiator, Poseidon composes algebraic rewriting with precision tuning for a $1.46\times$ speedup at relative error of $10^{-7}$. On DOE’s proxy application LULESH, Poseidon recovers optimal FP64 accuracy without substantially reducing performance.