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Floating-point arithmetic is a loosely standardized approximation of real arithmetic available on many computers today. Architectural and compiler differences can lead to diverse calculations across platforms, for the same input. If left untreated, platform dependence, called volatility in this paper, seriously interferes with result reproducibility and, ultimately, program portability. We present an approach to stabilizing floating-point programs against volatility. Our approach, dubbed provenance analysis, traces volatility observed in a given intermediate expression E back to volatility in preceding statements, and quantifies individual contributions to the volatility in E. Statements contributing the most are then stabilized, by disambiguating the arithmetic using expression rewriting and control pragmas. The benefit of local (as opposed to program-wide) stabilization is that compilers are free to engage performance- or precision-enhancing optimizations across program fragments that do not destabilize E. We have implemented our technique in a dynamic analysis tool that reports both volatility and provenance information. We demonstrate that local program stabilization often suffices to reduce platform dependence to an acceptable level.