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Despite being raised as a problem over ten years ago, the imprecision of floating point arithmetic continues to cause privacy failures in the implementations of differentially private noise mechanisms. In this paper, we highlight a new class of vulnerabilities, which we call \emph{precision-based attacks}, and which affect several open source libraries. To address this vulnerability and implement differentially private mechanisms on floating-point space in a safe way, we propose a novel technique, called \emph{interval refining}. This technique has minimal error, provable privacy, and broad applicability. We use interval refining to design and implement a variant of the Laplace mechanism that is equivalent to sampling from the Laplace distribution and rounding to a float. We report on the performance of this approach, and discuss how interval refining can be used to implement other mechanisms safely, including the Gaussian mechanism and the exponential mechanism.