Finite-precision arithmetic computations face an inherent tradeo between accuracy and e ciency. The points in this tradeo space are determined, among other factors, by di erent data types but also evaluation orders. To put it simply, the shorter a precision's bit-length, the larger the roundo error will be, but the faster the program will run. Similarly, the fewer arithmetic operations the program performs, the faster it will run; however, the e ect on the roundo error is less clear-cut. Manually optimizing the e ciency of nite-precision programs while ensuring that results remain accurate enough is challenging. The unintuitive and discrete nature of nite-precision makes estimation of roundo errors di cult; furthermore the space of possible data types and evaluation orders is prohibitively large.We present the rst fully automated and sound technique and tool for optimizing the performance of oating-point and xedpoint arithmetic kernels. Our technique combines rewriting and mixed-precision tuning. Rewriting searches through di erent evaluation orders to nd one which minimizes the roundo error at no additional runtime cost. Mixed-precision tuning assigns di erent nite precisions to di erent variables and operations and thus provides ner-grained control than uniform precision. We show that when these two techniques are designed and applied together, they can provide higher performance improvements than each alone.

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