## Pavel Panchekha, Alex Sanchez-Stern, James R. Wilcox, and Zachary Tatlock.
Automatically improving accuracy for floating point expressions.
In *2015 ACM SIGPLAN Conference on Programming Language Design
and Implementation*, PLDI '15. ACM, 2015.

Scientific and engineering applications depend on floating point arithmetic to approximate real
arithmetic. This approximation introduces rounding error, which can accumulate to produce unacceptable
results. While the numerical methods literature provides techniques to mitigate rounding error,
applying these techniques requires manually rearranging expressions and understanding the finer
details of floating point arithmetic.

We introduce Herbie, a tool which automatically improves
floating point accuracy by searching for error-reducing transformations. Herbie estimates and
localizes rounding error, applies a database of rules to generate improvements, takes series
expansions, and combines improvements for different input regions. We evaluated Herbie on every
example from a chapter in a classic numerical methods textbook, and found that Herbie was able to
improve accuracy on each example, some by up to 60 bits, while imposing a median performance overhead
of 40%. Colleagues in machine learning have applied Herbie to significantly improve the results of a
clustering algorithm, and a mathematical library has accepted a patch generated using Herbie.

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