Finding root causes of floating point error

Sanchez-SternAlex,; PanchekhaPavel,; LernerSorin,; TatlockZachary,

doi:10.1145/3296979.3192411

“…Mathematical rewriting tools are other alternatives to create correctly rounded functions. If the rounding error in the implementation is the root cause of an incorrect result, we can use tools that detect numerical errors to diagnose them [Benz et al 2012;Chowdhary et al 2020;Goubault 2001;Sanchez-Stern et al 2018;Yi et al 2019;Zou et al 2019]. Subsequently, we can rewrite them using tools such as Herbie [Panchekha et al 2015] or Salsa [Damouche and Martel 2018].…”

Section: Related Workmentioning

confidence: 99%

An approach to generate correctly rounded math libraries for new floating point variants

Lim

¹

,

Aanjaneya

²

,

Gustafson

³

et al. 2021

Proc. ACM Program. Lang.

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Given the importance of floating point (FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and posits). These representations do not have correctly rounded math libraries. Further, the use of existing FP libraries for these new representations can produce incorrect results. This paper proposes a novel approach for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods generate polynomials that approximate the real value of an elementary function 𝑓 (𝑥) and produce wrong results due to approximation errors and rounding errors in the implementation. In contrast, our approach generates polynomials that approximate the correctly rounded value of 𝑓 (𝑥) (i.e., the value of 𝑓 (𝑥) rounded to the target representation). It provides more margin to identify efficient polynomials that produce correctly rounded results for all inputs. We frame the problem of generating efficient polynomials that produce correctly rounded results as a linear programming problem. Using our approach, we have developed correctly rounded, yet faster, implementations of elementary functions for multiple target representations.

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“…Further, seminal research on range reduction has made such approximation feasible [2, 12, 43ś 46]. Simultaneously, there are verification efforts to prove bounds for math libraries [21ś23, 27, 41], identify numerical errors with expressions that can be used in the implementation of math libraries [1,11,16,18,40], and repair individual outputs of math libraries [38,48,50].…”

Section: Related Workmentioning

confidence: 99%

High performance correctly rounded math libraries for 32-bit floating point representations

Lim

¹

,

Nagarakatte

²

2021

Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

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This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLibm approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the context of 16-bit types to scale to 32-bit types. Specifically, this paper proposes new algorithms to (1) generate polynomials that produce correctly rounded outputs for all inputs using counterexample guided polynomial generation, (2) generate efficient piecewise polynomials with bit-pattern based domain splitting, and (3) deduce the amount of freedom available to produce correct results when range reduction involves multiple elementary functions. The resultant math library for the 32-bit float type is faster than state-of-the-art math libraries while producing the correct output for all inputs. We have also developed a set of correctly rounded elementary functions for 32-bit posits.

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“…Further, seminal research on range reduction has made such approximation feasible [2,12,[42][43][44][45]. Simultaneously, there are verification efforts to prove bounds for math libraries [21-23, 27, 40], identify numerical errors with expressions that can be used in the implementation of math libraries [1,11,16,18,39], and repair individual outputs of math libraries [37,47,49].…”

Section: Related Workmentioning

confidence: 99%

RLIBM-32: High Performance Correctly Rounded Math Libraries for 32-bit Floating Point Representations

Lim,

Nagarakatte

2021

Preprint

0

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This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLibm approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the context of 16-bit types to scale to 32-bit types. Specifically, this paper proposes new algorithms to (1) generate polynomials that produce correctly rounded outputs for all inputs using counterexample guided polynomial generation, (2) generate efficient piecewise polynomials with bit-pattern based domain splitting, and (3) deduce the amount of freedom available to produce correct results when range reduction involves multiple elementary functions. The resultant math library for the 32-bit float type is faster than state-of-the-art math libraries while producing the correct output for all inputs. We have also developed a set of correctly rounded elementary functions for 32-bit posits.

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Finding root causes of floating point error

Cited by 16 publications

References 24 publications

An approach to generate correctly rounded math libraries for new floating point variants

An approach to generate correctly rounded math libraries for new floating point variants

High performance correctly rounded math libraries for 32-bit floating point representations

RLIBM-32: High Performance Correctly Rounded Math Libraries for 32-bit Floating Point Representations

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