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Mixing Condition Numbers and Oracles for Accurate Floating-point Debugging

Bhargav Kulkarni, Pavel Panchekha

TL;DR

The paper addresses the challenge of accurately debugging floating-point computations by marrying two complementary approaches: condition-number-based error bounds and fast double-double oracles, augmented with a novel logarithmic overflow/underflow oracle. The proposed ExplaniFloat framework uses shadow memory to maintain both double-double and logarithmic representations, enabling first-order error propagation analysis and robust detection of rounding errors, as well as overflow/underflow events. Empirical results on 546 benchmarks show a precision of $80.0\%$ and a recall of $96.1\%$ at a threshold of $64$, outperforming both pure double-double oracle methods and an arbitrary-precision baseline while being significantly faster. The work contributes a practical debugging algorithm based on split condition numbers, a dedicated logarithmic oracle for out-of-range values, and a performant DSL-based implementation that leverages the qd library to realize high-precision yet efficient analysis.

Abstract

Recent advances have made numeric debugging tools much faster by using double-double oracles, and numeric analysis tools much more accurate by using condition numbers. But these techniques have downsides: double-double oracles have correlated error so miss floating-point errors while condition numbers cannot cleanly handle over- and under- flow. We combine both techniques to avoid these downsides. Our combination, EXPLANIFLOAT, computes condition numbers using double-double arithmetic, which avoids correlated errors. To handle over- and under- flow, it introduces a separate logarithmic oracle. As a result, EXPLANIFLOAT achieves a precision of 80.0% and a recall of 96.1% on a collection of 546 difficult numeric benchmarks: more accurate than double-double oracles yet dramatically faster than arbitrary-precision condition number computations.

Mixing Condition Numbers and Oracles for Accurate Floating-point Debugging

TL;DR

The paper addresses the challenge of accurately debugging floating-point computations by marrying two complementary approaches: condition-number-based error bounds and fast double-double oracles, augmented with a novel logarithmic overflow/underflow oracle. The proposed ExplaniFloat framework uses shadow memory to maintain both double-double and logarithmic representations, enabling first-order error propagation analysis and robust detection of rounding errors, as well as overflow/underflow events. Empirical results on 546 benchmarks show a precision of and a recall of at a threshold of , outperforming both pure double-double oracle methods and an arbitrary-precision baseline while being significantly faster. The work contributes a practical debugging algorithm based on split condition numbers, a dedicated logarithmic oracle for out-of-range values, and a performant DSL-based implementation that leverages the qd library to realize high-precision yet efficient analysis.

Abstract

Recent advances have made numeric debugging tools much faster by using double-double oracles, and numeric analysis tools much more accurate by using condition numbers. But these techniques have downsides: double-double oracles have correlated error so miss floating-point errors while condition numbers cannot cleanly handle over- and under- flow. We combine both techniques to avoid these downsides. Our combination, EXPLANIFLOAT, computes condition numbers using double-double arithmetic, which avoids correlated errors. To handle over- and under- flow, it introduces a separate logarithmic oracle. As a result, EXPLANIFLOAT achieves a precision of 80.0% and a recall of 96.1% on a collection of 546 difficult numeric benchmarks: more accurate than double-double oracles yet dramatically faster than arbitrary-precision condition number computations.

Paper Structure

This paper contains 15 sections, 16 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Debugging, Oracles, and Shadow Memory
  4. Error Taylor Series and Condition Numbers
  5. Condition Numbers for Rounding Error
  6. Oracles for Overflow and Underflows
  7. Implementation
  8. Shadow Memory
  9. Implementing Double-Double Computation
  10. Implementing Logarithmic Computation
  11. Evaluation
  12. RQ1: Predicting Floating-Point Error
  13. RQ2: Comparison to Oracle Method
  14. RQ3: Equal performance to arbitrary precision
  15. Conclusion

Figures (1)

  • Figure 1: A precision vs recall graph of ExplaniFloat, the arbitrary-precision baseline and the double-double oracle based debugging baseline. We run them on thresholds starting from 4 to 4096 doubling each time. Note that the precision and recall do not change drastically with change in the threshold.