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Combining Tools for Optimization and Analysis
- f Floating-Point Computations
Floating-Point Computations Are Ubiquitous
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for Optimization and Analysis of Floating-Point Computations Heiko - - PowerPoint PPT Presentation
for Optimization and Analysis of Floating-Point Computations Heiko - - PowerPoint PPT Presentation
Combining Tools for Optimization and Analysis of Floating-Point Computations Heiko Becker, Pavel Panchekha, Eva Darulova, Zachary Tatlock FM 2018, 17.07.2018 Floating-Point Computations Are Ubiquitous 2 Floating-Point Computations are Tricky
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Floating-Point Computations are Tricky
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Renewed Interest Lead to Numerous Tools
Precimonious Daisy Herbie VCFloat Precisa Gappa FPTuner
None of these tools have been connected together!
???
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- verify Herbies optimizations with Daisy
- compare Herbie’s and Daisy’s optimizations
⟹ best run together
+
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- connecting exposes tools to new inputs
- found bugs and inaccuracies
⟹ call to action to connect tools
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+
- unify input formats
- verify Herbies optimizations with Daisy
- compare optimization techniques
- fix bugs and inaccuracies
Concretely
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+𝐷𝐼𝐹𝐷𝐿
+𝑃𝑄𝑈
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Floating-Point Arithmetic
0.2 + 0.1 0.3 0.30000000000000004
0.2 + 0.1;
roundoff error
≠
Units in the Last Place: relative measure of lost bits 1 + 𝜀 abstraction: 𝑓1 + 𝑓2 = 𝑓1 + 𝑓2 ∗ 1 + 𝜀
in Daisy in Herbie
error of 𝟏 error of 𝟓𝒇−𝟐𝟔
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Heuristic Optimization
- f Floating-Point Programs
𝑦 + 1 − 𝑦 Optimizer 1 𝑦 + 1 + 𝑦 ⟹ dynamic analysis ⟹ (possibly) unsound optimizations ⟹ heuristic hill-climbing algorithm
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Sound Analysis
- f Floating-Point Programs
𝑔 𝑦 = 𝑦 + 1 − 𝑦 Static Analyzer 𝑦 ∈ [10, 100] ⟹ sound dataflow analysis ⟹ 2.34e−15 is a sound but possibly pessimistic upper bound error is 2.34e−15
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Sound Optimization
- f Floating-Point Programs
−𝑦1 ∗ 𝑦2 − 2 ∗ 𝑦2 ∗ 𝑦3 − 𝑦1 − 𝑦3 𝑦𝑗 ∈ [−15, 15] (−𝑦1 ∗ 𝑦2 − (𝑦1 + 𝑦3)) − (2 ∗ 𝑦2) ∗ 𝑦3 Static Analyzer + Rewriter ⟹ rewriting optimizes accuracy ⟹ improved error bound from 2.95e−13 to 1.98e−13
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+𝐷𝐼𝐹𝐷𝐿
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The FPBench Project
- collection of benchmarks
- new standardized input format
- converters to input formats
- FPCore → Gappa
- FPCore → C
- FPCore → Scala
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103 FPBench benchmarks in total
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Herbie times out on 34 Daisy raises an alarm for 22 13 alarms on source programs 9 alarms on Herbies result End-to-end result for 47:
- 8 have become worse
- 18 have the same worst-case bound
- 21 have a provable improvement
+𝐷𝐼𝐹𝐷𝐿
Herbie’s error improvement better
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+𝑃𝑄𝑈
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Error Improvement of Herbie Daisy Both better
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Evaluation of the optimization algorithms
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Herbie Daisy Both
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Additional Benefit: Improving Tool Robustness
Herbie
- Incorrect Typing rule for let-bindings
- Incorrect handling of duplicate fields
- Infinite loop for some preconditions
Daisy
- Improved analysis of elementary functions
- Improved error reporting
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Conclusion
- floating-point tools solve orthogonal problems
- connecting them is easy and exposes bugs
- Daisy is a good verification backend for Herbie
- Herbie’s and Daisy’s optimizations work best together
- first step on bigger vision of connecting tools
Questions?
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