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Equality Saturation: A Groundbreaking Optimization Method

This paper introduces a new optimization approach called Equality Saturation, which tackles the traditional Optimization Phase Ordering Problem in programming. By exploring equivalences and utilizing a Global Profitability Heuristic, our approach transforms programs efficiently. It offers benefits such as mitigating phase ordering issues, enabling non-destructive updates for exponential search, and enhancing translation validation. The method involves representing loops and equality analyses through Program Expression Graphs, allowing for complete representation without intermediate variables. With Equality Inference and Global Profitability Heuristic, the optimized Program Expression Graphs are achieved. The approach also includes Loop Induction Variable Strength Reduction and other emergent optimizations based on simple rules. Implementation details and case studies with Peggy, a Java Bytecode Optimizer, showcase significant improvements in program efficiency and runtime. Finally, this study concludes with insights on the extensibility and powerful nature of Equality Saturation in optimization and translation validation.

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Equality Saturation: A Groundbreaking Optimization Method

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  1. Equality Saturation:A New Approach to Optimization Ross Tate, Mike Stepp, Zach Tatlock, Sorin Lerner University of California, San Diego

  2. Traditional Optimization Phase Ordering Problem Original Program Optimized Program Optimizations Local Profitability Heuristics

  3. Traditional Optimization

  4. Exploring Equivalences

  5. Exploring Equivalences

  6. CFG→PEG Conversion Our Approach Program Expression Graph

  7. Equality Saturation Our Approach

  8. Our Approach Global Profitability Heuristic

  9. PEG→CFG Conversion Our Approach

  10. Benefits • Mitigates Phase Ordering Problem • Non-destructive updates allow exponential search • Global Profitability Heuristic • Explore first, then decide • Translation Validation • Verify translations using equality saturation

  11. Example sum = 0; for (i = 0; i < 10; i++) sum += 4 * i; return sum;

  12. Representing Loops sum = 0; for (i = 0; i < 10; i++) sum += 4 * i; return sum; * 4 θ1 • Complete Representation • Referentially Transparent • No Intermediate Variables 0 + 1

  13. Equality Analyses • Identify Equalities • PEG Node Granularity • Equality Axioms • ∀X. 4*X = (X≪2) ∀X. 4*X = (X≪2) << * 2 4 θ1 0 + 1

  14. Equality Inference ∀X. 4*X = (X≪2) ∀X,Y. 4*θ(X, Y) = θ(4*X, 4*Y) << θ1 * ∀ X,Y,Z. X*(Y+Z) = X*Y+ X*Z * 2 * + 4 θ1 4 * * 4 0 0 + ∀X. X*0 = 0 4 1 4 1 ∀X. X*1 = X

  15. E-PEG << * θ1 + * 2 4 θ1 * 4 * 0 + 4 0 1 4 1

  16. PEG Selection << * θ1 Global Profitability Heuristic + * 2 4 θ1 * 4 * 0 + 4 0 1 4 1

  17. Optimized PEG θ1 0 + 4

  18. Optimized Program sum = 0; for(j = 0; j < 40; j += 4) sum += j; return sum; θ1 0 + 4

  19. Optimized Program sum = 0; for(i = 0; i < 10; i++) sum += 4 * i; return sum; sum = 0; for(j = 0; j < 40; j += 4) sum += j; return sum; Loop Induction Variable Strength Reduction

  20. Emergent Optimizations Optimizations composed from simple rules • Loop Induction Variable Strength Reduction • Loop-Operation Factoring • Loop-Operation Distributing • Inter-Loop Strength Reduction • Temporary Object Removal • Partial Inlining

  21. Implementation

  22. Implementation • Algorithm provided in the Technical Report • Model heap with values having linear types

  23. Implementation • Tarjan’s Union-Find Algorithm • tracks equivalence classes in the E-PEG • Rete Pattern Matching Algorithm • incrementally finds significant nodes in the E-PEG • Equality Analyses: • PEG Operator Axioms • Language-Specific Axioms • Domain-Specific Axioms

  24. Implementation • Pseudo-Boolean Solver • Assign a cost to each operation in the E-PEG • Impose constraints for a well-formed PEG • Minimize the cost of the selected PEG

  25. Peggy: Java Bytecode Optimizer • Observed Emergent Optimizations • Traditionally need to be explicitly implemented • Domain-Specific Analyses: • 7% runtime improvement on Java ray tracer • Compilation of SpecJVM (per method): • 1030 programs found in less than 200MB memory • Average compilation time per stage: 14 msec 88 msec 1499 msec 1499 msec 43 msec

  26. CFG→PEG Conversion • CFG→PEG Conversion • Equality Saturation Translation Validator • Validation of Soot optimizer on SpecJVM: • 98% of optimized methods successfully validated • Optimization bug found within remaining 2% ?

  27. Conclusions • Powerful • Simultaneous Exponential Search • Emergent Optimizations • Extensible • Cooperative Equality Analyses • Domain-Specific Axioms • General • Optimization • Translation Validation

  28. Related Work • E-Graphs • Denali: Basic Block Assembly Superoptimizer • Simplify: Theorem Prover • Representations • Thinned-Gated SSA • Lucid programming language • Value Dependence Graph • Dependence Flow Graph • Program Dependence Graph/Web • Rewrite-Based Optimizers • TAMPR • ASF+SDF • Stratego

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