High-performance symbolic-numerics via multiple dispatch (2105.03949v3)

Abstract: As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

• The paper introduces Symbolics.jl, a Julia library achieving high-performance symbolic-numeric computing by leveraging dynamic multiple dispatch and flexible term representation.
• Symbolics.jl employs an Abstract Term Interface for integrating optimized data structures and demonstrates substantial speedups, including a 113x increase for symbolic transformations.
• The library supports advanced term rewriting systems, including novel e-graph methods, enabling practical applications like optimizing ODE models and automating PDE code generation.

High-Performance Symbolic-Numeric Computing with Symbolics.jl

The paper presents Symbolics.jl, a symbolic-numeric computing library implemented in the Julia programming language. This library leverages dynamic multiple dispatch to provide high-performance, extendable symbolic computation without sacrificing the benefits of a high-level language. The primary goal of Symbolics.jl is to democratize mathematical computing by making it accessible to domain scientists and engineers who may not have expertise in code optimization but require high-efficiency symbolic-numeric computations.

Key Contributions

The authors introduce several innovative features in Symbolics.jl that address central challenges in symbolic computing:

1. Dynamic Multiple Dispatch: By using Julia's multiple dispatch capabilities, Symbolics.jl allows for the specialization of term representations at runtime. This flexibility decouples data representations from operations, overcoming the Expression Problem commonly encountered in traditional symbolic systems.
2. Abstract Term Interface: A generic API is presented, abstracting term representation, which allows optimized data structures to be integrated without altering existing term rewriters. This interface facilitates the development of term-rewriting systems that coexist, such as classical and e-graph-based optimizers.
3. Performance Enhancements: The paper reports significant numerical improvements, such as a 113x increase in speed for symbolic transformations and a 157x acceleration in PDE simulations. These results are achieved through efficient term representations, constructor-based simplifications, and leveraging Julia's JIT compiler.

Term Representation and Rewriting Systems

Symbolics.jl introduces a parameterized Term{T} structure to represent mathematical expressions. This design supports varied algebraic properties and efficient simplifications crucial for high-performance computations. The library supports:

• Classical Term Rewriting: Driven by pattern-matching and rule-based transformations, this system offers deterministic expression manipulation.
• E-graph-based Rewriting: Utilizing equality saturation techniques, this approach optimizes expressions by applying non-terminating rewrite rules that minimize cost functions, a novel contribution to the field of symbolic computing.

Applications and Results

The practical applications showcased in the paper highlight the library's potential in optimizing scientific computations. Notably:

• Numerical Optimization: Symbolics.jl optimized a B Cell Antigen Signaling ODE model, halving the execution time of the right-hand side code.
• Automated Code Generation: For a reaction-diffusion PDE solver, the library's tools enable automatic conversion and optimization, resulting in substantial performance gains with minimal additional user input.

Implications and Future Directions

Symbolics.jl represents a significant step forward in the practical implementation of symbolic-numeric systems. By integrating intricate term rewriting capabilities with high-level programming constructs, it positions itself as a valuable tool for both theoretical exploration and practical applications in computational mathematics and engineering.

Future developments could focus on expanding the library's capabilities to other domains requiring symbolic manipulation, such as quantum computing and AI. Additionally, continued enhancements to the integration of symbolic methods with the larger numerical ecosystem will likely increase Symbolics.jl's utility in solving complex, real-world problems.