High-performance symbolic-numerics via multiple dispatch

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. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graphbased term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing 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.

Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression

“…On the lowest level, TermInterface.jl and Metatheory.jl packages [17,10,16] provide an efficient expression rewriting engine and rule definition eDSL. Rules are written as regular Julia expressions, thus parsed into S-expressions and then efficiently compiled to callable functions that perform pattern matching and support advanced conditional selection [18,10,16].…”

“…On the lowest level, TermInterface.jl and Metatheory.jl packages [17,10,16] provide an efficient expression rewriting engine and rule definition eDSL. Rules are written as regular Julia expressions, thus parsed into S-expressions and then efficiently compiled to callable functions that perform pattern matching and support advanced conditional selection [18,10,16]. JuliaSymbolics also supports the definition of equational (bidirectional) rewrite rules that can be executed through an e-graph rewriting engine (adopted from [19]), performing equality saturation.…”

“…Symbolic computation was added as an Embedded Domain Specific Language (eDSL) to ease the definition of ODE systems and to aid with automatic calculation of Jacobian and Hessian of such systems. With the expansion of the SciML ecosystem, the purely symbolic routines were decoupled from the ODE solvers and were collected under JuliaSymbolics [10]. Considering its history and origin, JuliaSymbolics is geared toward symbolic differentiation and numerical integration but lacks direct symbolic integration capabilities.…”

Automated Code Optimization with E-Graphs

“…• chapter 3 discusses a few real-world applications of the expression rewriting framework. We developed an example optimizer for functional streams in Julia, and we then discuss some results we have obtained when optimizing symbolic-numeric computations through e-graph rewriting, manipulating expressions from the Julia computer algebra system Symbolics.jl [19].…”

“…Up until version 4, the Julia CAS (Computer Algebra System) Symbolics.jl [19] (section 3.2) was designed with a partially monolithic architecture. The underlying package for term rewriting, SymbolicUtils.jl, provided a generic interface for symbolic terms that allowed programmers to use the classical term rewriting utilities offered by the package on their own expression types.…”