Crate symplex 

symplex — a fast, correct symbolic mathematics library for Rust.

Expressions are exact (Ratio<BigInt> arithmetic, hash-consed in a Context arena) and the library can differentiate, integrate (indefinite, definite, improper, numeric), sum, take limits and series, solve equations, systems, ODEs and recurrences, simplify with a public rewrite-rule engine, work with sets and boolean logic, do exact linear algebra, view expressions as polynomials with symbolic coefficients (Poly), put rational functions into normal form (ratsimp), solve linear programs exactly with dual values and Farkas certificates, compute Hermite and Smith normal forms of integer matrices, run numerical root finding and minimisation, apply Laplace/Fourier/Mellin/Z transforms, and generate optimized Rust or C99 code. See the README and The Symplex Book for a guided tour, and CHANGELOG.md for the 0.1 → 0.2 breaking changes and the 0.2 → 0.3 behaviour changes.

Symplex is designed around seven principles:

1. Construction is cheap, evaluation is explicit. Constructors only canonicalize (flatten, sort, combine). No expansion, no function evaluation, no identity application. Call .eval(), .expand(), or .simplify() when you choose.

2. Never silently wrong. Operations return Result::Err, an unevaluated node, or None instead of a guess: ∫₋₁¹ dx/x² is Err(Divergent), solve(x − x) is Err(InfiniteSolutions), re(z) stays re(z) until z is known to be real. Structural substitution by default.

3. One representation per concept. One assumption system. One polynomial type. One number type. One solve function.

4. Thread-safe from day one. Expression handles are Send + Sync.

5. No recursive tree walks. All traversals use explicit stacks.

6. The compiler is the API contract. pub = stable. pub(crate) = internal. Ex, BoolEx and SetEx are distinct types.

7. Extensible without inheritance. Custom functions via registered rules (Rule, RuleSet).

API model

• Operations for which “unevaluated” is a valid answer return Ex and have a try_ twin returning Result (integrate / try_integrate, integrate_definite / try_integrate_definite, summation / try_summation, …).
• Numeric boundaries (eval_f64, compile, to_rust_fn, to_c_fn, integrate_numeric) and structural preconditions (Matrix::inv, cholesky) return Result.
• Queries (is_positive, equals, SetEx::contains, Matrix::is_symmetric) return Option<bool>: yes, no, or unknown.

Quick Start

use symplex::prelude::*;
use symplex::syms;

let ctx = Context::new();
syms!(ctx; x, y);
let expr = &x * &x + &x * 2 + 1;
assert_eq!(format!("{expr}"), "x^2 + 2*x + 1");

// Differentiate, integrate over an infinite range, solve, compile.
assert_eq!(format!("{}", expr.diff(&x)), "2*x + 2");
let gauss = (-x.powi(2)).exp().integrate_definite(&x, &ctx.neg_infinity(), &ctx.infinity());
assert_eq!(format!("{gauss}"), "sqrt(pi)");
let roots = (&x.powi(2) - 4).solve(&x).unwrap();
assert_eq!(roots.len(), 2);
let f = expr.compile(&["x"]).unwrap();
assert_eq!(f(&[2.0]), 9.0);

Module map

The prelude re-exports everything most programs need. Domain modules are re-exported at the crate root: ntheory, diophantine, combinatorics, matrix, matrix_decomp, normalforms, linprog, optimize, vector, quaternion, control, robotics, dynamics, poly_ex, multipoly, polysys, groebner, factor_zassenhaus, definite, summation, formal_series, finite_diff, fourier_transform, mellin, z_transform, ode, rsolve, sets, logic, parse, tree, codegen, lambdify, units, assumptions, numeric, errors, config.

New in 0.3: poly_ex (the Poly view of an expression), linprog (exact simplex), normalforms (Hermite / Smith normal forms, integer kernels) and optimize (Brent, Nelder–Mead, differential evolution, least-squares fitting).

Re-exports

pub use base::assumptions;

pub use base::config;

pub use base::errors;

pub use base::extended::Extended;

pub use base::interval::Bounds;

pub use base::interval::Interval;

pub use base::interval::IntervalKind;

pub use base::numeric;

pub use num_bigint;

pub use num_complex;

pub use num_integer;

pub use num_rational;

pub use num_traits;

Modules

base

Foundation layer: expression nodes, arena, tree traversal, canonicalization, and core types.

certificates

Exact, machine-checkable non-negativity certificates: Handelman (boxes), half-lines, parametric polyhedra and sums of squares, with Lean export. Exact, machine-checkable certificates that a polynomial is non-negative on a box, on a half-line, or on a polyhedron whose facets depend on a parameter — each exportable as a Lean 4 / Mathlib proof.

codegen

Code-generation options and compiled numeric functions. Rust source code generation from symbolic expressions.

combinatorics

Combinatorics: Stirling numbers, multinomial coefficients, partition counting. Combinatorial functions: binomial and multinomial coefficients, Stirling, Bell and Catalan numbers, derangements, and integer partitions (counting and enumeration).

context

Expression context — arena, symbol table, configuration. The Context is the user-facing entry point for symplex.

control

Control systems: state-space models, transfer functions, stability analysis. Control systems analysis: state-space models, transfer functions, stability analysis, and controller design utilities.

data_export

Data export utilities: CSV, TSV, JSON, Markdown, HTML, LaTeX table output. Data export utilities for exporting tabular data in multiple formats.

decompositions

Named results of matrix decompositions (Qr, Lu, HermiteNormalForm, …) shared by Matrix, ZMatrix and QMatrix. Named results of matrix decompositions and normal forms.

definite

Definite and improper integration. Definite and improper integration: singularity checks, limits at endpoints, known-value tables.

diophantine

Diophantine equations. Diophantine equations: linear ax + by = c, Pell x² − Dy² = 1, sums of two squares, and Pythagorean triples.

discrete

Discrete transforms on exact sequences (convolution, NTT, Walsh–Hadamard, Möbius). Discrete transforms on exact sequences (SymPy’s sympy.discrete): linear, cyclic and subset convolutions, the number-theoretic transform, the Walsh–Hadamard transform and the Möbius (subset/superset-sum) transform. (0.9)

dynamics

Lagrangian dynamics: equations of motion, mass matrix, Coriolis, gravity. Lagrangian dynamics for robotic systems.

eq

Symbolic equation type (lhs = rhs). Symbolic equation type.

expr

The core expression handle and types. User-facing expression handle.

expr_complex

Complex-analysis methods on Ex (re, im, conjugate, arg, polar, …). Complex-analysis methods on Ex: re, im, conjugate, arg, polar form — plus the 0.2 special-function constructors (si, ci, ei, li, zeta, polygamma, kronecker_delta).

expr_ops

Operator overloads and scalar-conversion traits (ToEx, Scalar). Operator overloads, standard-library trait implementations, and numeric-ingestion helpers for Expr / Context.

expr_view

A non-locking, read-only view of an expression node for use in replace(). A non-locking, read-only view of an expression node.

factor_zassenhaus

Univariate factorization over ℤ via Berlekamp–Zassenhaus. Univariate factorization over ℤ via Berlekamp–Zassenhaus (mod-p factoring, Hensel lifting, recombination).

finite_diff

Finite difference methods: weights, application, and differentiation. Finite difference methods: weights, application, and differentiation.

formal_series

Formal power series representations and algorithms. Formal power series with exact, lazily computed coefficients.

fourier_transform

Symbolic Fourier transform. Symbolic Fourier transform and inverse Fourier transform (table + rules).

groebner

Gröbner basis computation via Buchberger’s algorithm with FGLM order conversion. Gröbner basis computation via Buchberger’s algorithm with Gebauer-Möller criteria.

integrate_api

Definite / improper / numeric integration methods on Ex. Definite / improper / numeric integration methods on Ex.

lambdify

Compiled numeric closures (CompiledFn, CompiledFnVec). Compile symbolic expressions to callable numerical functions.

lean

Lean 4 / Mathlib rendering (Ex::to_lean, LeanOpts). Lean 4 / Mathlib rendering of expressions (Ex::to_lean).

linprog

Exact linear programming over ℚ (two-phase simplex, duals, Farkas certificates). Exact linear programming over ℚ: two-phase simplex with Bland’s rule, dual values and Farkas infeasibility certificates.

logic

Boolean-logic simplification, normal forms, satisfiability. Boolean-logic simplification: flatten/absorb, CNF/DNF, satisfiability.

macros

Convenience macros for building expressions. Convenience macros for creating symbolic variables, plus the public rewrite-engine and simplification option types.

mathml

Presentation MathML rendering. Presentation MathML rendering of expressions (Ex::to_mathml). (0.9)

matrix

Symbolic matrix type and operations. Symbolic matrix type.

matrix_decomp

Additional matrix decompositions (QR, Gram–Schmidt) and structure tests. Additional matrix decompositions and structure tests.

mellin

Mellin transform. Mellin transform (table-based) and its inverse.

multipoly

Sparse multivariate polynomials over ℚ. Sparse multivariate polynomials over ℚ.

normalforms

Integer matrix normal forms: Hermite, Smith, unimodular transforms, integer kernels. Integer matrix normal forms: Hermite normal form, Smith normal form, unimodular transforms and integer kernels.

ntheory

Number theory: primality, factorization, divisors, modular arithmetic. Number theory: primality testing, integer factorization, divisors, modular arithmetic, quadratic residues, discrete logarithms, prime counting, continued fractions and classical integer sequences.

ode

Ordinary differential equation solver. Ordinary Differential Equation (ODE) solver.

optimize

Numerical optimisation and root bracketing (Brent, Nelder–Mead, polynomial fitting). Numerical optimisation and root bracketing.

parse

Runtime expression parser — convert strings to symbolic expressions. Runtime expression parser.

poly_api

Polynomial-algebra methods on Ex (resultant, discriminant, division, numeric roots, …). Polynomial-algebra methods on Ex: resultant, discriminant, square-free, division, numeric roots.

poly_ex

Public sparse polynomial view (Poly) over explicit generators. Public polynomial view: Poly — an expression seen as a sparse polynomial in an explicit list of generators, with symbolic or exact rational coefficients.

polysys

Polynomial system solving via Gröbner bases. Polynomial system solving via Gröbner bases.

polytope

Exact convex polyhedra in ℚⁿ from half-spaces: vertices, volume, containment, cutting. Exact convex polyhedra in ℚⁿ given by half-spaces.

prelude

The symplex prelude — one import to get started.

quaternion

Symbolic quaternion algebra for attitude representation. Symbolic quaternion algebra for attitude representation.

robotics

Robotics kinematics: DH parameters, forward kinematics, rotations. Robotics kinematics helpers: DH parameters, forward kinematics, Jacobian, and algebraic inverse kinematics.

rsolve

Recurrence-relation solver. Recurrence-relation solver (linear, constant-coefficient, with forcing).

rules

Public rewrite-rule engine: Rule, RuleSet, Bindings, RewriteOpts, Step. Public rewrite-rule engine and simplification extensions on Ex.

series_api

Summation, products, series and formal-power-series methods on Ex. Summation, products, and series extension methods on Ex.

sets

Set algebra on intervals, finite sets, unions. Set algebra: interval arithmetic, membership, subset tests, inf/sup, measure.

sets_api

Set-algebra and boolean-logic helpers (reduce_inequalities). Set-algebra, boolean-logic, and piecewise methods on SetEx / BoolEx / Ex.

solvers

Solver entry points beyond Ex::solve: linsolve, LinearSolution, GeneralSolution, Newton systems. Extended solving methods on Ex: general solutions, systems, recurrences, IVPs.

stats

Symbolic probability and statistics: random variables, exact moments, probabilities, densities. Symbolic probability and statistics (SymPy’s stats): random variables with named distributions, exact moments, probabilities of events, densities and distribution functions — all as expressions — plus conditioning, transformations and mixtures of distributions.

summation

Symbolic summation and products. Symbolic summation and products: Faulhaber, telescoping, hypergeometric, infinite sums, and closed-form products.

transforms_api

Integral transforms (Fourier, Mellin, Laplace helpers) and directional limits on Ex. Integral transforms (Fourier, Mellin) and directional limits on Ex.

tree

Serializable expression tree for interchange (JSON, etc.). Serializable expression tree for interchange.

units

Compile-time dimensional analysis for physical quantities. Compile-time dimensional analysis for physical quantities.

vector

Vector calculus: gradient, divergence, curl, laplacian. Vector calculus: gradient, divergence, curl, Laplacian (in Cartesian, cylindrical and spherical coordinates), directional derivatives, line integrals and scalar potentials.

z_transform

Z-transform for discrete-time signal analysis. Z-transform and inverse z-transform for discrete-time signal analysis.

Macros

assert_dim

Compile-time dimension checkpoint.

const_assert_dim

Compile-time formula dimension verification with custom error messages.

dim

Build a dimension-checked physical quantity using natural math syntax.

eq

Build a symbolic equation using natural math syntax.

expr

Build a symbolic expression using natural math syntax.

matrix

Build a symbolic matrix using natural math syntax.

rule

Define a rewrite rule with pattern/template syntax.

sym

Declare a symbol with mathematical assumptions.

syms

Declare multiple symbolic variables at once.

vars

Declare multiple symbolic variables at once (alias of syms!).