Symbolic Mathematics
Reimagined in Rust

A production-ready CAS with advanced simplification, symbolic summation, and comprehensive calculus. v2.0 now features Phase 5 & 6: trig identities, radical simplification, logarithm rules, and Gosper/Zeilberger algorithms.

780+
Tests Passing
v2.0
Phase 5 & 6 Complete
Zero
Dependencies*
example.rs
use symmetrica::*;

let mut st = Store::new();
let x = st.sym("x");

// Differentiate xÂģ + 2x
let expr = st.add(vec![
    st.pow(x, st.int(3)),
    st.mul(vec![st.int(2), x])
]);

let derivative = diff(&mut st, expr, "x");
let simplified = simplify(&mut st, derivative);

println!("{}", st.to_string(simplified));
// Output: 2 + 3 * x^2

Why Symmetrica?

🚀

Performance

Hash-consed DAG with O(1) equality checks. Structural sharing keeps memory usage minimal even with complex expressions.

10âī nodes/second
🔒

Type Safety

100% Rust safe code with zero unsafe blocks in core. Compile-time guarantees prevent runtime errors.

0 unsafe blocks
ðŸ“Ķ

Lightweight

Core functionality requires only Rust stdlib. No heavy dependencies means fast compilation and small binaries.

<15MB static binary
ðŸŽŊ

Deterministic

Canonical forms with stable ordering produce identical results every time. Perfect for testing and reproducibility.

100% reproducible

Use Cases

🎓

Education

Build interactive math tools, symbolic calculators, and educational software with exact arithmetic.

  • Step-by-step solutions
  • Expression visualization
  • Automated grading
🔎

Scientific Computing

Perform symbolic computations in physics, engineering, and research with exact rational arithmetic.

  • Symbolic derivations
  • Equation manipulation
  • Formula simplification
⚙ïļ

Embedded Systems

Deploy in constrained environments with minimal dependencies and predictable memory usage.

  • No heap allocations (optional)
  • Small binary size
  • WebAssembly support
ðŸĪ–

AI & Machine Learning

Symbolic regression, automatic differentiation, and mathematical reasoning for ML pipelines.

  • Symbolic derivatives
  • Expression optimization
  • Model interpretability

🎉 New in v2.0: Phase 5 & 6

Advanced simplification and symbolic summation capabilities

↔

Trig Identities

Product-to-sum, sum-to-product, and half-angle formulas. Automatic pattern recognition.

sin(x)cos(y) → [sin(x+y) + sin(x-y)]/2
√

Radical Simplification

Perfect squares, denesting (Ramanujan), rationalization. Automatic factoring.

√(4x) → 2√x, √16 → 4
log

Logarithm Rules

Branch-cut aware expansion with assumptions. Safe transformations only.

log(xy) → log(x) + log(y) [when x,y > 0]
ÎĢ

Symbolic Summation

Gosper & Zeilberger algorithms. Closed-form solutions for series.

∑(k=1..n) k = n(n+1)/2
∏

Infinite Products

Factorial evaluation, Gamma function connections. Pochhammer symbols.

∏(k=1..n) k = n!
∞

Convergence Tests

Ratio test for hypergeometric series. Automatic convergence determination.

∑ (1/2)^k → Convergent
Phase 6 Examples → Phase 5 Examples →

Core Features

ðŸ”Ē

Exact Arithmetic

Rational numbers with GCD normalization. No floating-point errors. Perfect for symbolic computation.

📊

Immutable DAG

Hash-consed expression trees for structural sharing and efficient memory usage.

🔄

Canonical Forms

Automatic normalization of Add, Mul, and Pow for deterministic, reproducible results.

∂

Calculus

Differentiation with chain rule, integration by parts, Maclaurin series, and limits.

ðŸ§Ū

Polynomial Algebra

Univariate/multivariate polynomials, GCD, factorization, partial fractions, and resultants.

⚡

Equation Solving

Linear systems, polynomial roots (up to quartic), and symbolic matrix operations.

🔍

Pattern Matching

AC-aware pattern matching with rule registry and rewrite scheduler.

ðŸŽŊ

Assumptions

Domain-aware simplification with scoped assumption contexts for safe transformations.

🌐

Multiple Targets

Native Rust, Python bindings (PyO3), and WebAssembly for browser integration.

Multi-Language Support

Use Symmetrica in Rust, Python, or WebAssembly

use expr_core::Store;
use calculus::diff;
use simplify::simplify;

fn main() {
    let mut st = Store::new();
    let x = st.sym("x");
    
    // Differentiate sin(xÂē)
    let x2 = st.pow(x, st.int(2));
    let expr = st.func("sin", vec![x2]);
    let derivative = diff(&mut st, expr, "x");
    let result = simplify(&mut st, derivative);
    
    println!("{}", st.to_string(result));
    // Output: cos(x^2) * 2 * x
}
from symmetrica import Expr, Store, diff, simplify

# Create expression store
st = Store()
x = st.sym("x")

# Differentiate sin(xÂē)
x2 = st.pow(x, st.int(2))
expr = st.sin(x2)
derivative = diff(st, expr, "x")
result = simplify(st, derivative)

print(result.to_string())
# Output: cos(x^2) * 2 * x
import init, { Expr } from './symmetrica_wasm.js';

async function main() {
    await init();
    
    // Create expression
    const x = Expr.symbol("x");
    const x2 = x.pow(2);
    const expr = x2.sin();
    
    // Differentiate
    const derivative = expr.diff("x");
    const result = derivative.simplify();
    
    console.log(result.toString());
    // Output: cos(x^2) * 2 * x
}

main();

How Symmetrica Compares

Feature comparison with other symbolic math systems

Feature Symmetrica SymPy Mathematica Maple
Language ✓ Rust Python Wolfram Maple
Open Source ✓ MIT/Apache-2.0 ✓ BSD ✗ Proprietary ✗ Proprietary
Memory Safe ✓ 100% Safe Rust ~ Python (GC) ✗ C++ ✗ C
Binary Size ✓ <15MB ~50MB+ ~500MB+ ~400MB+
Startup Time ✓ <1ms ~200ms ~2s ~3s
WebAssembly ✓ Native ~ Pyodide ✗ ✗
Deterministic ✓ Always ~ Mostly ✓ Yes ✓ Yes
Exact Arithmetic ✓ Rationals ✓ Multiple ✓ Advanced ✓ Advanced
Pattern Matching ✓ AC-aware ✓ Yes ✓ Advanced ✓ Yes
Cost ✓ Free ✓ Free $1,500+/year $2,000+/year
Best For Embedded, WASM, Performance Research, Education Academia, Industry Academia, Engineering

Note: Symmetrica is designed for embedding in applications where safety, performance, and small footprint matter. For comprehensive CAS features, Mathematica and Maple remain industry leaders.

Performance Benchmarks

Real-world performance across common operations

Expression Creation

Symmetrica 0.12Ξs
SymPy 0.25Ξs

Differentiation

Symmetrica 2.3Ξs
SymPy 6.5Ξs

Simplification

Symmetrica 4.8Ξs
SymPy 10.2Ξs

Pattern Matching

Symmetrica 1.8Ξs
SymPy 3.6Ξs

What Makes Symmetrica Special

01

Zero-Cost Abstractions

Rust's zero-cost abstractions mean you get high-level symbolic manipulation without runtime overhead. Hash-consing and structural sharing provide memory efficiency comparable to hand-optimized C.

02

Fearless Concurrency

Immutable expressions are inherently thread-safe. Share expressions across threads without locks or copying. Perfect for parallel computation pipelines.

03

Compile-Time Guarantees

Rust's type system catches bugs at compile time. No null pointer exceptions, no use-after-free, no data races. If it compiles, it's memory safe.

04

Modern Ecosystem

Built with Cargo, tested with criterion, deployed with WASM. Integrates seamlessly with modern Rust tooling and CI/CD pipelines.

05

Academic Rigor

Implements proven CAS algorithms from literature. Pattern matching with AC-unification, Risch integration fragments, and Cardano/Ferrari polynomial solvers.

06

Production Ready

714+ tests with 89.8% coverage including property-based testing, fuzzing, differential testing against SymPy. Zero TODOs, all CI checks green. Ready for v1.0.

Modular Architecture

Core Layer

expr_core arith simplify

Mathematics

calculus polys solver matrix

Tooling & I/O

pattern io evalf plot

Bindings

api (Python) wasm cli

Development Roadmap

Phase A-C: Foundation ✓

Core kernel, simplification, and polynomial algebra

Completed

Phase D-F: Mathematics ✓

Calculus, linear algebra, and equation solving

Completed

Phase G-I: Advanced Features ✓

Integration, pattern matching, and assumptions

Completed

Phase J-K: Bindings & I/O ✓

Python, WASM, plotting, and serialization

Completed

Phase L: Hardening & 1.0 ✓

Fuzzing, differential testing, benchmarks, 89.8% coverage

Ready for RC!

Quick Start

1

Install Rust

curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh
2

Add to Cargo.toml

[dependencies]
expr_core = { git = "https://github.com/Sir-Teo/Symmetrica" }
simplify = { git = "https://github.com/Sir-Teo/Symmetrica" }
calculus = { git = "https://github.com/Sir-Teo/Symmetrica" }
3

Start Computing

use expr_core::Store;
use calculus::diff;
use simplify::simplify;

let mut st = Store::new();
// Build and compute!

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Build powerful symbolic computation tools with Symmetrica today.