Symplex

Symbolic mathematics for Rust.

Pre-release (0.11). The API is stabilising but not stable: 0.7.0 reshaped the certificate API and 0.10.0 added variants to three fresh enums/structs; every breaking change has a one-line fix in the book's migration pages (0.6 → 0.7, 0.3 → 0.4, 0.1 → 0.2) and is listed first in CHANGELOG.md. Option structs are #[non_exhaustive] so that adding an option is never a break again. Feedback welcome.

Contributing? See CONTRIBUTING.md for architecture, the no-panic policy (enforced by a ratchet test), and conventions. The user guide is The Symplex Book; "What's New" pages cover each release.

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What This Is

symplex is a symbolic computation library — a computer algebra system for Rust with SymPy as the coverage reference. It manipulates mathematical expressions exactly (arbitrary-precision rational arithmetic, never floating point unless you ask) and can differentiate, integrate, sum, solve equations, systems, ODEs and recurrences, simplify, transform, factor, compute Gröbner bases and minimal polynomials, analyse functions, work with random variables and distributions exactly, do exact linear algebra, linear programming and polytope geometry, and generate Rust, C99, Python, NumPy or Julia code from symbolic results.

Two things set it apart from a port of SymPy: exact certificates — LP duals and Farkas vectors, Handelman / Pólya / sum-of-squares proofs of polynomial inequalities that are re-verified with exact arithmetic — and Lean 4 / Mathlib export of those proofs, so a Rust program can produce a theorem a proof assistant checks. It is designed for Rust developers in robotics, control, physics and signal processing, and for anyone generating machine-checked mathematics.

Quick Example

use symplex::prelude::*;

fn main() {
    let ctx = Context::new();
    let x = ctx.symbol("x");

    // Build an expression and differentiate
    let f = expr!(ctx, x^3 - 2*x + 1);
    let df = f.diff(&x);
    println!("f'(x) = {df}");                        // 3*x^2 - 2

    // Solve an equation (identities and contradictions are errors, not [])
    let roots = expr!(ctx, x^2 - 5*x + 6).solve(&x).unwrap();
    println!("roots: {roots:?}");                     // [Ex(3), Ex(2)]

    // Definite integral — improper and divergent cases are handled honestly
    let gauss = expr!(ctx, exp(-x^2)).integrate_definite(&x, &ctx.neg_infinity(), &ctx.infinity());
    println!("∫ e^(-x²) dx = {gauss}");                // sqrt(pi)

    // Simplify a trig identity
    println!("{}", expr!(ctx, sin(x)^2 + cos(x)^2).simplify());   // 1

    // Generate optimized Rust code from a symbolic result
    let code = df.to_rust_fn("gradient", &["x"]).unwrap();
    println!("{code}");
    // → pub fn gradient(x: f64) -> f64 { 3_f64.mul_add(x.powi(2), -2_f64) }

    // Or compile to a callable — no codegen, just fast evaluation
    let grad = df.compile(&["x"]).unwrap();
    println!("f'(2) = {}", grad(&[2.0]));             // 10.0
}

cargo add symplex

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When to Use This

• You need symbolic differentiation, integration, summation, or equation solving and want to stay in Rust.
• You are generating numerical code from symbolic derivations — Jacobians, transfer functions, filter coefficients, control laws — as Rust or C99.
• You need compile-time dimensional analysis for physical quantities.
• You need thread-safe symbolic computation without a GIL or global interpreter lock.
• You want exact rational arithmetic (1/3 stays as 1/3, not 0.33333...).
• You want a proof, not a number: a certificate that an inequality holds on a region, exportable to Lean 4 / Mathlib and compiled there.
• You need exact probability — P(Binomial(5, 1/3) > 2) = 17/81, E[X | X > 0] = √(2/π) — rather than Monte-Carlo estimates.

When Not to Use This

• You need a mature CAS with decades of community validation — use SymPy. It has broader coverage, more special functions, and a much larger test corpus.
• You need geometry, tensor algebra, quantum mechanics, or PDE solving — these are not available (planar/space geometry is next on the roadmap; statistics arrived in 0.11).
• You need interactive notebook-style exploration — symplex is a library, not an application. (Though see cargo run --example repl for a basic REPL.)
• You need results verified against extensive known-answer databases — symplex has ~11,400 tests including SymPy cross-validation fixtures and every 0.9+ test cites its SymPy reference value, but SymPy has orders of magnitude more coverage.
• You need large-scale or sparse numerical optimisation — the exact simplex is dense (O(m·n) integer operations per pivot, in i64/i128/256-bit/BigInt as the numbers grow; hundreds of rows, not hundreds of thousands), and the f64 routines are the classic derivative-free methods, not a replacement for a dedicated optimisation library.

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What You Can Do

Calculus

Differentiation handles the chain rule, product rule, all elementary functions, and the special functions (Bessel, orthogonal polynomials, digamma → polygamma). Indefinite integration uses 15+ strategies including by-parts, u-substitution, partial fractions, trig substitution, the Risch algorithm, Lazard–Rioboo–Trager log-to-real conversion, and heuristic integration. Radical coefficients (e.g., √5 from cyclotomic denominators) are handled exactly via algebraic number field arithmetic.

let ctx = Context::new();
syms!(ctx; x);

expr!(ctx, sin(x^2)).diff(&x);                      // 2*x*cos(x^2)
expr!(ctx, x * exp(x)).integrate(&x);               // x*exp(x) - exp(x)
expr!(ctx, sin(x) / x).limit(&x, &ctx.int(0));      // 1  (Gruntz algorithm)
expr!(ctx, exp(x)).series(&x, &ctx.int(0), 5);      // 1 + x + x^2/2 + x^3/6 + x^4/24

// One-sided limits; the two-sided limit stays a `Limit` node when they disagree
(1 / &x).limit_right(&x, &ctx.int(0));              // oo
(1 / &x).limit_left(&x, &ctx.int(0));               // -oo

Definite, Improper and Numeric Integration

integrate_definite locates interior singularities, treats infinite bounds and endpoint singularities as improper integrals via one-sided limits, resolves Abs/Heaviside/DiracDelta/Piecewise integrands, and consults a table of ~30 classical improper integrals (with symbolic parameters under assumptions). Divergence is reported, never hidden.

let ctx = Context::new();
syms!(ctx; x);
let (zero, one, inf) = (ctx.int(0), ctx.int(1), ctx.infinity());

x.powi(2).integrate_definite(&x, &zero, &one);                  // 1/3
(-&x).exp().integrate_definite(&x, &zero, &inf);                // 1
(&x.sin() / &x).integrate_definite(&x, &zero, &inf);            // 1/2*pi
x.ln().integrate_definite(&x, &zero, &one);                     // -1
x.abs().integrate_definite(&x, &ctx.int(-2), &ctx.int(3));      // 13/2

// ∫₋₁¹ dx/x² diverges: F(1) − F(−1) = −2 would be wrong, so it is an error
let r = x.powi(-2).try_integrate_definite(&x, &ctx.int(-1), &one);
assert!(matches!(r, Err(SymplexError::Divergent { .. })));

// Adaptive Gauss–Kronrod (G7/K15) quadrature when there is no closed form
let v = x.powi(2).exp().integrate_numeric(&x, &zero, &one).unwrap();   // 1.4626517459…

// Residues at poles of any order, and at infinity
let z = ctx.symbol("z");
(&z.exp() / &z.powi(3)).residue(&z, &zero);                     // 1/2
(1 / (&z.powi(2) + 1)).residue_at_infinity(&z);                 // 0

Summation, Products and Series

let ctx = Context::new();
syms!(ctx; k, x);
let n = ctx.symbol_with("n", &[Assumption::Integer, Assumption::Positive]);
let (zero, one, inf) = (ctx.int(0), ctx.int(1), ctx.infinity());

k.powi(5).summation(&k, &one, &n);                    // 1/6*n^6 + 1/2*n^5 + 5/12*n^4 - 1/12*n^2
(&k * &ctx.int(2).pow(&k)).summation(&k, &zero, &n);  // 2^(n + 1)*(n - 1) + 2   (Gosper)
n.binomial(&k).summation(&k, &zero, &n);              // 2^n
k.powi(-2).summation(&k, &one, &inf);                 // 1/6*pi^2
k.powi(-3).summation(&k, &one, &inf);                 // zeta(3)   (odd p: symbolic)
(&x.pow(&k) / &k.factorial()).summation(&k, &zero, &inf);   // exp(x)
(1 - k.powi(-2)).product_over(&k, &ctx.int(2), &inf); // 1/2

(1 / &k).is_convergent(&k);                           // Some(false)

// Formal power series: lazy exact coefficients and closed-form general terms
let s = x.sin().fps_maclaurin(&x);
s.coefficient(51);                                    // -1/1551118753287382280224243016469303211063259720016986112000000000000
s.general_term(&k);                                   // Some(sin(1/2*k*pi)/k!)
s.reversion().unwrap().coefficients(6);               // asin: [0, 1, 0, 1/6, 0, 3/40]

Function Analysis

SymPy's calculus.util on Ex (0.9): singularities, stationary points, extrema on an interval or union of intervals (one-sided limits at open or infinite endpoints, ±∞ allowed), monotonicity and convexity (exact for polynomial and rational derivatives via Sturm sequences; three-valued, never a guess), periodicity and the range of a function.

let ctx = Context::new();
syms!(ctx; x);
let f = &x.powi(3) - 3 * &x;
let interval = ctx.interval(&ctx.int(-2), &ctx.int(2), false, false);
f.stationary_points(&x, None).unwrap();                          // {-1, 1}
f.maximum(&x, &interval).unwrap();                               // 2
(1 / (&x.powi(2) - 1)).singularities(&x, None).unwrap();          // {-1, 1}
x.powi(3).is_increasing(&x, &ctx.reals());                       // Some(true)
((2 * &x).sin() + (3 * &x).cos()).periodicity(&x).unwrap();      // 2*pi

Also: minimum, is_decreasing, is_strictly_increasing/_decreasing, is_monotonic, is_convex, function_range.

Complex Analysis and Special Functions

re, im, conjugate, arg are honest about unknown realness: with no assumption on z, z.re() is the unevaluated re(z).

let ctx = Context::new();
let z = ctx.symbol("z");
let x = ctx.symbol_with("x", &[Assumption::Real]);
let y = ctx.symbol_with("y", &[Assumption::Real]);
let i = ctx.i_unit();

let w = &x + &i * &y;
w.conjugate();                                        // x - y*I
w.abs_squared();                                      // x^2 + y^2
w.exp().as_real_imag();                               // (cos(y)*exp(x), sin(y)*exp(x))
z.re();                                               // re(z)   — not assumed real
z.exp().re();                                         // cos(im(z))*exp(re(z))
(1 / &ctx.int(0)).eval();                             // zoo   (complex infinity)

// New constants and special functions with exact values and evalf
ctx.int(1).digamma().eval();                          // -EulerGamma
ctx.int(4).zeta().eval();                             // 1/90*pi^4
ctx.int(1).polygamma(&ctx.int(1)).eval();             // 1/6*pi^2
ctx.infinity().si().eval();                           // 1/2*pi
ctx.catalan().eval_decimal(30).unwrap();              // 0.915965594177219015054603514932

// 0.9: 24 more special functions — exact values, derivative rules, arbitrary-precision evalf
// (checked at 40 digits against mpmath), Display/LaTeX/parse, and integration results
x.powi(2).exp().integrate(&x);                        // 1/2*sqrt(pi)*erfi(x)      (was unevaluated before 0.9)
(x.sinh() / &x).integrate(&x);                        // Shi(x)
ctx.rational(1, 2).polylog(&ctx.int(2)).eval();       // -1/2*ln(2)^2 + 1/12*pi^2   (Li₂(½))
ctx.int(0).elliptic_k().eval();                       // 1/2*pi
x.airyai().diff(&x);                                  // airyaiprime(x)
x.assoc_legendre(&ctx.int(2), &ctx.int(1)).eval();    // -3*x*sqrt(-x^2 + 1)
expr!(ctx, airyai(x) + polylog(2, x));                // the macro knows them too

Also: Gamma, log-gamma, lowergamma/uppergamma, erf/erfc/erfi/erfinv/erfcinv, Beta, Lambert W, Bessel J/Y/I/K, Airy Ai/Bi and derivatives, elliptic K/E/F/Π, expint/E1, Shi/Chi, Fresnel S/C, polylog/dirichlet_eta, Legendre/Chebyshev/Hermite/Laguerre and the associated/Gegenbauer/Jacobi families, Si/Ci/Ei/li, Kronecker delta — all with arbitrary-precision evaluation.

Algebra and Factoring

Polynomial operations work over ℚ using arbitrary-precision rational arithmetic. Univariate factoring over ℤ uses Berlekamp–Zassenhaus (any degree); multivariate factoring uses Kronecker substitution. Gröbner bases use Buchberger's algorithm with FGLM order conversion.

let ctx = Context::new();
syms!(ctx; x, y);

expr!(ctx, x^12 - 1).factor(&x);                      // (x - 1)*(x + 1)*(x^2 + x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1)
expr!(ctx, x^3 - x*y^2 + x^2 - y^2).factor_all();     // (x + 1)*(x + y)*(x - y)
expr!(ctx, (x + 1)^3).expand();                       // x^3 + 3*x^2 + 3*x + 1
expr!(ctx, (x^2 - 1) / (x - 1)).cancel(&x);           // x + 1

// Polynomial algebra on Ex: resultant, discriminant, division, gcdex, real-root isolation, …
expr!(ctx, x^3 - x).discriminant(&x);                 // Some(4)
expr!(ctx, x^5 - x - 1).count_real_roots(&x);         // Some(1)
expr!(ctx, x^4 + 1).is_irreducible(&x);               // Some(true)

// 0.9: algebraic numbers, variable-free gcd, Gröbner bases, real roots as RootOf, GF(p), symbolic resultants
syms!(ctx; a, b, c);
(ctx.int(2).sqrt() + ctx.int(3).sqrt()).minimal_polynomial(&x).unwrap();   // x^4 - 10*x^2 + 1
(&x.powi(2) - &y.powi(2)).gcd_all(&(&x - &y)).unwrap();                    // x - y
Ex::groebner(&[&x.powi(2) + &y.powi(2) - 1, &x - &y], &[x.clone(), y.clone()], MonomialOrder::Lex).unwrap();
                                                      // [x - y, y^2 - 1/2]
(&x.powi(3) - 2 * &x).real_roots(&x).unwrap();        // [RootOf(x^2 - 2, 0), 0, RootOf(x^2 - 2, 1)]  (ascending)
(&x.powi(2) + 1).factor_mod(&x, 5).unwrap();          // (1, [(x + 2, 1), (x + 3, 1)])
(&a * &x.powi(2) + &b * &x + &c).discriminant_symbolic(&x).unwrap();       // -4*a*c + b^2

Polynomials as Data and Rational Normal Forms

Poly (0.3) views an expression as a sparse polynomial in an explicit list of generators. Coefficients are exact rationals or symbolic parameter expressions, terms come back in SymPy's lex-descending order, and nothing is approximated. degree/coeffs/leading_coeff on Ex accept symbolic coefficients too. ratsimp is a rational-function normal form — one cancelled fraction with integer-primitive numerator and denominator — and solve uses it for parametric linear and quadratic equations.

let ctx = Context::new();
syms!(ctx; x, y, a, j, r);

// Polynomial introspection on Ex with symbolic (var-free) coefficients
let e = &a * &x.powi(2) + &x * (&a + 1) + 3;
e.degree(&x);                                         // Some(2)
e.coeffs(&x);                                         // Some([3, a + 1, a])   (ascending)
e.leading_coeff(&x);                                  // Some(a)

// Poly: sparse terms over explicit generators, exact evaluation, calculus
let p = (&a * &x.powi(2) + &x * &y * 3 - &y + 1).as_poly(&[&x, &y]).unwrap();
p.terms();                                            // [([2, 0], a), ([1, 1], 3), ([0, 1], -1), ([0, 0], 1)]
p.coeff_monomial(&[1, 1]).unwrap();                   // 3
p.total_degree();                                     // Some(2)
p.eval_gen(&x, &ctx.int(2)).unwrap();                 // Poly(5*y + 4*a + 1, y)
p.derivative(&x).unwrap().to_ex();                    // 2*a*x + 3*y

// Rational normal form: nested fractions collapse to one cancelled fraction
(1 / (&x + 1 / &y) + 1 / (1 / &x + &y)).ratsimp();   // (x + y)/(x*y + 1)
((&r * 3 - 1) / (&j + 1) - (&r + 1) / (&j * 2)).solve(&r);   // Ok([(3*j + 1)/(5*j - 1)])

// Exact sign of a rational-coefficient polynomial on an interval (square-free part + Sturm)
(&x.powi(3) - &x).poly_is_nonnegative_on(&x, &ctx.int(2), &ctx.infinity());               // Some(true)
(&x.powi(2) - &x * 2 + 1).poly_is_positive_on(&x, &ctx.neg_infinity(), &ctx.infinity());  // Some(false) — touches 0 at x = 1

// Linear certificates: (x + 1)² = λ₁·(x + 1) + λ₂·(x² − 1) as an exact linear system
let (h1, h2) = ((&x + 1).as_poly(&[&x]).unwrap(), (&x.powi(2) - 1).as_poly(&[&x]).unwrap());
let goal = (&x + 1).powi(2).as_poly(&[&x]).unwrap();
let basis = Poly::monomial_basis(&[&h1, &h2, &goal]).unwrap();    // [[2], [1], [0]]
let m = Poly::coefficient_matrix(&[&h1, &h2], &basis).unwrap();   // [[0, 1], [1, 0], [1, -1]]
let b = Poly::coefficient_matrix(&[&goal], &basis).unwrap();      // [[1], [2], [1]]
linsolve_matrix(&m, &b);                              // Ok(Unique([(x1, 2), (x2, 1)]))

Also: Poly::{from_terms, all_coeffs, degree_list, eval, add/sub/mul/pow/scale, content_and_primitive, monic, to_multipoly, nroots}, and on MultiPoly a heuristic multivariate gcd/lcm, integer_content, clear_denominators.

Simplification and the Rule Engine

simplify() tries a dozen strategies and iterates to a fixpoint. The pattern-matching engine behind it is public in 0.2: build your own rules (symbols ending in _ are wildcards, rest__ absorbs the rest of a sum or product), rewrite with them, trace what fired, and interleave them with the built-in simplifier.

let ctx = Context::new();
syms!(ctx; x, y);
let (a, b) = (ctx.symbol("a_"), ctx.symbol("b_"));

let rules = RuleSet::from_rules(vec![
    Rule::new("sin_sq", &a.sin().powi(2), &(1 - &a.cos().powi(2))),
    Rule::new("ln_add", &(&a.ln() + &b.ln()), &(&a * &b).ln()),
]);
(&x.sin().powi(2) + 3).rewrite(&rules);               // -cos(x)^2 + 4
(&x.ln() + &y.ln()).rewrite(&rules);                  // ln(x*y)

let (result, steps) = (&x.sin().powi(2) + &x.cos().powi(2)).simplify_traced(&SimplifyOpts::default());
// result = 1; steps name the strategy and the rules that fired

x.powi(4).subs_algebraic(&x.powi(2), &y);             // y^2   (plain subs would leave x^4)
(ctx.int(5) + ctx.int(24).sqrt()).sqrt().sqrtdenest();  // sqrt(2) + sqrt(3)

Equation Solving

Polynomial equations are solved through quartic by radicals; degree ≥ 5 produces RootOf nodes with numerical evaluation. Transcendental equations use inversion peeling and Lambert W. solve never lies: identities are Err(InfiniteSolutions), contradictions and range violations are Err(NoSolution).

let ctx = Context::new();
syms!(ctx; x, y, z);

expr!(ctx, x^2 - 5*x + 6).solve(&x);                  // Ok([3, 2])
(&x.sin() - &ctx.rational(1, 2)).solve(&x);           // Ok([1/6*pi, 5/6*pi])
(&x.sin() - 2).solve(&x);                             // Err(NoSolution)
(&x - &x).solve(&x);                                  // Err(InfiniteSolutions)

// All periodic solutions, with an integer parameter
let fam = (&x.sin() - &ctx.rational(1, 2)).solve_general(&x).unwrap();
// fam.solutions = [2*n*pi + 1/6*pi, 2*n*pi + 5/6*pi], fam.parameters = [n]

// Linear systems: unique / parametric / inconsistent, symbolic coefficients allowed
let sol = linsolve(&[&x + &y + &z - 6, &x - &y - 2], &[x.clone(), y.clone(), z.clone()]).unwrap();
// LinearSolution::Parametric { solution: [x = -z/2 + 4, y = -z/2 + 2, z = z], free: [z] }

// Polynomial systems via Gröbner bases (algebraic solutions)
symplex::polysys::solve_system_ex(&[&x.powi(2) + &y.powi(2) - 1, &x - &y], &[x.clone(), y.clone()]);
// Ok([[√2/2, √2/2], [-√2/2, -√2/2]])

// Inequalities (sign-chart method), including absolute values
expr!(ctx, x^2 - 4).solve_gt(&x);                     // (-oo, -2) ∪ (2, oo)
(&(&x - 1).abs() - 2).solve_lt(&x);                   // (-1, 3)

Differential Equations and Recurrences

16 ODE classes (separable, linear, Bernoulli, Riccati, Euler–Cauchy, exact, integrating factor, Clairaut, nth-order constant-coefficient, variation of parameters, systems via matrix exponential, …), initial-value problems, and linear recurrences.

let ctx = Context::new();
syms!(ctx; x, n);
let y = ctx.symbol("y");
let (d1, d2) = (y.formal_diff(&x), y.formal_diff(&x).formal_diff(&x));

(&d2 + &y).solve_ode(&y, &x);                         // y = C1*sin(x) + C2*cos(x)
(&d2 + &y).solve_ode_ivp(&y, &x, &[(0, ctx.int(0), ctx.int(0)), (1, ctx.int(0), ctx.int(1))]);
                                                      // Ok(sin(x))
(&d1 * &x - &y - &d1.powi(2)).classify_ode(&y, &x);   // Clairaut

// a(n+2) = a(n+1) + a(n), a(0) = 0, a(1) = 1  →  Binet's formula
symplex::rsolve::rsolve_linear(&[ctx.int(-1), ctx.int(-1), ctx.int(1)], None, &n, &[ctx.int(0), ctx.int(1)]);

Sets and Logic

SetEx and BoolEx are first-class: intervals, finite sets, unions with a normal form, three-valued queries, and boolean normal forms with a DPLL satisfiability check.

let ctx = Context::new();
syms!(ctx; x, p, q);

let a = ctx.interval(&ctx.int(0), &ctx.int(5), false, false);    // [0, 5]
let b = ctx.interval(&ctx.int(3), &ctx.int(10), true, false);    // (3, 10]
a.intersection(&b).simplify();                        // (3, 5]
a.symmetric_difference(&b);                           // [0, 3] ∪ (5, 10]
a.contains(&ctx.int(7));                              // Some(false)
a.contains(&x);                                       // None
a.union(&b).measure();                                // Some(10)

let conds = [x.gt(&ctx.int(0)), x.le(&ctx.int(5)), (&x.powi(2) - 4).gt(&ctx.int(0))];
reduce_inequalities(&conds, &x);                      // Ok((2, 5])

let (pp, qq) = (p.gt(&ctx.int(0)), q.gt(&ctx.int(0)));
pp.and(&qq).or(&pp).simplify();                       // p > 0
pp.and(&qq).not().to_nnf();                           // 0 >= p | 0 >= q
pp.or(&pp.not()).is_tautology();                      // Some(true)

Linear Algebra

Symbolic matrices with exact decompositions. The eigen family needs no dummy variable in 0.2, structure tests are three-valued, and preconditions are Results.

use symplex::linprog::q;   // exact rational literal: q(1, 2) = 1/2

let ctx = Context::new();
syms!(ctx; t, n);
let m = matrix![ctx, [2, 1], [1, 2]];

m.det().unwrap();                                     // 3
m.eigenvals().unwrap();                               // [3, 1]
m.char_poly(&ctx.symbol("λ")).unwrap();               // λ^2 - 4*λ + 3
m.diagonalize().unwrap();                             // (P, D)
m.matrix_exp_t(&t).unwrap();                          // [[e^(3t)/2 + e^t/2, …], …]
m.matrix_pow_symbolic(&n).unwrap();                   // [[3^n/2 + 1/2, 3^n/2 - 1/2], …]
m.matrix_sqrt().unwrap();

let spd = matrix![ctx, [4, 12, -16], [12, 37, -43], [-16, -43, 98]];
spd.cholesky().unwrap();                              // [[2,0,0],[6,1,0],[-8,5,3]]
spd.is_positive_definite();                           // Some(true)
matrix![ctx, [1, 1, 0], [1, 0, 1], [0, 1, 1]].qr().unwrap();   // exact radicals

// Irreducible characteristic polynomials give exact, evaluable RootOf eigenvalues
matrix![ctx, [0, 1, 0], [0, 0, 1], [1, 1, 0]].eigenvals().unwrap();   // [RootOf(λ^3 - λ - 1, 0), …]

// 0.3: index-list extraction, exact rationals in and out, three-valued structure tests
m.extract(&[1, 0], &[0]).unwrap();                    // [[1], [2]]
Matrix::from_ratio(&ctx, &[vec![q(1, 2), q(3, 1)]]).unwrap();   // [[1/2, 3]]
(&m - &m.transpose()).is_zero();               // Some(true)   (m is symmetric)

// 0.3.5: QMatrix / ZMatrix — plain exact matrices over ℚ / ℤ, no expression arena.
// Fraction-free (Bareiss) elimination: a 30×30 rational inverse takes 10 ms, not 470.
let h = QMatrix::from_fn(4, 4, |i, j| q(1, (i + j + 1) as i64));   // Hilbert matrix
h.det().unwrap();                                     // 1/6048000
h.inv().unwrap()[(3, 3)];                             // 2800   (the inverse is integral)
let (r, pivots) = QMatrix::from_i64(&[&[1, 2, 3], &[4, 5, 6]]).unwrap().rref();
// r = [[1, 0, -1], [0, 1, 2]], pivots = [0, 1]
ZMatrix::from_i64(&[&[2, 4, 4], &[-6, 6, 12], &[10, -4, -16]]).unwrap().smith_normal_form();

Matrix::{rref, rank, nullspace, det, inv, solve}, linsolve/linsolve_matrix and the normal forms route through QMatrix/ZMatrix automatically whenever every entry is a rational literal, so existing code gets the speed-up without changes.

// 0.9: singular values and condition number, a pseudo-inverse defined for every matrix,
// rank decomposition, Hessenberg form, permanent, companion / Jordan blocks, exact LLL
matrix![ctx, [1, 2], [3, 4]].singular_values().unwrap();     // [sqrt(sqrt(221) + 15), sqrt(-sqrt(221) + 15)]
matrix![ctx, [1, 2], [2, 4]].pinv().unwrap();                // [[1/25, 2/25], [2/25, 4/25]]  (rank-deficient)
matrix![ctx, [1, 2, 3], [4, 5, 6], [7, 8, 9]].rank_decomposition().unwrap();   // (C, F) with A = C·F
matrix![ctx, [1, 2], [3, 4]].permanent().unwrap();           // 10
matrix![ctx, [1, 2], [3, 4]].inv_mod(5).unwrap();            // [[3, 1], [4, 2]]
ZMatrix::from_i64(&[&[1, 1, 1], &[-1, 0, 2], &[3, 5, 6]]).unwrap().lll_default().unwrap();   // [[0, 1, 0], [1, 0, 1], [-1, 0, 2]]

Also: LU, LDLᵀ, Gram–Schmidt, Jordan form, matrix_log, Kronecker product, rank/nullspace/rowspace, norms, least squares, Hessian, Wronskian, Casoratian, hessenberg, companion, jordan_block, row/column insert/delete/permute, quaternions, vector calculus in Cartesian/cylindrical/spherical coordinates, state-space ↔ transfer function; select_rows/select_cols, from_bigint/from_f64_rows, to_rational_rows/to_bigint_rows, is_integer_matrix, subs_map, nnz.

Exact Optimization and Integer Lattices

Linear programs are solved over ℚ by a two-phase simplex: optima, shadow prices and Farkas infeasibility certificates are exact, never "infeasible to within tolerance". The tableau is fraction-free (integers with a common denominator) and runs on i64, then i128, then 256-bit, then BigInt cells as the numbers grow — the same pivot path in every width, so results are identical and small problems never touch the heap. Dantzig's rule is used until twelve consecutive degenerate pivots, then Bland's until the next improving step (finite, and not condemned to Bland's slow walk on the degenerate certificate LPs). A Budget (deadline and/or pivot cap, checked at every pivot) turns a runaway solve into LpStatus::BudgetExhausted. Integer matrices get Hermite and Smith normal forms with unimodular transforms, ℤ-bases of integer kernels, and exact LLL reduction.

use symplex::linprog::{feasible_nonneg, q, qi};
use symplex::normalforms::hermite_normal_form_with_transform;
let ctx = Context::new();

// max 5x + 4y  s.t.  6x + 4y ≤ 24,  x + 2y ≤ 6,  x, y ≥ 0
let sol = LpProblem::maximize(vec![qi(5), qi(4)])
    .le(vec![qi(6), qi(4)], qi(24))
    .le(vec![qi(1), qi(2)], qi(6))
    .solve().unwrap();
sol.status;                                           // Optimal
sol.x;                                                // [3, 3/2]
sol.objective;                                        // Some(21)
sol.duals;                                            // [3/4, 1/2]   shadow prices: yᵀb = 21 = cᵀx*

// x + y ≤ 1 and x + y ≥ 2 cannot both hold — here is the proof
let bad = LpProblem::minimize(vec![qi(0), qi(0)])
    .le(vec![qi(1), qi(1)], qi(1))
    .ge(vec![qi(1), qi(1)], qi(2))
    .solve().unwrap();
bad.status;                                           // Infeasible
bad.farkas;                                           // Some([1, -1])   Aᵀy = 0, yᵀb = −1 < 0

// "Is there μ ≥ 0 with Aμ = b?", exactly (Farkas / Carathéodory searches)
feasible_nonneg(&[vec![q(1, 3), q(1, 7)], vec![qi(1), qi(-1)]], &[qi(1), qi(0)]);   // Ok(Some([21/10, 21/10]))

// Integer normal forms: H = U·A (row style), S = U·A·V, ℤ-basis of the kernel
let a = matrix![ctx, [2, 4, 4], [-6, 6, 12], [10, -4, -16]];
let (h, u) = hermite_normal_form_with_transform(&a).unwrap();
h;                                                    // [[2, 4, 4], [0, 6, 0], [0, 0, 12]]
(&u * &a).eval() == h;                                // true  (det U = −1)
a.smith_normal_form().unwrap();                       // [[2, 0, 0], [0, 6, 0], [0, 0, 12]]
matrix![ctx, [2, 1, 1]].integer_nullspace().unwrap(); // [(1, 0, −2)ᵀ, (0, 1, −1)ᵀ] — generates every integer solution

Also: linprog (SciPy-shaped), linprog_matrix (from Matrix data), per-variable bounds and free variables, nonneg_combination / feasible_nonneg_certified (cone membership with the separating Farkas vector on failure), column_hermite_normal_form (SymPy's convention), smith_normal_form_with_transforms, is_unimodular, lattice_determinant.

Certified Inequalities and Lean Export

certificates::prove_nonnegative_on_box proves goal ≥ 0 on a box with a Handelman certificate — an exact identity goal = Σ λₖ·Π(xᵢ − lᵢ)^a(uᵢ − xᵢ)^b with λ ≥ 0 found by the exact LP and re-verified with exact polynomial arithmetic — or refutes the claim with an exact counterexample. The certificate exports as a Lean 4 / Mathlib theorem whose proof is nlinarith over exactly those products; Ex::to_lean() renders any elementary expression in Mathlib syntax.

use symplex::certificates::{prove_nonnegative_on_box, BoxOutcome};
let ctx = Context::new();
syms!(ctx; x, y);
let square = [(x.clone(), ctx.int(0), ctx.int(1)), (y.clone(), ctx.int(0), ctx.int(1))];

let cert = match prove_nonnegative_on_box(&(1 - &x * &y), &square, 2).unwrap() {
    BoxOutcome::Proved(c) => c,
    other => panic!("{other:?}"),
};
cert.to_string();                                     // -x*y + 1 = -y + y*(-x + 1) + 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
cert.verify();                                        // true — exact re-check, independent of the LP
cert.to_lean("one_minus_xy").unwrap();
// theorem one_minus_xy (x y : ℝ) (_h_x_lo : (0 : ℝ) ≤ x) (h_x_hi : x ≤ (1 : ℝ)) (h_y_lo : (0 : ℝ) ≤ y)
//     (h_y_hi : y ≤ (1 : ℝ)) :
//     0 ≤ -(x * y) + 1 := by
//   nlinarith [sub_nonneg.mpr h_y_hi, mul_nonneg (sub_nonneg.mpr h_x_hi) (sub_nonneg.mpr h_y_lo)]

prove_nonnegative_on_box(&(&x * &y - ctx.rational(1, 2)), &square, 2).unwrap();
                                                      // Refuted { point: [(x, 0), (y, 0)], value: -1/2, .. }
((&x - 1) / (2 * &x)).to_lean().unwrap();             // "(x - 1) / (2 * x)"
x.sqrt().gt(&ctx.int(0)).to_lean().unwrap();          // "0 < Real.sqrt x"

// 0.4: a polyhedron whose facets depend on a parameter j ≥ j₀.  On { t ≥ r, t + j·r ≥ j + 1 }
// the goal t − 1 ≥ 0 needs the multiplier λ(j) = 1 + j: (j + 1)(t − 1) = j·h₀ + h₁.
use symplex::certificates::{prove_nonnegative_on_polyhedron, PolyhedronOpts};
syms!(ctx; j, r, t);
let hyps = [&t - &r, &t + &j * &r - &j - 1];
let out = prove_nonnegative_on_polyhedron(&(&t - 1), &hyps, Some((&j, &ctx.int(0))), &PolyhedronOpts::default()).unwrap();
out.certificate().unwrap().to_string();               // (j + 1)*(t - 1) = j*h0 + h1; h0 = -r + t, h1 = j*r - j + t - 1; j ≥ 0
out.certificate().unwrap().to_lean("needs_lambda").unwrap();
//   … have h0J := mul_nonneg hJ0 h0
//   have hg : (0 : ℝ) ≤ (j + 1) * (t - 1) := by linarith only [h0J, h1]
//   have hg' := nonneg_of_mul_nonneg_right hg (by linarith only [hJ0])
//   linarith only [hg']

// 0.6: sums of squares — non-negativity on all of ℝⁿ, interior zeros included.  The Gram SDP is
// solved by a built-in interior-point method, rounded, projected and checked exactly (rational LDLᵀ).
use symplex::certificates::{prove_sos, SosOpts};
syms!(ctx; x, y, z);
let amgm = x.powi(4) + y.powi(4) + z.powi(4) - &x * &y * &z * 4 + 1;
let cert = prove_sos(&amgm, &[x.clone(), y.clone(), z.clone()], &SosOpts::default()).unwrap();
cert.certificate().unwrap().to_string();
// x^4 + y^4 + z^4 - 4*x*y*z + 1 = (-1/3*x^2 - 1/3*y^2 - 1/3*z^2 + 1)^2 + 2/3*(-y*z + x)^2 + 2/3*(-x*z + y)^2
//   + 2/3*(-x*y + z)^2 + 2/3*(-x^2 + y^2)^2 + 8/9*(-1/2*x^2 - 1/2*y^2 + z^2)^2
cert.certificate().unwrap().to_lean("amgm3").unwrap();   // have h : … := by ring;  rw [h];  positivity

Every prover returns one Outcome<C, U> — Proved(certificate), Refuted { point, value, .. } with an exact counterexample, or Unknown(what was tried), never a wrong Proved — and every certificate type implements the Certificate trait (goal, verify, to_lean, JSON round trip with re-verification). A prover call can be given a budget (PolyhedronOpts::default().with_time_limit(Duration::from_secs(150)), or with_max_pivots) and returns Unknown naming the limit rather than running on. For assembling many certificates into one lemma, lean::{Block, Tactic, Decl} is a small structured model of a tactic proof that renders haves, bullets and by blocks from their tactic column — no hand-counted indentation — and its output for a real generator's proof shape is pinned to text that compiled against Mathlib.

Also: prove_polyhedron_empty (the same identity with goal −1: a cell is empty for every j), PolyhedronProver (parse the hypotheses once, prove many goals; prove_poly takes an exact Poly directly; used_hyps names the hypotheses a certificate needs), prove_nonnegative_on_halfline / prove_nonnegative_on_reals (univariate, Pólya multipliers and square factors), lean_steps / block / lean_hints for dropping a proof into an existing skeleton, LeanOpts::{prefer_subtraction, single_fraction, symbol_text}, and symplex::polytope::{Polytope, ParametricPolytope} for the exact geometry of the cells — vertices in integer arithmetic with tight sets, clip (both sides of a cut from the cached vertices, no re-enumeration), volume in any dimension, is_full_dimensional (one LP), redundancy.

Transforms

let ctx = Context::new();
syms!(ctx; t, w, s, x);
let a = ctx.symbol_with("a", &[Assumption::Positive]);

(-&a * t.abs()).exp().fourier_transform(&t, &w);      // Ok(2*a/(a^2 + w^2))
(-t.powi(2)).exp().fourier_transform(&t, &w);         // Ok(sqrt(pi)*exp(-1/4*w^2))
(1 / (1 + &x)).mellin_transform(&x, &s);              // Ok((pi/sin(s*pi), re(s) > 0 & 1 > re(s)))
t.sin().laplace(&t, &s);                              // 1/(s^2 + 1)
((&s * -2).exp() / &s).inverse_laplace(&s, &t);       // H(t - 2)
x.sign().fourier_series_on(&x, &(-ctx.pi()), &ctx.pi(), 5).unwrap().truncate(5);
                                                      // 4*sin(x)/pi + 4*sin(3*x)/(3*pi) + 4*sin(5*x)/(5*pi)

Number Theory and Combinatorics

Pollard–Brent rho + ECM factorization, BPSW primality, modular square roots and discrete logarithms, continued fractions, Diophantine equations, and integer sequences.

use symplex::ntheory::*;
use symplex::diophantine;
use symplex::combinatorics::*;
use symplex::linprog::qi;
use symplex::num_bigint::BigInt;

isprime(561);                                         // false (Carmichael number)
factorint(1_099_532_599_387u64);                      // [(1048583, 1), (1048589, 1)]  — ~1 ms
sqrt_mod(2, 7);                                       // Some(3)
discrete_log(3, 13, 17);                              // Some(4)
primepi(1_000_000);                                   // Some(78498)
continued_fraction_periodic(23);                      // Some(([4], [1, 3, 1, 8]))
diophantine::pell(61);                                // Some((1766319049, 226153980))
diophantine::sum_of_two_squares(65);                  // Some((4, 7))
stirling2(10, 4);                                     // Some(34105)
partition_count(100);                                 // Some(190569292)
crt_i64(&[2, 3, 2], &[3, 5, 7]);                      // Some(23)
igcd(&[12i64, 18, 30]);                               // 6    (gcd_many / lcm_many take BigInt slices)
ilcm(&[4i64, 6, 10]);                                 // 60

// 0.10: n-th roots and polynomial congruences for any modulus, and exact discrete transforms
nthroot_mod(11, 4, 19, true);                         // Some([8, 11])
polynomial_congruence(&[1, 0, -3, 5].map(BigInt::from), 1000003);   // [488045, 745229, 766732]  (x³ − 3x + 5 mod p)
is_carmichael(561);                                   // true
symplex::discrete::convolution(&[qi(1), qi(2), qi(3)], &[qi(4), qi(5), qi(6)]);       // [4, 13, 28, 27, 18]
symplex::discrete::ntt(&[1, 2, 3, 4].map(BigInt::from), BigInt::from(998244353));    // number-theoretic transform

Also: quadratic_residues, is_nthpow_residue, multiplicity, primenu/primeomega, primorial, continued_fraction_reduce (finite and periodic → quadratic surd), is_amicable, binomial_coefficients; discrete::{convolution_cyclic, convolution_subset, intt, fwht, mobius_transform}.

Numerical Toolbox

Deterministic, budgeted f64 routines — bracketed roots, derivative-free minimisation, global search in a box, least-squares fits — usable on plain closures or directly on expressions (which are compiled first). Bad input is Err(InvalidArgument), a non-finite value is Err(ComputationFailed); nothing panics.

use symplex::optimize::{DeOpts, brent_root, nelder_mead, poly_fit};

let ctx = Context::new();
syms!(ctx; x, y);

// Bracketed roots (Brent–Dekker), on a closure or on a compiled expression
brent_root(|t| t * t - 2.0, 0.0, 2.0, &RootOpts::default()).unwrap();   // 1.41421356237…
(x.cos() - &x).find_root_bracket(&x, 0.0, 1.0).unwrap();                 // 0.739085133215…

// Nelder–Mead: local minimum from a starting point
nelder_mead(|p| (p[0] - 1.0).powi(2) + (p[1] + 2.0).powi(2), &[0.0, 0.0], &MinimizeOpts::default()).unwrap();   // x ≈ [1, −2]
let rosen = (1 - &x).powi(2) + 100 * (&y - &x.powi(2)).powi(2);
let r = rosen.minimize_numeric(&[&x, &y], &[-1.2, 1.0]).unwrap();       // r.x ≈ [1, 1], r.fun ≈ 1e-18, r.converged

// Differential evolution: global minimum in a box, deterministic for a given seed
let himmelblau = (&x.powi(2) + &y - 11).powi(2) + (&x + &y.powi(2) - 7).powi(2);
himmelblau.minimize_global_numeric(&[&x, &y], &[(-5.0, 5.0), (-5.0, 5.0)], &DeOpts::default()).unwrap();   // fun < 1e-8

// Brent scalar minimisation, and least-squares fits (f64 via Householder QR, or exact rational)
(&x * x.ln()).minimize_scalar_numeric(&x, 0.1, 2.0).unwrap();            // (0.36787944…, −0.36787944…) = (1/e, −1/e)
poly_fit(&[0.0, 1.0, 2.0, 3.0], &[1.0, 3.0, 9.0, 19.0], 2).unwrap();     // ≈ [1, 0, 2]   (ascending: 1 + 2x²)
let pts = [(ctx.int(0), ctx.int(1)), (ctx.int(1), ctx.int(0)), (ctx.int(2), ctx.int(4)), (ctx.int(3), ctx.int(2))];
Ex::poly_fit_points(&ctx, &pts, &x, 1).unwrap();                          // 7/10*x + 7/10   (exact least-squares line)

Also: bisect, newton_root, golden_section, minimize_scalar, poly_fit_exact, linear_fit, trapezoid, eval_poly; RootOpts/MinimizeOpts/DeOpts for tolerances, budgets and seeds.

Code Generation: Rust, C99 and Compiled Closures

Symbolic expressions compile to optimized Rust or C functions with common subexpression elimination, mul_add/fma, integer powers as multiplications, optional domain assertions, and a self-contained special-function runtime.

let ctx = Context::new();
syms!(ctx; x, y);
let f = &x.sin().powi(2) + &(&x * 2 + &y).exp() * 3;

f.to_rust_fn("f", &["x", "y"]).unwrap();
// pub fn f(x: f64, y: f64) -> f64 { 3_f64.mul_add(2_f64.mul_add(x, y).exp(), x.sin().powi(2)) }

f.to_c_fn("f", &["x", "y"]).unwrap();
// #include <math.h>
// double f(double x, double y) { return fma(3.0, exp(fma(2.0, x, y)), pow(sin(x), 2.0)); }

// Special functions embed only the helpers they need (Rust: `mod symplex_rt`; C: `static inline`)
x.lambertw().to_c_fn("w0", &["x"]).unwrap();          // contains symplex_lambert_w0

let opts = CodegenOptions { precision: Precision::F32, checked_domain: true, ..Default::default() };
x.ln().to_c_fn_with_options("g", &["x"], &opts).unwrap();   // float g(float x) { return assert(x > 0.0f), logf(x); }

// Compiled closures: Result, arity-checked, Send + Sync; gradients share one CSE pass
let cf = f.compile(&["x", "y"]).unwrap();
cf(&[0.5, 0.25]);
let grad = Ex::compile_many(&[&f.diff(&x), &f.diff(&y)], &["x", "y"]).unwrap();
grad.call_vec(&[0.5, 0.25]);

f.to_latex();                                          // \sin^{2}\left(x\right) + 3\exp\left(2x + y\right)

// 0.10: more targets and interchange formats
(x.sin().powi(2) + x.exp()).to_python().unwrap();      // math.sin(x)**2 + math.exp(x)   (executable; tested against eval_f64)
x.sin().to_numpy().unwrap();                           // numpy.sin(x)
(x.exp() + x.sin().powi(2)).to_julia().unwrap();       // sin(x)^2 + exp(x)
(&x.powi(2) + 1).to_mathml().unwrap();                 // <math xmlns=…><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></math>
(2 * &x + 1).to_srepr();                               // Add(Integer(1), Mul(Integer(2), Symbol('x')))
f.to_dot();                                            // Graphviz digraph of the expression tree

// and the other direction: relations, Boolean connectives, implicit multiplication
ctx.parse_bool("x > 0 and x < 1").unwrap().to_lean().unwrap();   // 0 < x ∧ x < 1
ctx.parse_implicit("2x + 3(x - 1)").unwrap();          // 5*x - 3

For build.rs pipelines and no_std targets see symplex-build; for the browser see symplex-wasm.

Probability and Statistics

symplex::stats (0.11) is the counterpart of sympy.stats: a RandomVariable is a symbol with a Distribution, and mean, variance, moments, probabilities of events, density, CDF, moment generating function, quantile and entropy are computed exactly — closed-form moments for polynomial expectations, the family's CDF for probabilities, exact integration or summation over the support otherwise. Fourteen continuous families (Normal, Uniform, Exponential, Gamma, χ², Beta, Cauchy, Laplace, Logistic, LogNormal, Student t, Weibull, Pareto, Triangular), nine discrete ones (Bernoulli, Binomial, Poisson, Geometric, Negative Binomial, Hypergeometric, DiscreteUniform, Die, and explicit finite tables), independence algebra over several variables, and seeded sampling.

use symplex::stats::{self, Distribution, RandomVariable, Rng};
let ctx = Context::new();
let x = RandomVariable::new(&ctx, "X", Distribution::normal(ctx.int(0), ctx.int(1)));
let y = RandomVariable::new(&ctx, "Y", Distribution::exponential(ctx.int(3)));
let b = RandomVariable::new(&ctx, "B", Distribution::binomial(ctx.int(5), ctx.rational(1, 3)));

x.expectation(&(x.symbol().powi(2) + 3 * x.symbol()));   // 1
y.probability(&y.symbol().gt(&ctx.int(1))).unwrap();     // exp(-3)
b.probability(&b.symbol().gt(&ctx.int(2))).unwrap();     // 17/81
(y.skewness(), b.kurtosis());                            // (2, 27/10)
x.cdf(&ctx.symbol("t"));                                 // 1/2*erf(1/2*t*sqrt(2)) + 1/2
y.quantile(&ctx.symbol("p")).unwrap();                   // -1/3*ln(-p + 1)
stats::conditional_expectation(&x, x.symbol(), &x.symbol().gt(&ctx.int(0))).unwrap();   // sqrt(2)*pi^(-1/2)
stats::covariance(&[&x], x.symbol(), &(2 * x.symbol())).unwrap();                        // 2
let z = RandomVariable::new(&ctx, "Z", Distribution::normal(ctx.int(1), ctx.int(2)));
stats::sum_distribution(&x, &z).unwrap();                // Normal(1, sqrt(5))
x.entropy();                                             // 1/2*ln(2*pi*E)
let coin = Distribution::try_finite(&ctx, vec![(ctx.int(1), ctx.rational(2, 3)), (ctx.int(0), ctx.rational(1, 3))]).unwrap();
RandomVariable::new(&ctx, "C", coin).variance();         // 2/9
y.sample(20_000, &mut Rng::new(1)).unwrap();             // reproducible f64 samples; mean ≈ 0.33

Also: std, moment(n), central_moment, median, mgf, characteristic_function, density, support; stats::{expectation, variance, correlation, probability} over several independent variables (rectangles and X < Y), conditional_probability; every constructor has a try_ twin validating numeric parameters. All test values come from SymPy 1.14.

Compile-Time Dimensional Analysis

Physical quantity types are checked at compile time. Adding a Mass to a Length is a compiler error. Differentiation respects dimensions: d(Length)/d(Time) produces Velocity.

use symplex::units::*;

let ctx = Context::new();
let m = Mass::symbol(&ctx, "m");
let a = Acceleration::symbol(&ctx, "a");

// dim! macro: the `: Force` annotation is a compile-time assertion
let force = dim!(ctx, Force: m * a);               // F = m·a [N]

// Typed calculus: d(Length)/d(Time) → Velocity
syms!(ctx; g, t);                                  // raw symbols for expr!
let t_var = Time::symbol(&ctx, "t");
let position = Length::from_ex(expr!(ctx, 1/2 * g * t^2));
let velocity: Velocity = position.diff_wrt(&t_var);   // g·t [m/s]

// 30 named quantity types, ~100 unit conversions (all exact rationals)
// Mass + Length → compile error

---

Design Principles

1. Exact by default. Every number is Ratio<BigInt>. No floating-point contamination. 0.1 + 0.2 == 3/10, not 0.30000000000000004. Floats only appear on explicit eval_f64(), compile(), or integrate_numeric(). Context::from_f64 converts a float to its exact dyadic rational; from_f64_approx to the nearest bounded-denominator rational.

2. Explicit contexts. Every expression belongs to a Context. No hidden global state. Mixing expressions from different contexts is caught immediately (compiler-enforced private field + runtime guard).

3. Type-safe expressions. Ex (numeric), BoolEx (boolean), SetEx (set-valued) are distinct types. sin(bool_expr) is a compile error.

4. Thread-safe. Context is Clone (Arc-based), Ex is Send + Sync. Multiple threads can share a context safely.

5. No recursion. All tree traversals use explicit stacks. Deep expressions don't blow the call stack.

6. Never silently wrong. Numerical evaluation returns Result. Operations that can't produce a closed form return unevaluated symbolic nodes — ∫x^x dx returns Integral(x^x, x), not garbage. ∫₋₁¹ dx/x² is Err(Divergent), not −2. re(z) stays re(z) unless z is known to be real. A certificate prover says Unknown with what it tried, never a wrong Proved.

7. No panics in library code. Failure is a Result, absence an Option, invariants debug_assert!; the two documented exceptions (mixing contexts, an empty Sum of Ex) are logic errors like indexing out of bounds. A ratchet test over src/ fails the build if a panicking construct is added — see CONTRIBUTING.md for the policy and why error plumbing costs nothing measurable.

8. One crate, no knobs. There are no Cargo features to combine; every capability is always present. Compile time is not the constraint, capability is.

---

The API Model

Every symbolic operation that might not produce a closed-form result has two entry points:

Intent	Method	Returns	When to use
Give me math	integrate(&x)	Ex (always — may contain Integral nodes)	Interactive exploration, chaining
Fail if you can't	try_integrate(&x)	Result<Ex>	Pipelines, codegen, safety-critical

try_ twins exist for diff, integrate, integrate_definite, limit, limit_left/right/dir, series, series_at_infinity, summation, product_over, laplace, inverse_laplace, residue, gosper_sum, solve_ode, solve_gt/ge/lt/le, char_poly, wronskian, and every stats::Distribution constructor. Check any expression for unevaluated forms:

let anti = hard_expr.integrate(&x);
if anti.has_unevaluated() {
    println!("integration produced formal result: {anti}");
}

(RootOf and RootSum are not unevaluated: they are complete algebraic answers.)

Operations that always succeed (simplify, expand, eval, factor, subs, rewrite) return Ex with no try_ variant — "unchanged" is a valid answer.

Result boundaries. Crossing from symbols to numbers (eval_f64, eval_decimal, compile, to_rust_fn, to_c_fn, integrate_numeric) always returns Result. So do operations with structural preconditions (Matrix::inv, cholesky, lu, minor, matmul) and solvers whose failure is a mathematical fact: solve returns Err(InfiniteSolutions) for identities and Err(NoSolution) for contradictions, try_integrate_definite returns Err(Divergent), laplace_final_value returns Err(Divergent) for unstable poles. Transform APIs without an unevaluated node (fourier_transform, mellin_transform, z_transform) are Result-only.

Three-valued queries. is_positive, equals, is_convergent, SetEx::contains, is_subset, Matrix::is_symmetric, is_diagonalizable, is_positive_definite, BoolEx::is_tautology, vector::is_conservative, is_increasing, is_convex … return Option<bool>: yes, no, or unknown. degree, resultant, discriminant, hypergeometric_ratio, minimal_polynomial, periodicity return Option<T>.

Certificate outcomes. The four inequality provers return Outcome<C, U>: Proved(C) (re-verified), Refuted { point, value, .. } (an exact point where the goal is negative) or Unknown(U) (what was tried, including a budget that ran out). is_proved(), certificate(), refutation(), unknown() and map_certificate are shared; Refuted and the Unknown payloads are #[non_exhaustive].

Budgets. Long-running exact algorithms accept a deadline and/or pivot cap (linprog::Budget, PolyhedronOpts::with_time_limit, SosOpts::with_time_limit); running out is a status, not an error.

---

Comparison with SymPy

Feature	symplex 0.11	SymPy 1.14
Arithmetic	Exact Ratio<BigInt>	Exact (similar)
Differentiation	Complete, incl. Bessel/Airy/orthogonal/polygamma/erf family	Complete
Indefinite integration	15+ strategies incl. Risch + LRT log-to-real; results in erf/erfi/Si/Shi/Ei/…	Risch + heurisch + Meijer G (broader)
Definite / improper integration	Singularity detection, ~30-entry improper table, divergence reported as Err	Meijer G-based; much broader table
Numeric integration	Adaptive G7/K15 quadrature	via mpmath (more algorithms)
Summation	Faulhaber, Gosper, telescoping, binomial, p-series, power-series recognition	+ Zeilberger, hypergeometric closed forms (broader)
Polynomial solving	Through quartic + RootOf; real_roots as ordered RootOfs	Through quartic + CRootOf
General solutions	solve_general (periodic families)	solveset with ImageSet
Linear systems	linsolve (unique / parametric / inconsistent, symbolic)	linsolve (similar)
Polynomial systems	Gröbner + FGLM, algebraic solutions; Ex::groebner/reduce_modulo	Gröbner, more strategies
Polynomial algebra	Poly with symbolic coefficients, MultiPoly over ℚ, resultants/discriminants (also with symbolic coefficients), gcd_all, factor_mod over GF(p), minimal_polynomial	Poly with domains, algebraic extensions, primitive_element (broader)
Function analysis	singularities, stationary_points, maximum/minimum, monotonicity/convexity (exact via Sturm), periodicity, function_range	calculus.util (similar)
Rational simplification	ratsimp/cancel: heuristic multivariate GCD	cancel, ratsimp, together, apart (similar)
Linear programming	Exact simplex on hybrid i64/i128/256-bit/BigInt cells; duals, Farkas certificates, budgets	sympy.solvers.simplex (exact; optimum and argmin only)
Polynomial inequalities	Handelman (boxes), Pólya (half-lines), parametric polyhedra with a goal multiplier, sums of squares (built-in SDP + exact rounding); all re-verified exactly; Lean export	—
Integer normal forms	Row/column HNF with transform, SNF with transforms, integer nullspace, lattice index, exact LLL	hermite_normal_form, smith_normal_form (no transforms), lll
Polytopes	Exact vertices (integer arithmetic, tight sets), clip, volume in any dimension, parametric families	—
Numerical optimisation	Brent, bisection, Newton, Nelder–Mead, differential evolution, QR least squares, exact rational fits	Defers to SciPy / mpmath; far broader via SciPy
Series expansion	Taylor / Laurent / at ∞ / formal power series with general terms	+ O() notation, Puiseux
Limits	Gruntz with work budget, one-sided	Gruntz (more mature)
Simplification	Multi-strategy fixpoint + public rule engine with AC matching, tracing	More strategies; replace/Wild patterns
Factoring	Berlekamp–Zassenhaus (any degree), multivariate via Kronecker, GF(p)	Zassenhaus + Wang (faster multivariate), algebraic extensions
Matrices	Eigen/Jordan/exp/log/sqrt/pow, QR, Cholesky, LDL, LU, Hessenberg, singular values, condition number, rank-deficient pseudo-inverse, permanent, RootOf eigenvalues; exact QMatrix/ZMatrix (Bareiss)	SVD, Schur, sparse, matrix expressions (broader)
Statistics	23 distribution families + finite tables; exact moments, probabilities, CDF/MGF/quantile/entropy; independence algebra, conditional expectation, sum closures; seeded sampling	sympy.stats (broader: joint/compound/stochastic processes, matrix distributions)
ODE solving	16 classes, IVPs, systems	More classes, hints, series solutions
Recurrences	Linear constant-coefficient, first-order	rsolve (poly/rational/hyper)
Transforms	Laplace, Fourier (3 conventions), Mellin (with strip), Z, Fourier series; discrete: convolutions, NTT, Walsh–Hadamard, Möbius	Broader tables, Hankel, cosine/sine; discrete (+ float FFT)
Sets & logic	Interval algebra, three-valued queries, NNF/CNF/DNF, DPLL; parse_bool	Richer set types (ImageSet, ConditionSet), satisfiable models
Number theory	rho/ECM, BPSW, sqrt_mod, nthroot_mod (any modulus), polynomial_congruence, dlog, CRT, Pell, two/four squares, continued fractions, Carmichael/amicable	Broader (quadratic forms, general Diophantine)
Combinatorics	Stirling, Bell, partitions, derangements, multinomial	Broader (permutation groups, etc.)
Special functions	Γ family (incl. incomplete), ψ⁽ⁿ⁾, erf/erfi/erfinv, B, W, Bessel, Airy, elliptic K/E/F/Π, expint, Si/Ci/Shi/Chi/Ei/li, Fresnel, polylog/η/ζ, classical + associated orthogonal polynomials — all arbitrary precision	Many more (hypergeometric, Meijer G, Mathieu, …)
Algebraic numbers	ℚ(α) field with exact zero/sign testing; minimal_polynomial	AlgebraicNumber + ANP, primitive_element
Code generation	Rust, C99, Python, NumPy, Julia with CSE and an embedded special-function runtime (Rust/C); compiled closures	Python / C / Fortran / Rust / Julia / Octave / JS via codegen
Interchange	LaTeX, Unicode pretty-print, Presentation MathML, srepr, Graphviz DOT, JSON tree, Lean 4	LaTeX, MathML, srepr, dotprint, pretty
Parsing	Numeric expressions, relations and Boolean connectives, implicit multiplication	sympify/parse_expr (+ LaTeX, Mathematica)
Dimensional analysis	Compile-time type checking	Runtime physics.units
Thread safety	Send + Sync, no GIL	GIL-bound
Expression type safety	Ex / BoolEx / SetEx at compile time	Runtime only
Failure model	No panics (ratchet-enforced); Result/Option/unevaluated forms; budgets are statuses	Exceptions
Language	Rust (compiled, ~206K lines, 91 node types)	Python (interpreted)

Where SymPy is stronger: geometry, tensor algebra, quantum mechanics, general Diophantine equations, PDE solving, hypergeometric/Meijer-G machinery, stochastic processes and joint distributions beyond independence, and 30 years of community contributions and testing.

Where symplex is different: exact certificates as first-class results (LP duals and Farkas vectors, Handelman/Pólya/SOS proofs re-verified with exact arithmetic, Sturm-verified polynomial signs, unimodular HNF/SNF transforms) and their export to Lean 4 / Mathlib as theorems that compile; compile-time dimensional analysis; thread safety; Rust and C code generation with an embedded runtime; algebraic number field arithmetic with exact zero/sign testing; a library verified not to panic. Operations that can't complete return honest unevaluated forms, Err, or Unknown rather than guessing.

---

Migrating

Every breaking change has a one-line fix in the book:

From → to	Page	The gist
0.6 → 0.7	migrating-0.7	one Outcome<C, U> for every prover (patterns need ..; Unknown(u)), the Handelman struct is BoxCertificate and Certificate is the trait, LeanOpts/PolyhedronOpts/SosOpts are #[non_exhaustive] (builders instead of literals), control/robotics/dynamics shape failures return Result
0.9 → 0.10	CHANGELOG	LpStatus::BudgetExhausted, Tactic::Apply, Decl.preamble (use Decl::new + builders)
0.3 → 0.4	migrating-0.4	roots_count_real removed, LeanOpts gained a field
0.1 → 0.2	migrating-0.2	Result boundaries, three-valued queries, integrate_definite, eigen family without dummy variables

0.2 → 0.3, 0.4 → 0.6, 0.7 → 0.9 and 0.10 → 0.11 were additive (cargo semver-checks clean).

---

Examples

Every example is self-contained and runs in a few seconds; CI runs all of them.

Getting started:

cargo run --example quickstart              # Tour of core operations
cargo run --example repl                    # Interactive expression evaluation
cargo run --example readme_snippets         # Every code block in this README, executed

Certificates, Lean and exact geometry (0.3–0.10):

cargo run --example certificates_to_lean    # Handelman / half-line / SOS certificates exported as Mathlib theorems (writes a .lean file)
cargo run --example polyhedron_certificates # Parametric polyhedra: λ(j) goal multiplier, staged exact LP, emptiness, lean_steps
cargo run --example exact_matrices          # QMatrix / ZMatrix: fraction-free elimination, rank, nullspace, Smith form
cargo run --example polynomials             # Poly views with symbolic coefficients, ratsimp, linear certificates, exact sign on an interval
cargo run --example exact_lp                # Exact simplex: optima, shadow prices, Farkas certificates, feasible_nonneg, linprog_matrix
cargo run --example integer_lattices        # Row/column HNF with transforms, Smith normal form, integer nullspace, unimodularity, lattice index
cargo run --example numeric_optimization    # Brent/Newton roots, Nelder–Mead, differential evolution, polynomial fits (f64 and exact)

New in 0.2:

cargo run --example definite_integration    # Improper integrals, divergence detection, quadrature, residues
cargo run --example summation_and_series    # Σ/Π closed forms, convergence, formal power series, finite differences
cargo run --example complex_analysis        # re/im/conjugate/arg, new constants, Si/Ci/Ei/ζ/polygamma
cargo run --example rule_engine             # Custom rewrite rules, tracing, subs_algebraic, targeted simplifiers
cargo run --example linear_systems_and_ivp  # solve semantics, linsolve, solve_general, ODE IVPs, rsolve
cargo run --example sets_and_logic          # Interval algebra, reduce_inequalities, CNF/DNF, tautology
cargo run --example factoring_and_ntheory   # Zassenhaus factoring, factorint, sqrt_mod, Pell, continued fractions
cargo run --example transforms              # Fourier, Mellin, Laplace, Fourier series, Z, one-sided limits
cargo run --example matrix_decompositions   # QR, Cholesky, LDL, Jordan, matrix_exp_t, RootOf eigenvalues
cargo run --example c_codegen               # C99 backend, embedded runtime, compile_many (compiles the C if cc exists)

Engineering workflows:

cargo run --example pid_controller          # PID design → stability → Rust codegen
cargo run --example robotics_codegen        # DH parameters → Jacobian → optimized Rust
cargo run --example control_system          # State-space, transfer functions, pole placement, ZOH
cargo run --example signal_filter           # Bilinear transform → digital filter → codegen
cargo run --example dynamics                # Lagrangian mechanics, equations of motion
cargo run --example inverse_kinematics      # 2-DOF IK via Gröbner bases

Mathematics and science:

cargo run --example calculus                # Differentiation, integration, limits, series
cargo run --example equation_solving        # Polynomial, transcendental, system solving
cargo run --example matrix_algebra          # Eigenvalues, Jordan form, codegen
cargo run --example ode_solving             # ODE classification and solving
cargo run --example combinatorics_counting  # Stirling numbers, partitions, multinomials
cargo run --example optimization            # Gradient, Hessian, critical points
cargo run --example complex_numbers         # Euler's formula, complex roots
cargo run --example laplace_transforms      # Forward, inverse, z-transforms

Applied problems:

cargo run --example gradient_descent        # Symbolic gradient → compiled optimization loop
cargo run --example crypto_rsa              # RSA with number theory primitives
cargo run --example number_theory           # Primality, factorization, CRT

Dimensional analysis:

cargo run --example units_physics           # Compile-time unit checking
cargo run --example units_electrical        # Circuit analysis with units
cargo run --example units_engineering       # Motor design, imperial conversions
cargo run --example units_kinematics        # Kinematics with typed quantities
cargo run --example units_lagrangian        # Lagrangian mechanics with units

Output:

cargo run --example latex_output            # LaTeX rendering
cargo run --example physics_constants       # Physical constants (symbolic + exact)

---

Dependencies

All MIT or Apache-2.0 licensed. No C bindings. No LGPL.

Core: num-bigint, num-rational, num-traits, num-integer, smallvec, rustc-hash, bitflags, parking_lot, thiserror, astro-float, serde, serde_json, tracing, typenum.

Proc macros: syn, quote, proc-macro2.

Companion crates: symplex-build (build-time codegen for no_std firmware), symplex-wasm (browser bindings).

Requirements

Rust 1.93+ (Edition 2024). No Cargo features by design; pure Rust on every platform Rust targets, including wasm32-unknown-unknown. ~11,400 tests (cargo nextest run), ~800 doctests; every emitted Lean shape is pinned to text compiled against Mathlib (Lean 4.30).

License

Dual-licensed under MIT and Apache 2.0.