Enum Expression 

pub enum Expression {
    Number(Number),
    Symbol(Symbol),
    Add(Arc<Vec<Expression>>),
    Mul(Arc<Vec<Expression>>),
    Pow(Arc<Expression>, Arc<Expression>),
    Function {
        name: Arc<str>,
        args: Arc<Vec<Expression>>,
    },
    Constant(MathConstant),
    Set(Arc<Vec<Expression>>),
    Complex(Arc<ComplexData>),
    Matrix(Arc<Matrix>),
    Relation(Arc<RelationData>),
    Piecewise(Arc<PiecewiseData>),
    Interval(Arc<IntervalData>),
    Calculus(Arc<CalculusData>),
    MethodCall(Arc<MethodCallData>),
}

Memory-optimized Expression enum (target: 32 bytes)

Uses Arc for O(1) clone performance - cloning is just an atomic increment. Hot-path variants (frequently used) are kept inline for performance. Cold-path variants (less common) use Arc to maintain small enum size.

Variants

Number(Number)

Symbol(Symbol)

Add(Arc<Vec<Expression>>)

Mul(Arc<Vec<Expression>>)

Pow(Arc<Expression>, Arc<Expression>)

Function

Fields

name: Arc<str>

args: Arc<Vec<Expression>>

Constant(MathConstant)

Set(Arc<Vec<Expression>>)

Complex(Arc<ComplexData>)

Matrix(Arc<Matrix>)

Relation(Arc<RelationData>)

Piecewise(Arc<PiecewiseData>)

Interval(Arc<IntervalData>)

Calculus(Arc<CalculusData>)

MethodCall(Arc<MethodCallData>)

Implementations

impl Expression

pub fn factorial(arg: Expression) -> Expression

Create factorial expression

pub fn ln(arg: Expression) -> Expression

Create natural logarithm expression

impl Expression

pub fn is_constant(&self) -> bool

Check if an expression is constant (contains no variables)

impl Expression

pub fn separate_constants(&self) -> (Expression, Expression)

Separate variables and constants

impl Expression

pub fn real(&self) -> Expression

Extract the real part of a complex number

Returns the real component of a complex expression. For non-complex expressions, returns the expression itself.

Examples

use mathhook_core::{Expression, expr};

let z = Expression::complex(expr!(3), expr!(4));
let real_part = z.real();
assert_eq!(real_part, expr!(3));

pub fn imag(&self) -> Expression

Extract the imaginary part of a complex number

Returns the imaginary component of a complex expression. For non-complex expressions, returns zero.

Examples

use mathhook_core::{Expression, expr};

let z = Expression::complex(expr!(3), expr!(4));
let imag_part = z.imag();
assert_eq!(imag_part, expr!(4));

pub fn conjugate(&self) -> Expression

Compute the complex conjugate

Returns the complex conjugate (a + bi → a - bi). For non-complex expressions, returns the expression itself.

Examples

use mathhook_core::{Expression, expr};

let z = Expression::complex(expr!(3), expr!(4));
let conjugate = z.conjugate();
if let Expression::Complex(data) = conjugate {
    assert_eq!(data.real, expr!(3));
    assert_eq!(data.imag, expr!(-4));
}

pub fn abs(&self) -> Expression

Compute the absolute value (modulus) of a complex number

Returns |z| = √(re² + im²). For complex numbers, this is the magnitude. For real numbers, this is the absolute value.

Examples

use mathhook_core::{Expression, expr};

let z = Expression::complex(expr!(3), expr!(4));
let magnitude = z.abs();

pub fn arg(&self) -> Expression

Compute the argument (phase angle) of a complex number

Returns the angle θ = atan2(im, re) in radians, in the range (-π, π]. This is the principal value of the argument.

Examples

use mathhook_core::{Expression, expr};

let z = Expression::complex(expr!(1), expr!(1));
let angle = z.arg();

pub fn to_polar(&self) -> (Expression, Expression)

Convert to polar form (magnitude, angle)

Returns (r, θ) where z = r·e^(iθ). The angle is in radians, in the range (-π, π].

Examples

use mathhook_core::{Expression, expr};

let z = Expression::complex(expr!(3), expr!(4));
let (magnitude, angle) = z.to_polar();

pub fn from_polar(magnitude: Expression, angle: Expression) -> Expression

Create a complex number from polar form

Converts polar coordinates (magnitude, angle) to rectangular form (a + bi). The angle should be in radians.

Arguments

• magnitude - The magnitude (r) of the complex number
• angle - The angle (θ) in radians

Examples

use mathhook_core::{Expression, expr};

let magnitude = expr!(5);
let angle = Expression::pi();
let z = Expression::from_polar(magnitude, angle);

pub fn from_polar_form(magnitude: Expression, angle: Expression) -> Expression

Create a complex number from polar form

Converts polar coordinates (magnitude, angle) to rectangular form (a + bi).

Examples

use mathhook_core::{Expression, expr};

let magnitude = expr!(5);
let angle = Expression::pi();
let z = Expression::from_polar_form(magnitude, angle);

pub fn simplify_complex(expr: &Expression) -> Expression

Simplify complex expressions by removing zero parts

Converts complex numbers to their simplest form by removing zero real or imaginary components.

Examples

use mathhook_core::{Expression, expr};
use mathhook_core::simplify::Simplify;

let z = Expression::complex(expr!(3), expr!(0));
let simplified = z.simplify();

impl Expression

pub fn expand_binomial( &self, a: &Expression, b: &Expression, n: u32, ) -> Expression

Expand binomial expressions: (a + b)^n using binomial theorem

For commutative terms, uses binomial theorem: C(n,k) * a^k * b^(n-k) For noncommutative terms, uses direct multiplication to preserve order

impl Expression

pub fn factor_numeric_coefficient(&self) -> (BigInt, Expression)

Factor out numeric coefficients

impl Expression

pub fn factor_perfect_square(&self, terms: &[Expression]) -> Option<Expression>

Factor perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2

pub fn factor_difference_of_squares( &self, a: &Expression, b: &Expression, ) -> Expression

Factor difference of squares: a^2 - b^2 = (a + b)(a - b)

impl Expression

pub fn add_rationals(&self, other: &Expression) -> Expression

Add rational expressions: a/b + c/d = (ad + bc)/(bd)

pub fn multiply_rationals(&self, other: &Expression) -> Expression

Multiply rational expressions: (a/b) * (c/d) = (ac)/(bd)

pub fn simplify_complex_rational(&self) -> Expression

Simplify complex rational expressions

pub fn extract_rational_coefficient(&self) -> (BigRational, Expression)

Extract rational coefficient from expression

pub fn extract_rational_parts(&self) -> (String, String, Expression)

Extract rational coefficient as string parts (FFI-friendly, no BigRational)

Returns (numerator, denominator, remainder) where numerator/denominator is the rational coefficient and remainder is the non-rational part.

impl Expression

pub fn is_valid_expression(&self) -> bool

Check if expression is a valid mathematical expression

pub fn solver_to_latex(&self) -> String

Convert to LaTeX representation for solvers (avoid conflict)

pub fn flatten_add_terms(&self) -> Vec<Expression>

pub fn negate(&self) -> Expression

Negate an expression

impl Expression

pub fn detect_advanced_zero_patterns(&self) -> bool

Advanced zero detection for specific algebraic patterns

impl Expression

pub fn number<T: Into<Number>>(value: T) -> Self

Create a number expression

Examples

use mathhook_core::{Expression, Number};

let expr = Expression::number(42);
let expr = Expression::number(3.14);

pub fn integer(value: i64) -> Self

Create an integer expression

Examples

use mathhook_core::Expression;

let expr = Expression::integer(42);

pub fn big_integer(value: BigInt) -> Self

Create an integer expression from BigInt

Examples

use mathhook_core::Expression;
use num_bigint::BigInt;

let big_val = BigInt::from(42);
let expr = Expression::big_integer(big_val);

pub fn rational(numerator: i64, denominator: i64) -> Self

Create a rational number expression

Examples

use mathhook_core::Expression;

let expr = Expression::rational(3, 4); // 3/4
let expr = Expression::rational(-1, 2); // -1/2

pub fn float(value: f64) -> Self

Create a float expression

Examples

use mathhook_core::Expression;

let expr = Expression::float(3.14159);

pub fn symbol<T: Into<Symbol>>(symbol: T) -> Self

Create a symbol expression

Examples

use mathhook_core::{symbol, Expression};

let expr = Expression::symbol(symbol!(x));

pub fn add(terms: Vec<Expression>) -> Self

Create an addition expression in canonical form

This constructor automatically produces a canonical form expression by:

• Flattening nested additions: (a + b) + c → a + b + c
• Removing identity elements: x + 0 → x
• Combining like terms: 2x + 3x → 5x
• Sorting terms in canonical order: y + x → x + y
• Evaluating constant subexpressions: 2 + 3 → 5

Examples

use mathhook_core::{Expression, expr};

// Constant folding
let expression = Expression::add(vec![
    Expression::integer(1),
    Expression::integer(2),
]);
assert_eq!(expression, Expression::integer(3));

// Identity element removal
let x = expr!(x);
let expression = Expression::add(vec![x.clone(), Expression::integer(0)]);
assert_eq!(expression, x);

// Commutativity (canonical ordering)
let y = expr!(y);
let expr1 = Expression::add(vec![x.clone(), y.clone()]);
let expr2 = Expression::add(vec![y.clone(), x.clone()]);
assert_eq!(expr1, expr2); // Both produce x + y in canonical order

pub fn mul(factors: Vec<Expression>) -> Self

Create a multiplication expression in canonical form

This constructor automatically produces a canonical form expression by:

• Flattening nested multiplications: (a * b) * c → a * b * c
• Removing identity elements: x * 1 → x
• Handling zero: x * 0 → 0
• Sorting factors in canonical order: y * x → x * y
• Evaluating constant subexpressions: 2 * 3 → 6
• Converting division to multiplication: a / b → a * b^(-1)

Examples

use mathhook_core::{Expression, expr};

// Constant folding
let expression = Expression::mul(vec![
    Expression::integer(2),
    Expression::integer(3),
]);
assert_eq!(expression, Expression::integer(6));

// Identity element removal
let x = expr!(x);
let expr = Expression::mul(vec![x.clone(), Expression::integer(1)]);
assert_eq!(expr, x);

// Zero handling
let expression = Expression::mul(vec![x.clone(), Expression::integer(0)]);
assert_eq!(expression, Expression::integer(0));

// Commutativity (canonical ordering)
let y = expr!(y);
let expr1 = Expression::mul(vec![x.clone(), y.clone()]);
let expr2 = Expression::mul(vec![y.clone(), x.clone()]);
assert_eq!(expr1, expr2); // Both produce x * y in canonical order

pub fn pow(base: Expression, exponent: Expression) -> Self

Create a power expression in canonical form

This constructor automatically produces a canonical form expression by:

• Applying power identities: x^0 → 1, x^1 → x, 1^x → 1
• Evaluating constant powers: 2^3 → 8
• Converting negative exponents to rationals: x^(-1) → 1/x
• Flattening nested powers: (x^a)^b → x^(a*b)
• Handling special cases: 0^n → 0 for positive n

Examples

use mathhook_core::{Expression, expr};

// Power identities
let x = expr!(x);
let expression = Expression::pow(x.clone(), Expression::integer(1));
assert_eq!(expression, x);

let expression = Expression::pow(x.clone(), Expression::integer(0));
assert_eq!(expression, Expression::integer(1));

// Constant evaluation
let expression = expr!(2 ^ 3);
assert_eq!(expression, Expression::integer(8));

// Nested power flattening
let expression = Expression::pow(
    Expression::pow(x.clone(), Expression::integer(2)),
    Expression::integer(3),
);
// Produces x^6 in canonical form

pub fn constant(constant: MathConstant) -> Self

Create a constant expression

Examples

use mathhook_core::{Expression, core::MathConstant};

let expr = Expression::constant(MathConstant::Pi);

pub fn pi() -> Self

Create a pi constant expression

Examples

use mathhook_core::Expression;

let pi = Expression::pi();

pub fn e() -> Self

Create an e constant expression

Examples

use mathhook_core::Expression;

let e = Expression::e();

pub fn i() -> Self

Create an imaginary unit expression

Examples

use mathhook_core::Expression;

let i = Expression::i();

pub fn infinity() -> Self

Create an infinity expression

Examples

use mathhook_core::Expression;

let inf = Expression::infinity();

pub fn negative_infinity() -> Self

Create a negative infinity expression

Examples

use mathhook_core::Expression;

let neg_inf = Expression::negative_infinity();

pub fn undefined() -> Self

Create an undefined expression

Examples

use mathhook_core::Expression;

let undef = Expression::undefined();

pub fn golden_ratio() -> Self

Create a golden ratio (phi) expression

Examples

use mathhook_core::Expression;

let phi = Expression::golden_ratio();

pub fn euler_gamma() -> Self

Create an Euler-Mascheroni constant (gamma) expression

Examples

use mathhook_core::Expression;

let gamma = Expression::euler_gamma();

pub fn equation(left: Expression, right: Expression) -> Self

Create an equation (equality relation)

Examples

use mathhook_core::{Expression, expr};

let expr = Expression::equation(
    expr!(x),
    expr!(5),
);

pub fn relation( left: Expression, right: Expression, relation_type: RelationType, ) -> Self

Create a relation expression

Examples

use mathhook_core::{Expression, expr};
use mathhook_core::core::expression::RelationType;

let relation = Expression::relation(
    expr!(x),
    expr!(5),
    RelationType::Greater,
);

pub fn div(numerator: Expression, denominator: Expression) -> Self

Create a division expression (symbolic, always succeeds)

This constructor is for symbolic division where the denominator may be unknown or symbolic. It converts division to multiplication by the reciprocal: a / b → a * b^(-1)

For numerical evaluation contexts where you need to detect division by zero, use div_checked() instead.

Examples

use mathhook_core::{Expression, symbol};

// Symbolic division (denominator is unknown)
let x = symbol!(x);
let expr = Expression::div(Expression::integer(1), Expression::symbol(x));
// Produces: 1 * x^(-1)

// Constant division (still symbolic context)
let expr = Expression::div(Expression::integer(3), Expression::integer(4));
// Produces: 3 * 4^(-1), which simplifies to 3/4

pub fn div_checked( numerator: Expression, denominator: Expression, ) -> Result<Self, MathError>

Create a division expression with division-by-zero checking

This constructor checks if the denominator is zero and returns an error if so. Use this in evaluation contexts where division by zero should be detected.

For symbolic contexts where the denominator is unknown or symbolic, use div() instead.

Errors

Returns MathError::DivisionByZero if the denominator is exactly zero.

Examples

use mathhook_core::{Expression, MathError};

// Valid division
let result = Expression::div_checked(
    Expression::integer(10),
    Expression::integer(2),
);
assert!(result.is_ok());

// Division by zero
let result = Expression::div_checked(
    Expression::integer(1),
    Expression::integer(0),
);
assert!(matches!(result, Err(MathError::DivisionByZero)));

impl Expression

pub fn function<S: AsRef<str>>(name: S, args: Vec<Expression>) -> Self

Create a function expression

Examples

use mathhook_core::{Expression, expr};

let expression = Expression::function("sin", vec![expr!(x)]);

pub fn sqrt(arg: Expression) -> Self

Create a square root expression

Examples

use mathhook_core::Expression;

let sqrt_2 = Expression::sqrt(Expression::integer(2));

pub fn derivative(expression: Expression, variable: Symbol, order: u32) -> Self

Create a derivative expression

Examples

use mathhook_core::{Expression, symbol, expr};

let expr = Expression::derivative(
    Expression::pow(expr!(x), Expression::integer(2)),
    symbol!(x),
    1,
);

pub fn integral(integrand: Expression, variable: Symbol) -> Self

Create an integral expression

Examples

use mathhook_core::{Expression, symbol, expr};

let expr = Expression::integral(
    expr!(x),
    symbol!(x),
);

pub fn definite_integral( integrand: Expression, variable: Symbol, start: Expression, end: Expression, ) -> Self

Create a definite integral expression

Examples

use mathhook_core::{Expression, symbol, expr};

let expr = Expression::definite_integral(
    expr!(x),
    symbol!(x),
    Expression::integer(0),
    Expression::integer(1),
);

pub fn limit( expression: Expression, variable: Symbol, point: Expression, ) -> Self

Create a limit expression

Examples

use mathhook_core::{Expression, symbol, expr};

let expr = Expression::limit(
    expr!(x),
    symbol!(x),
    Expression::integer(0),
);

pub fn sum( expression: Expression, variable: Symbol, start: Expression, end: Expression, ) -> Self

Create a sum expression

Examples

use mathhook_core::{Expression, symbol, expr};

let expr = Expression::sum(
    expr!(i),
    symbol!(i),
    Expression::integer(1),
    Expression::integer(10),
);

pub fn product( expression: Expression, variable: Symbol, start: Expression, end: Expression, ) -> Self

Create a product expression

Examples

use mathhook_core::{Expression, symbol, expr};

let expr = Expression::product(
    expr!(i),
    symbol!(i),
    Expression::integer(1),
    Expression::integer(10),
);

impl Expression

pub fn complex(real: Expression, imag: Expression) -> Self

Create a complex number expression

Examples

use mathhook_core::Expression;

let expr = Expression::complex(
    Expression::integer(3),
    Expression::integer(4),
);

pub fn set(elements: Vec<Expression>) -> Self

Create a set expression

Examples

use mathhook_core::Expression;

let set = Expression::set(vec![
    Expression::integer(1),
    Expression::integer(2),
    Expression::integer(3),
]);

pub fn interval( start: Expression, end: Expression, start_inclusive: bool, end_inclusive: bool, ) -> Self

Create an interval expression

Examples

use mathhook_core::Expression;

let interval = Expression::interval(
    Expression::integer(0),
    Expression::integer(10),
    true,
    false,
);

pub fn piecewise( pieces: Vec<(Expression, Expression)>, default: Option<Expression>, ) -> Self

Create a piecewise function expression

Examples

use mathhook_core::{Expression, symbol};

let piecewise = Expression::piecewise(
    vec![(Expression::symbol(symbol!(x)), Expression::integer(1))],
    Some(Expression::integer(0)),
);

pub fn matrix(rows: Vec<Vec<Expression>>) -> Self

Create a matrix expression from rows

Examples

use mathhook_core::Expression;

let matrix = Expression::matrix(vec![
    vec![Expression::integer(1), Expression::integer(2)],
    vec![Expression::integer(3), Expression::integer(4)]
]);

pub fn identity_matrix(size: usize) -> Self

Create an identity matrix expression

Examples

use mathhook_core::Expression;
use mathhook_core::matrices::operations::MatrixOperations;

let identity = Expression::identity_matrix(3);
assert!(identity.is_identity_matrix());

pub fn method_call( object: Expression, method_name: impl Into<String>, args: Vec<Expression>, ) -> Self

Create a method call expression

Examples

use mathhook_core::{Expression, symbol};

let matrix = Expression::symbol(symbol!(A));
let det_call = Expression::method_call(matrix, "det", vec![]);
let trace_call = Expression::method_call(
    Expression::symbol(symbol!(B)),
    "trace",
    vec![]
);

pub fn zero_matrix(rows: usize, cols: usize) -> Self

Create a zero matrix expression

Examples

use mathhook_core::matrices::operations::MatrixOperations;
use mathhook_core::Expression;

let zero = Expression::zero_matrix(2, 3);
assert!(zero.is_zero_matrix());

pub fn diagonal_matrix(diagonal_elements: Vec<Expression>) -> Self

Create a diagonal matrix expression

Examples

use mathhook_core::Expression;

let diag = Expression::diagonal_matrix(vec![
    Expression::integer(1),
    Expression::integer(2),
    Expression::integer(3)
]);

pub fn scalar_matrix(size: usize, scalar_value: Expression) -> Self

Create a scalar matrix expression (c*I)

Examples

use mathhook_core::Expression;

let scalar = Expression::scalar_matrix(3, Expression::integer(5));
// Creates 5*I (5 times the 3x3 identity matrix)

pub fn matrix_from_arrays<const R: usize, const C: usize>( arrays: [[i64; C]; R], ) -> Self

Create matrix from nested arrays (convenience method)

Examples

use mathhook_core::Expression;

let matrix = Expression::matrix_from_arrays([
    [1, 2, 3],
    [4, 5, 6]
]);

pub fn commutator(a: Expression, b: Expression) -> Self

Create a commutator: [A, B] = AB - BA

The commutator measures the failure of two operators to commute. It is zero for commutative operators and nonzero for noncommutative.

Examples

use mathhook_core::{Expression, symbol};

let A = Expression::symbol(symbol!(A; matrix));
let B = Expression::symbol(symbol!(B; matrix));
let comm = Expression::commutator(A.clone(), B.clone());
// Represents: AB - BA

Mathematical Properties

• [A, B] = -[B, A] (antisymmetry)
• [A, A] = 0 (self-commutator is zero)
• Jacobi identity: [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

Quantum Mechanics Example

use mathhook_core::{Expression, symbol};

let x = Expression::symbol(symbol!(x; operator));
let p = Expression::symbol(symbol!(p; operator));
let comm = Expression::commutator(x, p);
// In quantum mechanics: [x, p] = ih (canonical commutation relation)

pub fn anticommutator(a: Expression, b: Expression) -> Self

Create an anticommutator: {A, B} = AB + BA

The anticommutator is the symmetric combination of two operators.

Examples

use mathhook_core::{Expression, symbol};

let A = Expression::symbol(symbol!(A; matrix));
let B = Expression::symbol(symbol!(B; matrix));
let anticomm = Expression::anticommutator(A.clone(), B.clone());
// Represents: AB + BA

Mathematical Properties

• {A, B} = {B, A} (symmetry)
• {A, A} = 2A^2 (self-anticommutator)

Physics Example

use mathhook_core::{Expression, symbol};

let sigma_x = Expression::symbol(symbol!(sigma_x; operator));
let sigma_y = Expression::symbol(symbol!(sigma_y; operator));
let anticomm = Expression::anticommutator(sigma_x, sigma_y);
// For Pauli matrices: {sigma_x, sigma_y} = 0

impl Expression

pub fn evaluate(&self) -> Result<Expression, MathError>

Evaluate expression with domain checking

Computes numerical values from expressions while validating mathematical domain constraints. Returns Result<Expression, MathError> to handle domain violations gracefully.

Evaluation vs Simplification

Use evaluate() when: You need numerical results with domain validation Use simplify() when: You need algebraic reduction without domain checking Use evaluate_with_context() when: You need variable substitution + computation

Domain Constraints Checked

• sqrt(x): Requires x >= 0 in real domain
• log(x): Requires x > 0 (pole at 0)
• tan(x): Has poles at π/2 + nπ
• arcsin(x), arccos(x): Require |x| <= 1 in real domain
• csc(x), sec(x), cot(x): Have poles where sin/cos/tan = 0
• Division by zero: Checked in x/y and x^(-n) for n > 0

Returns

• Ok(Expression): Evaluated result (numerical or symbolic if can’t evaluate)
• Err(MathError::DomainError): Domain constraint violated
• Err(MathError::DivisionByZero): Division by zero detected

Examples

Successful Evaluation

use mathhook_core::{Expression, MathError};

// Constants evaluate to numbers
let sum = Expression::add(vec![Expression::integer(2), Expression::integer(3)]);
assert_eq!(sum.evaluate().unwrap(), Expression::integer(5));

let product = Expression::mul(vec![Expression::integer(2), Expression::integer(3), Expression::integer(4)]);
assert_eq!(product.evaluate().unwrap(), Expression::integer(24));

// Special values
let sin_zero = Expression::function("sin".to_string(), vec![Expression::integer(0)]);
assert_eq!(sin_zero.evaluate().unwrap(), Expression::integer(0));

Domain Errors

use mathhook_core::{expr, Expression, MathError};

// sqrt requires non-negative input
let sqrt_neg = Expression::function("sqrt".to_string(), vec![Expression::integer(-1)]);
assert!(matches!(
    sqrt_neg.evaluate(),
    Err(MathError::Pole { .. })
));

// log has pole at 0
assert!(matches!(
    expr!(log(0)).evaluate(),
    Err(MathError::Pole { .. })
));

// Division by zero
assert!(matches!(
    expr!(1 / 0).evaluate(),
    Err(MathError::DivisionByZero)
));

Symbolic Results (No Variables to Substitute)

use mathhook_core::{expr, symbol};
use mathhook_core::simplify::Simplify;

let x = symbol!(x);

// Can't evaluate without variable value - returns simplified symbolic
let result = expr!(x + 1).evaluate().unwrap();
assert_eq!(result, expr!(x + 1).simplify());

// For variable substitution, use evaluate_with_context() instead

Handling Errors

use mathhook_core::{Expression, MathError};

let sqrt_neg = Expression::function("sqrt".to_string(), vec![Expression::integer(-1)]);
match sqrt_neg.evaluate() {
    Ok(result) => println!("Result: {}", result),
    Err(MathError::DomainError { operation, value, reason }) => {
        eprintln!("Domain error in {}: {} ({})", operation, value, reason);
    }
    Err(e) => eprintln!("Other error: {:?}", e),
}

pub fn evaluate_with_context( &self, context: &EvalContext, ) -> Result<Expression, MathError>

High-level evaluation with context

This is the PRIMARY user-facing evaluation method following SymPy’s two-level architecture. It handles:

1. Variable substitution
2. Optional symbolic simplification
3. Optional numerical evaluation

This mirrors SymPy’s evalf(subs={...}, ...) high-level API.

Arguments

• context - Evaluation context (variables, numeric mode, precision, etc.)

Returns

Evaluated expression

Errors

Returns MathError for domain violations, undefined operations, etc.

Examples

use mathhook_core::{expr, symbol};
use mathhook_core::core::expression::eval_numeric::EvalContext;
use std::collections::HashMap;

// Symbolic evaluation (no substitution)
let x = symbol!(x);
let f = expr!(x ^ 2);
let result = f.evaluate_with_context(&EvalContext::symbolic()).unwrap();
assert_eq!(result, expr!(x ^ 2)); // Unchanged

// Numerical evaluation with substitution
let mut vars = HashMap::new();
vars.insert("x".to_string(), expr!(3));
let ctx = EvalContext::numeric(vars);
let result = f.evaluate_with_context(&ctx).unwrap();
assert_eq!(result, expr!(9));

pub fn evaluate_to_f64(&self) -> Result<f64, MathError>

Convert evaluated expression to f64

First evaluates the expression, then attempts to convert the result to f64. Returns error if the result is non-numerical (symbolic).

Returns

f64 value if expression evaluates to a number

Errors

Returns MathError::NonNumericalResult if evaluation produces symbolic expression

Examples

use mathhook_core::{expr, symbol};

// Numerical expression
let e = expr!(2 + 3);
assert_eq!(e.evaluate_to_f64().unwrap(), 5.0);

// Symbolic expression fails
let x = symbol!(x);
assert!(x.evaluate_to_f64().is_err());

impl Expression

pub fn substitute( &self, substitutions: &HashMap<String, Expression>, ) -> Expression

Substitute variables with expressions

Recursively replaces all occurrences of symbols with provided expressions.

Arguments

• substitutions - Map from symbol name to replacement expression

Returns

New expression with substitutions applied

Examples

use mathhook_core::{expr, symbol};
use std::collections::HashMap;

let x = symbol!(x);
let y = symbol!(y);
let e = expr!(x + y);

let mut subs = HashMap::new();
subs.insert("x".to_string(), expr!(3));
subs.insert("y".to_string(), expr!(4));

let result = e.substitute(&subs);
assert_eq!(result, expr!(3 + 4));

pub fn substitute_and_simplify( &self, substitutions: &HashMap<String, Expression>, ) -> Expression

Substitute and simplify in one step

Convenience method that applies substitutions and then simplifies the result.

Arguments

• substitutions - Map from symbol name to replacement expression

Returns

New simplified expression with substitutions applied

impl Expression

pub fn transpose(&self) -> Expression

Compute transpose of matrix expression

For symbolic matrix expressions, this implements proper order reversal according to the mathematical rule: (AB)^T = B^T A^T

Mathematical Rules

• For products: (AB)^T = B^T A^T (order reverses)
• For sums: (A+B)^T = A^T + B^T (distributes)
• For scalars: scalar^T = scalar (no change)
• For matrix symbols: A^T creates transpose function

Examples

use mathhook_core::{Expression, symbol};

let a = symbol!(A; matrix);
let b = symbol!(B; matrix);

let product = Expression::mul(vec![
    Expression::symbol(a.clone()),
    Expression::symbol(b.clone()),
]);

let transposed = product.transpose();

pub fn inverse(&self) -> Expression

Compute inverse of matrix expression

For symbolic matrix expressions, this implements proper order reversal according to the mathematical rule: (AB)^(-1) = B^(-1) A^(-1)

Mathematical Rules

• For products: (AB)^(-1) = B^(-1) A^(-1) (order reverses)
• For matrix symbols: A^(-1) creates inverse function
• For identity: I^(-1) = I
• For scalars: a^(-1) = 1/a (reciprocal)

Examples

use mathhook_core::{Expression, symbol};

let a = symbol!(A; matrix);
let b = symbol!(B; matrix);

let product = Expression::mul(vec![
    Expression::symbol(a.clone()),
    Expression::symbol(b.clone()),
]);

let inverse = product.inverse();

impl Expression

pub fn commutativity(&self) -> Commutativity

Compute commutativity of this expression

Commutativity is inferred from the symbols and operations:

• Numbers, constants: Commutative
• Symbols: Depends on symbol type (Scalar -> Commutative, Matrix/Operator/Quaternion -> Noncommutative)
• Mul: Noncommutative if ANY factor is noncommutative
• Add, Pow, Function: Depends on subexpressions

Examples

Basic scalar symbols (commutative):

use mathhook_core::core::symbol::Symbol;
use mathhook_core::core::expression::Expression;
use mathhook_core::core::commutativity::Commutativity;

let x = Symbol::scalar("x");
let y = Symbol::scalar("y");
let expr = Expression::mul(vec![
    Expression::symbol(x.clone()),
    Expression::symbol(y.clone()),
]);
assert_eq!(expr.commutativity(), Commutativity::Commutative);

Matrix symbols (noncommutative):

use mathhook_core::core::symbol::Symbol;
use mathhook_core::core::expression::Expression;
use mathhook_core::core::commutativity::Commutativity;

let a = Symbol::matrix("A");
let b = Symbol::matrix("B");
let expr = Expression::mul(vec![
    Expression::symbol(a.clone()),
    Expression::symbol(b.clone()),
]);
assert_eq!(expr.commutativity(), Commutativity::Noncommutative);

pub fn count_variable_occurrences(&self, variable: &Symbol) -> usize

Count occurrences of a variable in the expression

Recursively counts how many times a specific variable symbol appears in the expression tree. This is useful for:

• Determining if an expression is polynomial in a variable
• Analyzing variable dependencies
• Checking if a variable appears in an equation

Arguments

• variable - The symbol to count occurrences of

Returns

The number of times the variable appears in the expression

Examples

Basic counting in simple expressions:

use mathhook_core::{Expression, symbol};

let x = symbol!(x);
let expr = Expression::mul(vec![
    Expression::integer(2),
    Expression::symbol(x.clone()),
]);
assert_eq!(expr.count_variable_occurrences(&x), 1);

Counting multiple occurrences:

use mathhook_core::{Expression, symbol};
use std::sync::Arc;

let x = symbol!(x);
// x^2 + 2*x + 1 has 2 occurrences of x (in x^2 and in 2*x)
let expr = Expression::Add(Arc::new(vec![
    Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
    Expression::mul(vec![Expression::integer(2), Expression::symbol(x.clone())]),
    Expression::integer(1),
]));
assert_eq!(expr.count_variable_occurrences(&x), 2);

Counting in power expressions:

use mathhook_core::{Expression, symbol};

let x = symbol!(x);
// x^x has 2 occurrences (base and exponent)
let expr = Expression::pow(
    Expression::symbol(x.clone()),
    Expression::symbol(x.clone())
);
assert_eq!(expr.count_variable_occurrences(&x), 2);

Counting in functions:

use mathhook_core::{Expression, symbol};

let x = symbol!(x);
// sin(x)
let expr = Expression::function("sin", vec![Expression::symbol(x.clone())]);
assert_eq!(expr.count_variable_occurrences(&x), 1);

// f(x, x, 2) has 2 occurrences
let expr2 = Expression::function("f", vec![
    Expression::symbol(x.clone()),
    Expression::symbol(x.clone()),
    Expression::integer(2),
]);
assert_eq!(expr2.count_variable_occurrences(&x), 2);

Zero occurrences when variable is not present:

use mathhook_core::{Expression, symbol};

let x = symbol!(x);
let y = symbol!(y);
let expr = Expression::symbol(y.clone());
assert_eq!(expr.count_variable_occurrences(&x), 0);

pub fn contains_variable(&self, symbol: &Symbol) -> bool

pub fn is_simple_variable(&self, var: &Symbol) -> bool

Check if expression is just the variable itself

pub fn is_symbol_matching(&self, symbol: &Symbol) -> bool

Check if this expression is a specific symbol

Convenience method for pattern matching against a specific symbol. More readable than inline matches! pattern in complex conditions.

Arguments

• symbol - The symbol to check against

Returns

True if this expression is exactly the given symbol

Examples

use mathhook_core::{Expression, symbol};

let x = symbol!(x);
let y = symbol!(y);
let expr = Expression::symbol(x.clone());

assert!(expr.is_symbol_matching(&x));
assert!(!expr.is_symbol_matching(&y));

pub fn as_pow(&self) -> Option<(&Expression, &Expression)>

Extract the base and exponent from a Pow expression

Returns Some((base, exp)) if this is a Pow expression, None otherwise. This is a helper method for pattern matching with the Arc-based structure.

pub fn as_function(&self) -> Option<(&str, &[Expression])>

Extract the name and args from a Function expression

Returns Some((name, args)) if this is a Function expression, None otherwise. This is a helper method for pattern matching with the Arc-based structure.

impl Expression

pub fn gcd(&self, other: &Expression) -> Expression

Compute the greatest common divisor of two expressions

For integer expressions, computes the mathematical GCD. For polynomial expressions with integer coefficients, uses fast IntPoly algorithm. For other expressions, returns 1.

Arguments

• other - The expression to compute GCD with

Examples

use mathhook_core::expr;

let a = expr!(12);
let b = expr!(8);
let gcd = a.gcd(&b);
assert_eq!(gcd, expr!(4));

use mathhook_core::expr;

// GCD of zero with any number is that number
let a = expr!(0);
let b = expr!(15);
let gcd = a.gcd(&b);
assert_eq!(gcd, expr!(15));

pub fn lcm(&self, other: &Expression) -> Expression

Compute the least common multiple of two expressions

For integer expressions, computes the mathematical LCM. For other expressions, returns the first expression.

Arguments

• other - The expression to compute LCM with

Examples

use mathhook_core::expr;

let a = expr!(12);
let b = expr!(8);
let lcm = a.lcm(&b);
assert_eq!(lcm, expr!(24));

use mathhook_core::expr;

// LCM with zero is zero
let a = expr!(0);
let b = expr!(15);
let lcm = a.lcm(&b);
assert_eq!(lcm, expr!(0));

pub fn factor_gcd(&self) -> Expression

Factor out the GCD from an expression

Currently a stub that returns the original expression. Will be implemented to extract common factors.

Examples

use mathhook_core::{expr, Expression};

let expr = Expression::add(vec![
    expr!(6*x),
    expr!(9),
]);
let factored = expr.factor_gcd();

pub fn cofactors( &self, other: &Expression, ) -> (Expression, Expression, Expression)

Compute GCD and cofactors

Returns a tuple of (gcd, cofactor_a, cofactor_b) where:

• gcd is the greatest common divisor
• cofactor_a = a / gcd
• cofactor_b = b / gcd

Arguments

• other - The expression to compute cofactors with

Examples

use mathhook_core::expr;

let a = expr!(12);
let b = expr!(8);
let (gcd, cofactor_a, cofactor_b) = a.cofactors(&b);
assert_eq!(gcd, expr!(4));
assert_eq!(cofactor_a, expr!(3));
assert_eq!(cofactor_b, expr!(2));

pub fn find_variables(&self) -> Vec<Symbol>

Find all variables in expression

Returns a vector of all unique Symbol nodes found in the expression tree. This is used by GCD algorithms to detect univariate polynomials.

impl Expression

pub fn solve(&self, variable: &Symbol) -> SolverResult

Solve equation for a variable with auto-detection

Solves the equation self = 0 for the given variable. Automatically detects equation type (linear, quadratic, ODE, etc.) and routes to the appropriate solver. For performance-critical code where you already know the equation type, use the fast path methods instead:

• solve_linear()
• solve_quadratic()
• solve_polynomial()
• solve_ode()
• solve_pde()
• solve_system()

Arguments

• variable - The variable to solve for

Returns

The solver result containing solutions or error information

Examples

use mathhook_core::{Expression, symbol};
use mathhook_core::algebra::solvers::SolverResult;

let x = symbol!(x);
let equation = Expression::add(vec![
    Expression::mul(vec![Expression::integer(2), Expression::symbol(x.clone())]),
    Expression::integer(-6),
]);

let result = equation.solve(&x);
match result {
    SolverResult::Single(solution) => {
        assert_eq!(solution, Expression::integer(3));
    }
    _ => panic!("Expected single solution"),
}

pub fn solve_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Solve equation with step-by-step educational explanation

Solves the equation self = 0 for the given variable and generates an educational explanation showing each solving step. Use this when you need to teach or explain the solving process. For performance-critical code where you only need the answer, use solve() instead.

Arguments

• variable - The variable to solve for

Returns

A tuple of (solver result, step-by-step explanation)

Examples

use mathhook_core::{Expression, symbol};

let x = symbol!(x);
let equation = Expression::add(vec![
    Expression::mul(vec![Expression::integer(2), Expression::symbol(x.clone())]),
    Expression::integer(-6),
]);

let (result, explanation) = equation.solve_with_steps(&x);
// The explanation contains educational content showing how to solve the equation

impl Expression

pub fn solve_linear(&self, variable: &Symbol) -> SolverResult

Fast path: solve as linear equation (skip classification)

Directly solves equations of the form ax + b = 0 without running equation type classification.

Arguments

• variable - The variable to solve for

Returns

The solver result containing the solution x = -b/a

pub fn solve_linear_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Fast path: solve as linear equation with steps (skip classification)

pub fn solve_quadratic(&self, variable: &Symbol) -> SolverResult

Fast path: solve as quadratic equation (skip classification)

Directly solves equations of the form ax^2 + bx + c = 0 using the quadratic formula without running equation type classification.

Arguments

• variable - The variable to solve for

Returns

The solver result containing solutions from the quadratic formula

pub fn solve_quadratic_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Fast path: solve as quadratic equation with steps (skip classification)

pub fn solve_polynomial(&self, variable: &Symbol) -> SolverResult

Fast path: solve as polynomial equation (skip classification)

Directly solves polynomial equations of degree 3+ (cubic, quartic, etc.) without running equation type classification.

Arguments

• variable - The variable to solve for

Returns

The solver result containing polynomial roots

pub fn solve_polynomial_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Fast path: solve as polynomial equation with steps (skip classification)

pub fn solve_ode( &self, dependent: &Symbol, independent: &Symbol, ) -> SolverResult

Fast path: solve as ordinary differential equation (skip classification)

Directly solves ODEs using the cached ODE solver registry without running equation type classification. Supports separable, linear, and homogeneous first-order ODE types.

Uses a static cached registry for optimal performance on repeated calls.

Arguments

• dependent - The dependent variable (e.g., y)
• independent - The independent variable (e.g., x)

Returns

The solver result containing the ODE solution

pub fn solve_ode_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Fast path: solve as ODE with steps (skip classification)

pub fn solve_pde( &self, dependent: &Symbol, independent_vars: &[Symbol], ) -> SolverResult

Fast path: solve as partial differential equation (skip classification)

Directly solves PDEs using the cached PDE solver registry without running equation type classification. Supports heat equation, wave equation, Laplace equation, and other types.

Uses a static cached registry for optimal performance on repeated calls.

Arguments

• dependent - The dependent variable (e.g., u)
• independent_vars - The independent variables (e.g., [x, t])

Returns

The solver result containing the PDE solution

pub fn solve_pde_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Fast path: solve as PDE with steps (skip classification)

pub fn solve_system( equations: &[Expression], variables: &[Symbol], ) -> SolverResult

Fast path: solve system of equations (skip classification)

Directly solves systems of equations without running equation type classification. Supports linear systems (Gaussian elimination) and polynomial systems (Groebner basis).

Arguments

• equations - Slice of equations to solve
• variables - Slice of variables to solve for

Returns

The solver result containing system solutions

pub fn solve_system_with_steps( equations: &[Expression], variables: &[Symbol], ) -> (SolverResult, StepByStepExplanation)

Fast path: solve system of equations with steps (skip classification)

pub fn solve_matrix_equation(&self, variable: &Symbol) -> SolverResult

Fast path: solve matrix/noncommutative equation (skip classification)

Directly solves equations involving matrices, operators, or quaternions where multiplication order matters.

Handles:

• Left multiplication: A*X = B (solution: X = A^(-1)*B)
• Right multiplication: XA = B (solution: X = BA^(-1))

Arguments

• variable - The matrix/operator variable to solve for

Returns

The solver result containing the matrix equation solution

pub fn solve_matrix_equation_with_steps( &self, variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Fast path: solve matrix equation with steps (skip classification)

impl Expression

pub fn is_zero(&self) -> bool

Check if the expression is zero (robust version with simplification)

This method simplifies the expression first before checking if it equals zero. It correctly detects zero for expressions like:

• Literal zeros: 0, 0.0, 0/1
• Symbolic zeros: x - x, 0 * y, sin(0)
• Simplified zeros: (x + 1) - (x + 1)

Performance Note

This method calls simplify(), which may be expensive for complex expressions. For performance-critical code where you only need to check literal zeros, use is_zero_fast() instead.

Examples

use mathhook_core::simplify::Simplify;
use mathhook_core::Expression;

// Literal zero
assert!(Expression::integer(0).is_zero());

// Symbolic zero after simplification
let expr = Expression::mul(vec![Expression::integer(0), Expression::integer(5)]);
assert!(expr.is_zero());

pub fn is_zero_fast(&self) -> bool

Fast literal zero check without simplification

This is a performance-optimized version that only checks if the expression is literally Number(0). It does NOT simplify the expression first.

Use this in performance-critical loops where you know the expression is already in simplified form, or where you specifically want to check for literal zeros only.

Examples

use mathhook_core::Expression;

// Detects literal zero
assert!(Expression::integer(0).is_zero_fast());

// Mul constructor auto-simplifies, so this IS detected
let expr = Expression::mul(vec![Expression::integer(0), Expression::integer(5)]);
assert!(expr.is_zero_fast()); // Simplified to 0 by constructor

// Example of what is_zero_fast() does NOT detect (without simplification):
// If we had a raw unsimplified Mul expression, is_zero_fast() wouldn't detect it

pub fn is_one(&self) -> bool

Check if the expression is one (robust version with simplification)

This method simplifies the expression first before checking if it equals one. It correctly detects one for expressions like:

• Literal ones: 1, 1.0, 2/2
• Symbolic ones: x / x, x^0, cos(0)
• Simplified ones: (x + 1) / (x + 1)

Performance Note

This method calls simplify(), which may be expensive for complex expressions. For performance-critical code where you only need to check literal ones, use is_one_fast() instead.

Examples

use mathhook_core::simplify::Simplify;
use mathhook_core::Expression;

// Literal one
assert!(Expression::integer(1).is_one());

// Symbolic one after simplification
let expr = Expression::pow(Expression::integer(5), Expression::integer(0));
assert!(expr.is_one());

pub fn is_one_fast(&self) -> bool

Fast literal one check without simplification

This is a performance-optimized version that only checks if the expression is literally Number(1). It does NOT simplify the expression first.

Use this in performance-critical loops where you know the expression is already in simplified form, or where you specifically want to check for literal ones only.

Examples

use mathhook_core::Expression;

// Detects literal one
assert!(Expression::integer(1).is_one_fast());

// Pow constructor auto-simplifies, so x^0 = 1 IS detected
let expr = Expression::pow(Expression::integer(5), Expression::integer(0));
assert!(expr.is_one_fast()); // Simplified to 1 by constructor

// is_one_fast() checks ONLY for literal Number(1)
// It does not simplify complex expressions first

pub fn as_number(&self) -> Option<&Number>

Get the numeric coefficient if this is a simple numeric expression

pub fn as_symbol(&self) -> Option<&Symbol>

Get the symbol if this is a simple symbol expression

pub fn is_negative_number(&self) -> bool

Check if this expression is a negative number

Returns true if the expression is a negative integer, rational, or float. Returns false for symbolic expressions (even if they might evaluate to negative).

Examples

use mathhook_core::Expression;

assert!(Expression::integer(-5).is_negative_number());
assert!(Expression::rational(-1, 2).is_negative_number());
assert!(!Expression::integer(5).is_negative_number());
assert!(!Expression::symbol("x").is_negative_number()); // Symbolic, not a number

pub fn is_positive_number(&self) -> bool

Check if this expression is a positive number

Returns true if the expression is a positive integer, rational, or float. Returns false for symbolic expressions (even if they might evaluate to positive).

Examples

use mathhook_core::Expression;

assert!(Expression::integer(5).is_positive_number());
assert!(Expression::rational(1, 2).is_positive_number());
assert!(!Expression::integer(-5).is_positive_number());
assert!(!Expression::symbol("x").is_positive_number()); // Symbolic, not a number

pub fn evaluate_method_call(&self) -> Expression

Evaluate method calls on expressions

This handles method calls like matrix.det(), matrix.trace(), etc. by calling the appropriate methods on the underlying objects.

pub fn is_function(&self) -> bool

Check if this expression represents a mathematical function

Returns true for expressions like sin(x), cos(x), etc. Now integrated with Universal Function Intelligence System

pub fn get_function_intelligence(&self) -> Option<&FunctionProperties>

Get function intelligence properties if available

Seamless integration between core expressions and function intelligence

pub fn explain_function(&self) -> Vec<Step>

Generate educational explanation for function expressions

Perfect integration with the educational system

impl Expression

Convenient formatting methods for Expression without requiring context

pub fn format(&self) -> Result<String, FormattingError>

Format expression using default LaTeX formatting

This is the most convenient way to format expressions when you don’t need specific formatting options. Always uses LaTeX format.

Examples

use mathhook_core::core::Expression;
use mathhook_core::{expr};

let x_expr = expr!(x);
let formatted = x_expr.format().unwrap();
// Returns LaTeX formatted string

pub fn format_as( &self, language: MathLanguage, ) -> Result<String, FormattingError>

Format expression with specific language/format

Examples

use mathhook_core::core::Expression;
use mathhook_core::formatter::MathLanguage;
use mathhook_core::expr;

let x_expr = expr!(x);
let latex = x_expr.format_as(MathLanguage::LaTeX).unwrap();
let simple = x_expr.format_as(MathLanguage::Simple).unwrap();
let wolfram = x_expr.format_as(MathLanguage::Wolfram).unwrap();

impl Expression

pub fn matrix_dimensions(&self) -> Option<(usize, usize)>

Get matrix dimensions for any matrix type

Returns (rows, columns) for all matrix types.

Examples

use mathhook_core::Expression;

let matrix = Expression::matrix(vec![
    vec![Expression::integer(1), Expression::integer(2)],
    vec![Expression::integer(3), Expression::integer(4)]
]);
assert_eq!(matrix.matrix_dimensions(), Some((2, 2)));

let identity = Expression::identity_matrix(3);
assert_eq!(identity.matrix_dimensions(), Some((3, 3)));

pub fn is_matrix(&self) -> bool

Check if expression is any kind of matrix

Trait Implementations

impl Add<&Expression> for Expression

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: &Expression) -> Expression

Performs the + operation.

impl Add<&Expression> for i32

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: &Expression) -> Expression

Performs the + operation.

impl Add<Expression> for &Expression

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: Expression) -> Expression

Performs the + operation.

impl Add<Expression> for i32

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: Expression) -> Expression

Performs the + operation.

impl Add<i32> for &Expression

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: i32) -> Expression

Performs the + operation.

impl Add<i32> for Expression

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: i32) -> Expression

Performs the + operation.

impl Add for &Expression

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: &Expression) -> Expression

Performs the + operation.

impl Add for Expression

type Output = Expression

The resulting type after applying the + operator.

fn add(self, rhs: Expression) -> Expression

Performs the + operation.

impl AdvancedPolynomial for Expression

fn polynomial_divide(&self, divisor: &Self) -> (Expression, Expression)

Polynomial division using optimized algorithms

fn polynomial_remainder(&self, divisor: &Self) -> Expression

Polynomial remainder

fn polynomial_degree(&self, var: &Symbol) -> Option<i64>

Polynomial degree computation

fn polynomial_leading_coefficient(&self, var: &Symbol) -> Expression

Leading coefficient extraction

fn polynomial_content(&self) -> Expression

Content extraction (GCD of coefficients)

fn polynomial_primitive_part(&self) -> Expression

Primitive part extraction (polynomial / content)

fn polynomial_resultant(&self, other: &Self, var: &Symbol) -> Expression

Resultant computation for polynomial elimination

fn polynomial_discriminant(&self, var: &Symbol) -> Expression

Discriminant computation

impl AdvancedSimplify for Expression

fn advanced_simplify(&self) -> Self

Perform advanced simplification including special functions

fn simplify_factorial(&self) -> Self

Simplify factorial expressions

fn simplify_logarithms(&self) -> Self

Simplify logarithmic expressions

fn simplify_trigonometric(&self) -> Self

Simplify trigonometric expressions

fn simplify_special_functions(&self) -> Self

Simplify other special functions

impl Clone for Expression

fn clone(&self) -> Expression

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Collect for Expression

fn collect(&self, var: &Symbol) -> Self

Collect terms with respect to a specific variable

fn collect_terms(&self) -> Self

Collect and combine all like terms

fn combine_like_terms(&self) -> Self

Combine like terms in the expression

impl ComplexAnalysis for Expression

fn is_analytic_at(&self, variable: &Symbol, point: &Expression) -> bool

Check if function is analytic at a point

fn pole_order(&self, variable: &Symbol, pole: &Expression) -> u32

Determine the order of a pole

fn is_removable_singularity( &self, variable: &Symbol, point: &Expression, ) -> bool

Check if point is a removable singularity

fn is_essential_singularity( &self, variable: &Symbol, point: &Expression, ) -> bool

Check if point is an essential singularity

impl ComplexOperations for Expression

fn complex_add(&self, other: &Expression) -> Expression

Add two complex expressions

fn complex_subtract(&self, other: &Expression) -> Expression

Subtract two complex expressions

fn complex_multiply(&self, other: &Expression) -> Expression

Multiply two complex expressions

fn complex_divide(&self, other: &Expression) -> Expression

Divide two complex expressions

fn complex_conjugate(&self) -> Expression

Get the complex conjugate

fn complex_modulus(&self) -> Expression

Get the modulus (absolute value)

fn complex_argument(&self) -> Expression

Get the argument (angle in radians)

fn to_polar_form(&self) -> (Expression, Expression)

Convert to polar form (magnitude, angle)

fn is_real(&self) -> bool

Check if the expression is real

fn is_imaginary(&self) -> bool

Check if the expression has an imaginary component

fn is_pure_imaginary(&self) -> bool

Check if the expression is pure imaginary

impl Debug for Expression

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Derivative for Expression

fn derivative(&self, variable: Symbol) -> Expression

Compute the derivative with respect to a variable

fn nth_derivative(&self, variable: Symbol, order: u32) -> Expression

Compute higher-order derivatives

fn is_differentiable(&self, variable: Symbol) -> bool

Check if expression is differentiable with respect to variable

impl DerivativeWithSteps for Expression

fn derivative_with_steps( &self, variable: &Symbol, order: u32, ) -> StepByStepExplanation

Compute derivative with step-by-step explanation

impl<'de> Deserialize<'de> for Expression

fn deserialize<__D>(__deserializer: __D) -> Result<Self, __D::Error>

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Display for Expression

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl EvalNumeric for Expression

fn eval_numeric(&self, _precision: u32) -> Result<Expression, MathError>

Evaluate expression to numerical form

impl Expand for Expression

fn expand(&self) -> Self

Expand the expression by distributing multiplication over addition

impl<C: FormattingContext> ExpressionFormatter<C> for Expression

fn format(&self, context: &C) -> Result<String, FormattingError>

impl Factor for Expression

fn factor(&self) -> Self

Factor the expression by extracting common factors

fn factor_out_gcd(&self) -> Self

Factor out the GCD from an expression

fn factor_common(&self) -> Self

Factor common elements

impl From<&str> for Expression

fn from(name: &str) -> Self

Converts to this type from the input type.

impl From<Symbol> for Expression

fn from(symbol: Symbol) -> Self

Converts to this type from the input type.

impl From<f64> for Expression

fn from(value: f64) -> Self

Converts to this type from the input type.

impl From<i32> for Expression

fn from(value: i32) -> Self

Converts to this type from the input type.

impl From<i64> for Expression

fn from(value: i64) -> Self

Converts to this type from the input type.

impl Integration for Expression

fn integrate(&self, variable: Symbol, depth: usize) -> Expression

Compute indefinite integral with recursion depth tracking

fn definite_integrate( &self, variable: Symbol, lower: Expression, upper: Expression, ) -> Result<Expression, MathError>

Compute definite integral

impl LaTeXFormatter for Expression

fn to_latex_with_depth( &self, context: &LaTeXContext, depth: usize, ) -> Result<String, FormattingError>

Format with explicit recursion depth tracking

fn function_to_latex_with_depth( &self, name: &str, args: &[Expression], context: &LaTeXContext, depth: usize, ) -> Result<String, FormattingError>

Convert function to LaTeX with context and depth tracking

fn to_latex<C>(&self, context: C) -> Result<String, FormattingError>

where C: Into<Option<LaTeXContext>>,

Format an Expression as LaTeX mathematical notation

fn function_to_latex( &self, name: &str, args: &[Expression], context: &LaTeXContext, ) -> Result<String, FormattingError>

Convert function to LaTeX (convenience method)

impl Limits for Expression

fn limit(&self, variable: &Symbol, point: &Expression) -> Expression

Compute two-sided limit

fn limit_directed( &self, variable: &Symbol, point: &Expression, direction: LimitDirection, ) -> Expression

Compute one-sided limit

fn limit_at_infinity(&self, variable: &Symbol) -> Expression

Compute limit at infinity

fn limit_at_negative_infinity(&self, variable: &Symbol) -> Expression

Compute limit at negative infinity

impl Matchable for Expression

fn matches(&self, pattern: &Pattern) -> Option<PatternMatches>

Match this expression against a pattern

fn replace(&self, pattern: &Pattern, replacement: &Pattern) -> Expression

Replace all occurrences of a pattern with a replacement expression

impl MatrixOperations for Expression

fn matrix_add(&self, other: &Expression) -> Expression

Add two matrices

fn matrix_subtract(&self, other: &Expression) -> Expression

Subtract two matrices

fn matrix_multiply(&self, other: &Expression) -> Expression

Multiply two matrices

fn matrix_scalar_multiply(&self, scalar: &Expression) -> Expression

Multiply matrix by scalar

fn matrix_determinant(&self) -> Expression

Get matrix determinant

fn matrix_transpose(&self) -> Expression

Get matrix transpose

fn matrix_inverse(&self) -> Expression

Get matrix inverse

fn matrix_trace(&self) -> Expression

Get matrix trace (sum of diagonal elements)

fn matrix_power(&self, exponent: &Expression) -> Expression

Raise matrix to a power

fn is_identity_matrix(&self) -> bool

Check if matrix is identity matrix

fn is_zero_matrix(&self) -> bool

Check if matrix is zero matrix

fn is_diagonal(&self) -> bool

Check if matrix is diagonal

fn simplify_matrix(&self) -> Expression

Simplify matrix expression

impl Mul<&Expression> for Expression

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: &Expression) -> Expression

Performs the * operation.

impl Mul<&Expression> for i32

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: &Expression) -> Expression

Performs the * operation.

impl Mul<Expression> for &Expression

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: Expression) -> Expression

Performs the * operation.

impl Mul<Expression> for i32

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: Expression) -> Expression

Performs the * operation.

impl Mul<i32> for &Expression

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: i32) -> Expression

Performs the * operation.

impl Mul<i32> for Expression

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: i32) -> Expression

Performs the * operation.

impl Mul for &Expression

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: &Expression) -> Expression

Performs the * operation.

impl Mul for Expression

type Output = Expression

The resulting type after applying the * operator.

fn mul(self, rhs: Expression) -> Expression

Performs the * operation.

impl Neg for &Expression

type Output = Expression

The resulting type after applying the - operator.

fn neg(self) -> Expression

Performs the unary - operation.

impl Neg for Expression

type Output = Expression

The resulting type after applying the - operator.

fn neg(self) -> Expression

Performs the unary - operation.

impl PartialEq for Expression

fn eq(&self, other: &Expression) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PolynomialArithmetic for Expression

fn poly_div( &self, divisor: &Expression, var: &Symbol, ) -> Result<(Expression, Expression), PolynomialError>

Perform polynomial long division

fn poly_quo( &self, divisor: &Expression, var: &Symbol, ) -> Result<Expression, PolynomialError>

Get quotient of polynomial division

fn poly_rem( &self, divisor: &Expression, var: &Symbol, ) -> Result<Expression, PolynomialError>

Get remainder of polynomial division

fn is_divisible_by(&self, divisor: &Expression, var: &Symbol) -> bool

Check if one polynomial divides another exactly

impl PolynomialClassification for Expression

fn is_polynomial(&self) -> bool

Check if expression is a valid polynomial

fn is_polynomial_in(&self, vars: &[Symbol]) -> bool

Check if polynomial in specific variables

fn polynomial_variables(&self) -> Vec<Symbol>

Get polynomial variables (empty if not polynomial)

fn classify(&self) -> ExpressionClass

Classify expression type for routing

fn is_intpoly_compatible(&self) -> bool

Check if expression can be represented as IntPoly

fn try_as_intpoly(&self) -> Option<(IntPoly, Symbol)>

Try to convert to IntPoly

impl PolynomialEducational for Expression

fn explain_poly_division( &self, divisor: &Expression, var: &Symbol, ) -> StepByStepExplanation

Generate step-by-step explanation for polynomial division

fn explain_poly_gcd(&self, other: &Expression) -> StepByStepExplanation

Generate step-by-step explanation for GCD computation

fn explain_poly_factorization(&self, var: &Symbol) -> StepByStepExplanation

Generate step-by-step explanation for factorization

impl PolynomialGcd for Expression

fn gcd(&self, other: &Self) -> Self

GCD using IntPoly fast-path (primary path)

Converts to IntPoly at API boundary, performs pure numeric GCD, converts result back. NO Expression tree walking in fast path.

fn lcm(&self, other: &Self) -> Self

Least Common Multiple

fn factor_gcd(&self) -> Self

Factor out GCD from expression

fn cofactors(&self, other: &Self) -> (Expression, Expression, Expression)

Compute GCD and cofactors: returns (gcd, a/gcd, b/gcd)

fn div_polynomial( &self, divisor: &Expression, var: &Symbol, ) -> (Expression, Expression)

Divides this polynomial by another, returning (quotient, remainder).

fn quo_polynomial(&self, divisor: &Expression, var: &Symbol) -> Expression

Returns the quotient of polynomial division.

fn rem_polynomial(&self, divisor: &Expression, var: &Symbol) -> Expression

Returns the remainder of polynomial division.

impl PolynomialGcdOps for Expression

fn polynomial_gcd(&self, other: &Self) -> Result<Expression, PolynomialError>

Compute GCD of two polynomials

fn polynomial_lcm(&self, other: &Self) -> Result<Expression, PolynomialError>

Compute LCM of two polynomials

fn polynomial_cofactors( &self, other: &Self, ) -> Result<(Expression, Expression, Expression), PolynomialError>

Compute GCD and cofactors

impl PolynomialProperties for Expression

fn degree(&self, var: &Symbol) -> Option<i64>

Compute degree with respect to a variable

fn total_degree(&self) -> Option<i64>

Compute total degree (sum of degrees in all variables)

fn leading_coefficient(&self, var: &Symbol) -> Expression

Get leading coefficient with respect to a variable

fn content(&self) -> Expression

Extract content (GCD of all coefficients)

fn primitive_part(&self) -> Expression

Extract primitive part (polynomial divided by content)

impl RationalSimplify for Expression

fn simplify_rational(&self) -> Self

Simplify rational expressions by canceling common factors

fn to_rational_form(&self) -> (Expression, Expression)

Convert to rational form (numerator/denominator)

fn rationalize(&self) -> Self

Rationalize denominators (remove radicals from denominators)

impl ResidueCalculus for Expression

fn residue(&self, variable: &Symbol, pole: &Expression) -> Expression

Compute residue at a pole

fn find_poles(&self, variable: &Symbol) -> Vec<Expression>

Find all poles of the function

fn contour_integral(&self, variable: &Symbol) -> Expression

Compute contour integral using residue theorem

impl Serialize for Expression

fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl SeriesExpansion for Expression

fn taylor_series( &self, variable: &Symbol, point: &Expression, order: u32, ) -> Expression

Compute Taylor series expansion

fn laurent_series( &self, variable: &Symbol, point: &Expression, order: u32, ) -> Expression

Compute Laurent series expansion

fn maclaurin_series(&self, variable: &Symbol, order: u32) -> Expression

Compute Maclaurin series (Taylor around 0)

fn power_series_coefficients( &self, variable: &Symbol, point: &Expression, order: u32, ) -> Vec<Expression>

Compute power series coefficients

impl SimpleFormatter for Expression

fn to_simple_with_depth( &self, context: &SimpleContext, depth: usize, ) -> Result<String, FormattingError>

Format with explicit recursion depth tracking

fn to_simple(&self, context: &SimpleContext) -> Result<String, FormattingError>

Format an Expression as simple mathematical notation

impl Simplify for Expression

fn simplify(&self) -> Self

Simplify expression using algebraic reduction rules

impl SolverStepByStep for Expression

fn solve_with_steps( &self, _variable: &Symbol, ) -> (SolverResult, StepByStepExplanation)

Solve with complete step-by-step explanation

fn explain_solving_steps(&self, variable: &Symbol) -> StepByStepExplanation

Generate step-by-step explanation for solving process

impl StepByStep for Expression

fn explain_simplification(&self) -> StepByStepExplanation

Generate step-by-step explanation for simplification

fn explain_expansion(&self) -> StepByStepExplanation

Generate step-by-step explanation for expansion

fn explain_factorization(&self) -> StepByStepExplanation

Generate step-by-step explanation for factorization

impl Sub<&Expression> for Expression

type Output = Expression

The resulting type after applying the - operator.

fn sub(self, rhs: &Expression) -> Expression

Performs the - operation.

impl Sub<Expression> for &Expression

type Output = Expression

The resulting type after applying the - operator.

fn sub(self, rhs: Expression) -> Expression

Performs the - operation.

impl Sub<Expression> for i32

type Output = Expression

The resulting type after applying the - operator.

fn sub(self, rhs: Expression) -> Expression

Performs the - operation.

impl Sub<i32> for Expression

type Output = Expression

The resulting type after applying the - operator.

fn sub(self, rhs: i32) -> Expression

Performs the - operation.

impl Sub for &Expression

type Output = Expression

The resulting type after applying the - operator.

fn sub(self, rhs: &Expression) -> Expression

Performs the - operation.

impl Sub for Expression

type Output = Expression

The resulting type after applying the - operator.

fn sub(self, rhs: Expression) -> Expression

Performs the - operation.

impl Substitutable for Expression

fn subs(&self, old: &Expression, new: &Expression) -> Expression

Substitute a single expression with another

fn subs_multiple( &self, substitutions: &[(Expression, Expression)], ) -> Expression

Apply multiple substitutions simultaneously

impl Summation for Expression

fn finite_sum( &self, variable: &Symbol, start: &Expression, end: &Expression, ) -> Expression

Compute finite sum

fn infinite_sum(&self, variable: &Symbol, start: &Expression) -> Expression

Compute infinite sum

fn finite_product( &self, variable: &Symbol, start: &Expression, end: &Expression, ) -> Expression

Compute finite product

fn infinite_product(&self, variable: &Symbol, start: &Expression) -> Expression

Compute infinite product

impl SummationEducational for Expression

fn explain_finite_sum( &self, variable: &Symbol, start: &Expression, end: &Expression, ) -> StepByStepExplanation

Explain finite sum computation with step-by-step guidance

fn explain_infinite_sum( &self, variable: &Symbol, start: &Expression, ) -> StepByStepExplanation

Explain infinite sum computation with convergence analysis

impl WolframFormatter for Expression

fn format_function_with_depth( &self, name: &str, args: &[Expression], context: &WolframContext, depth: usize, ) -> Result<String, FormattingError>

Convert function to Wolfram Language with depth tracking

fn to_wolfram_with_depth( &self, context: &WolframContext, depth: usize, ) -> Result<String, FormattingError>

Format with explicit recursion depth tracking

fn to_wolfram( &self, context: &WolframContext, ) -> Result<String, FormattingError>

Format an Expression as Wolfram Language notation

impl ZeroDetection for Expression

fn is_algebraic_zero(&self) -> bool

Detect if an expression is algebraically equivalent to zero

fn detect_zero_patterns(&self) -> bool

Detect common patterns that equal zero

fn simplify_to_zero(&self) -> Option<Expression>

Try to simplify expression to zero if it’s algebraically zero

impl StructuralPartialEq for Expression