Expression Evaluation

MathHook provides two fundamental operations for working with expressions: 1. **Evaluation** - Compute numerical values with domain checking 2. **Simplification** - Algebraic reduction while staying symbolic Understanding when to use each operation is critical for correct mathematical computation.

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Mathematical Definition

Function Evaluation:

f(a) = f(x)|_{x=a}

Evaluation with Context: For expression $f(x_1, \ldots, x_n)$ and substitutions $\{x_i \mapsto v_i\}$ :

\text{evaluate}(f, \{x_i \mapsto v_i\}) = f(v_1, \ldots, v_n)

Domain Constraints: - $\sqrt{x}$ requires $x \geq 0$ in $\mathbb{R}$ - $\log(x)$ requires $x > 0$ (pole at 0) - $\tan(x)$ has poles at $\frac{\pi}{2} + n\pi$ - $\arcsin(x), \arccos(x)$ require $|x| \leq 1$ in $\mathbb{R}$

Code Examples

Constants Evaluate to Numbers

Direct evaluation of constant expressions

use mathhook::prelude::*;

let sum = expr!(2 + 3);
assert_eq!(sum.evaluate().unwrap(), expr!(5));

// Domain checking catches errors
let sqrt_neg = expr!(sqrt(-1));
assert!(matches!(sqrt_neg.evaluate(), Err(MathError::DomainError { .. })));

Evaluation with Context (Substitution)

Substitute variable values and evaluate

use mathhook_core::core::expression::eval_numeric::EvalContext;
use mathhook::prelude::*;
use std::collections::HashMap;

let x = symbol!(x);

// Substitute x = 3 and evaluate
let mut vars = HashMap::new();
vars.insert("x".to_string(), Expression::integer(3));
let ctx = EvalContext::numeric(vars);

let expr = Expression::pow(x.clone(), Expression::integer(2));
assert_eq!(expr.evaluate_with_context(&ctx).unwrap(), Expression::integer(9));

Simplification Without Domain Checking

Simplify operates purely symbolically without domain validation

use mathhook::prelude::*;
use mathhook_core::simplify::Simplify;

let x = symbol!(x);

// Combine like terms
let sum = expr!(x + x);
assert_eq!(sum.simplify(), expr!(2 * x));

// Apply identities
assert_eq!(expr!(x * 1).simplify(), expr!(x));
assert_eq!(expr!(0 * x).simplify(), expr!(0));

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