Simplification Rules

This module defines the specific simplification rules used by the SimplificationEngine. The rules are organized into an object, keyed by the omdNode type they apply to.

Rule Structure

Each rule is an instance of SimplificationEngine.SimplificationRule and has three main parts:

1. match function: Determines if the rule can be applied to a given node.
2. transform function: Executes the simplification and returns a new, transformed node.
3. message function: Generates a human-readable description of the transformation.

Available Rules

The rules object contains arrays of rules for each of the following node types:

binary (for omdBinaryExpressionNode)

• Same Term Cancellation: a - a simplifies to 0.
• Opposite Term Cancellation: a + (-a) simplifies to 0.
• Cancel Constants in Sums: x + 2 - 2 simplifies to x + 0.
• Constant Folding: Evaluates operations on constants (e.g., 2 + 3 -> 5). Includes rules for add, subtract, multiply, and divide.
• Identity Operations: Simplifies operations with identity elements (e.g., x + 0 -> x, y * 1 -> y).
• Zero Multiplication: x * 0 simplifies to 0.
• Combine Coefficients: 2 * 3x simplifies to 6x.
• Distributive Property: 2 * (x + 3) expands to 2x + 6.
• Expand Polynomial Multiplication: (x + 2)(x + 3) expands to x^2 + 5x + 6.
• Multiply Monomials: 2x * 3x simplifies to 6x^2.
• Combine Like Factors: x * x simplifies to x^2.
• Combine Multiple Constants in Sums: x + 2 + 3 simplifies to x + 5.
• Combine Like Terms: 2x + 3x simplifies to 5x.

rational (for omdRationalNode)

• Variable Self Division: x / x simplifies to 1.
• Power Base Division: x^n / x simplifies to x^(n-1).
• Monomial Variable Division: cx / x simplifies to c.
• Simplify Fraction: Reduces constant fractions to their simplest form (e.g., 6/8 -> 3/4).
• Simplify Multiplication in Rational: (4x)/3 restructures to (4/3) * x.
• Simplify Rational Division: (4x + 6) / 2 distributes the division to 2x + 3.

parenthesis (for omdParenthesisNode)

• Remove Redundant Parentheses: ((x)) simplifies to (x).

unary (for omdUnaryExpressionNode)

• Simplify Double Negation: -(-x) simplifies to x.
• Remove Unary Plus: +x simplifies to x.

power (for omdPowerNode)

• Calculate Powers: Evaluates constant powers (e.g., 2^3 -> 8).
• Power of Zero: x^0 simplifies to 1.
• Power of One: x^1 simplifies to x.
• Expand Polynomial Power: (x - 2)^2 expands to x^2 - 4x + 4.
• Square of Square Root: (sqrt(x))^2 simplifies to x.
• Fractional Exponent to Square Root: x^(1/2) converts to sqrt(x).

sqrt (for omdSqrtNode)

• Simplify Square Root of Constant: sqrt(9) simplifies to 3.
• Square Root of Square: sqrt(x^2) simplifies to x (assuming x >= 0).

function (for omdFunctionNode)

• Function Inverse Simplification: sin(arcsin(x)) simplifies to x.
• Function to Sqrt Node: Converts sqrt(x) from a function representation to the dedicated omdSqrtNode.

Rule Lookup

The getRulesForNode(node) function is used by the simplification engine to retrieve the appropriate list of rules for a given node based on its type.

See Also

• omdSimplification: The top-level module that uses these rules.
• SimplificationEngine: The engine that provides the helpers to create and apply these rules.