Strength Reduction: Optimizing Multiplication with Addition

Strength reduction is a compiler optimization technique that replaces expensive operations with equivalent, less expensive ones. This challenge focuses on a common form of strength reduction: replacing multiplications by constants with equivalent additions or shifts. This optimization can significantly improve performance, especially in loops.

Problem Description

Your task is to implement a function in Go that takes an Abstract Syntax Tree (AST) representing a simple arithmetic expression and performs strength reduction for multiplications by a constant. Specifically, you need to identify expressions of the form x * C where C is a constant integer, and if C is a power of 2, replace it with a left bit shift (x << log2(C)). If C is not a power of 2, you should identify opportunities to replace x * C with a series of additions and shifts if C can be expressed as a sum of powers of 2 (e.g., x * 5 can be (x << 2) + x).

Key Requirements:

1. AST Traversal: You will be provided with a simplified AST structure. Your solution should traverse this AST.
2. Multiplication by Constant Detection: Identify binary expressions where the operator is multiplication (*) and the right-hand operand is a constant integer.
3. Power of 2 Optimization: If the constant multiplier is a power of 2 (e.g., 2, 4, 8, 16), replace the multiplication with an equivalent left bit shift. For example, x * 8 becomes x << 3.
4. Sum of Powers of 2 Optimization: If the constant multiplier is not a power of 2 but can be represented as a sum of powers of 2 (e.g., 5 = 4 + 1, 6 = 4 + 2), replace the multiplication with a combination of left shifts and additions. For example, x * 5 can become (x << 2) + x.
5. Preserve Semantics: The transformed AST must produce the same results as the original AST for all valid inputs.
6. Handle Edge Cases: Consider cases where the multiplier is 0 or 1.

Expected Behavior:

The function should return a new AST that has undergone the strength reduction. The structure of the AST should be modified in place if feasible, or a new AST can be returned.

Important Considerations:

• This challenge assumes a simplified AST structure for demonstration purposes.
• We are focusing on multiplying a variable or another expression by a constant.
• We are not concerned with floating-point numbers or other complex types.

Examples

Example 1:

Input AST:
Multiply(Variable("x"), Constant(8))

Output AST:
ShiftLeft(Variable("x"), Constant(3))

Explanation: 8 is 2^3, so x * 8 is optimized to x << 3.

Example 2:

Input AST:
Multiply(Variable("y"), Constant(5))

Output AST:
Add(ShiftLeft(Variable("y"), Constant(2)), Variable("y"))

Explanation: 5 = 4 + 1. So y * 5 becomes (y * 4) + (y * 1), which is (y << 2) + y.

Example 3:

Input AST:
Multiply(Constant(0), Variable("z"))

Output AST:
Constant(0)

Explanation: Multiplying by 0 always results in 0, regardless of the other operand.

Example 4:

Input AST:
Multiply(Variable("a"), Constant(1))

Output AST:
Variable("a")

Explanation: Multiplying by 1 does not change the value.

Example 5:

Input AST:
Add(Multiply(Variable("b"), Constant(6)), Constant(3))

Output AST:
Add(Add(ShiftLeft(Variable("b"), Constant(2)), ShiftLeft(Variable("b"), Constant(1))), Constant(3))

Explanation: 6 = 4 + 2. So b * 6 becomes (b * 4) + (b * 2), which is (b << 2) + (b << 1). The constant addition remains.

Constraints

• Operands for multiplication will be variables or constants.
• Constant multipliers will be non-negative integers.
• The AST depth is at most 10.
• The value of constant multipliers will not exceed 255.
• The solution should aim for an efficient traversal, ideally O(N) where N is the number of nodes in the AST.

Notes

• You'll need to define the AST node structures yourself. Think about Expression, Variable, Constant, BinaryOp (like Add, Multiply, ShiftLeft), etc.
• For checking if a number is a power of 2, you can use bitwise operations.
• For decomposing a number into powers of 2 for addition, you can iterate through the bits of the number.
• Consider how to represent the operations in your AST nodes.
• The goal is to transform the AST. You do not need to evaluate it.