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Okay, let's dive deep into evaluating mathematical expressions given as strings. This is a classic problem with a few different approaches, ranging from simple (but limited) to robust (but more complex). We'll cover a common method: the *Shunting-Yard Algorithm* combined with a stack-based evaluation.

*1. Understanding the Problem*

We want to take a string like `"3 + 4 * 2 / (1 - 5) ^ 2"` and produce the numerical result, which in this case is `5.0`. The challenge lies in handling:

*Operator precedence:* Multiplication/division before addition/subtraction, and parentheses to override the usual order.
*Associativity:* Left-to-right for most operators (e.g., `3 - 2 - 1`), right-to-left for exponentiation (e.g., `2 ^ 3 ^ 2`).
*Functions:* Handling built-in functions like `sin()`, `cos()`, `sqrt()`.
*Parentheses:* Properly managing nested parentheses.
*Unary operators:* Handling cases like `-5` or `+3`.

*2. The Shunting-Yard Algorithm (For Parsing)*

The Shunting-Yard Algorithm (named after a railroad shunting yard) is a method for converting an infix expression (the way we normally write math) into a postfix expression (also called Reverse Polish Notation or RPN). RPN is much easier to evaluate using a stack.

*How it works:*

1. *Input:* The infix expression string.
2. *Output:* A queue of tokens in postfix order.
3. *Data Structures:*
*Output Queue:* Stores the RPN tokens as we generate them.
*Operator Stack:* Temporarily holds operators, ensuring proper precedence and associativity.

*Algorithm Steps:*

1. *Tokenize:* Split the input string into a list of tokens (numbers, operators, parentheses, function names).
2. *Iterate through the tokens:*
*Number:* Add the number to the output queue.
*Function:* Push the function onto the operator stack.
*Operator:*
While there's an operator at the top of the stack and either:
The cu ...

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