The order of operations, or operator precedence, determines how operators are parsed concerning one another. Operators with higher precedence are evaluated before operators with lower precedence. Changing operator precedence is essential in controlling how expressions are evaluated within code, ensuring that calculations are performed as intended.

Basics of Operator Precedence

Operator precedence refers to the rules that define the order in which different operators in an expression are evaluated. Generally, higher-precedence operators bind more tightly and are evaluated first. The typical order of precedence in many programming languages follows the mathematical convention: multiplication and division before addition and subtraction.

Example in Arithmetic Operations

Consider the arithmetic expression:
`3 + 4 * 5`

Using standard precedence rules (multiplication before addition), the above is evaluated as:
`3 + (4 * 5) = 3 + 20 = 23`

However, if you intend to add `3` and `4` together before multiplying, changing the precedence with parentheses would alter the evaluation order:
`(3 + 4) * 5 = 7 * 5 = 35`

Changing Operator Precedence

To override the natural precedence of operators, especially when operations need to be carried out in a non-standard order, parentheses are utilized. Parentheses have the highest precedence and can be used to manipulate the evaluation order of operators effectively.

Technical Explanation

Various programming languages have their operator precedence rules and the ability to override them using parentheses, making it crucial to check specific language documentation. For instance, languages such as Python, JavaScript, or C++ adhere to a similar precedence hierarchy but may differ in their implementation details.

Here’s a table summarizing the operator precedence for a common language (like Python):

Note: The above table is indicative of Python's precedence rules; other languages might differ slightly.

Complex Example

Consider the following mixed operation:
`2 + 3 * 2 ** 2`

Without changing precedence, it evaluates as follows due to the rules:
`2 + (3 * (2 ** 2)) = 2 + (3 * 4) = 2 + 12 = 14`

You can change the precedence using parentheses:
`(2 + 3) * 2 ** 2 = 5 * 4 = 20`

Real-World Application and Best Practices

Understanding and controlling operator precedence is pivotal when developing complex programs, ensuring that logic is executed as intended. Here are some best practices:

1. Clarity Over Cleverness: Even when a precedence rule is known, using parentheses can make code more readable and less error-prone.
2. Consistency Across Languages: While precedence rules are often similar, confirm them within the manual of the language being used to prevent mistakes.
3. Complex Expressions: Break down complex expressions into smaller parts, using intermediate variables to improve readability.

Advanced Use: Override with Functions

Advanced applications may involve function overriding or custom operator implementation to achieve desired precedence in languages that support such features. For example, in languages like Python, operator overloading allows redefining the behavior of operators for user-defined classes which indirectly lets you control how expressions involving these operators are evaluated.

In conclusion, operator precedence is fundamental to programming, ensuring appropriate execution order for mixed operators in expressions. Being intentional about its application using tools like parentheses can transform program logic from typical to intentional, matching the developer’s intent with execution flow.