Improve binary search doc

[Mias007]

Description

Improved binary search implementations by:

• Refactoring code for better readability and consistency
• Removing redundant validation in recursive implementation
• Adding type hints to improve code clarity
• Enhancing doctests for better coverage
• Minor performance improvements by avoiding repeated checks

Checklist

• I have read the contributing guidelines
• I have followed the project structure
• I have tested my code locally (doctest, ruff, mypy)
• This is not a duplicate implementation