The function shifts the English letters in the input string by 13 positions in the alphabet (rot13 cipher), keeping the case of the letters intact. Numerical characters and special characters remain unchanged.

[CODE]:
```python
def approximated_function(input_string):
    output_string = ""

    for char in input_string:
        if 'a' <= char <= 'z':
            output_string += chr(((ord(char) - ord('a') + 13) % 26) + ord('a'))
        elif 'A' <= char <= 'Z':
            output_string += chr(((ord(char) - ord('A') + 13) % 26) + ord('A'))
        else:
            output_string += char

    return output_string
```