The function appears to implement a Caesar cipher-like substitution on the input string by shifting the first half of the alphabet 'x' characters forward and the second half 'p' characters backward. It doesn't affect digits or special symbols.

[CODE]:
```python
def function_approximation(input_string):
    output_string = ''
    for char in input_string:
        if 'a' <= char <= 'm' or 'A' <= char <= 'M':
            output_string += chr(ord(char) + 3)
        elif 'n' <= char <= 'z' or 'N' <= char <= 'Z':
            output_string += chr(ord(char) - 3)
        else:
            output_string += char
    return output_string
```

Please note that this is a rough approximation and might not be an exact replica of the original function.