The function seems to shift characters in a string by a certain number of positions in the ASCII table, and the pattern varies depending on the input character's case and position in the alphabet.

[CODE]: 
Here's an approximation of the function's behavior in Python:

```python
def approx_function(s: str) -> str:
    result = []
    for c in s:
        if 'a' <= c <= 'z':
            result.append(chr(((ord(c) - ord('a') + 5) % 26) + ord('a')))
        elif 'A' <= c <= 'Z':
            result.append(chr(((ord(c) - ord('A') + 7) % 26) + ord('A')))
        else:
            result.append(c)
    return "".join(result)
```

Note that this approximation may not be 100% accurate for all inputs, but it captures the general behavior observed in the given input-output pairs.