The function converts uppercase letters to lowercase, maintains numbers and some characters, and alters certain characters by a consistent pattern.

[CODE]: 
```python
def approximate_function(input_str):
    altered_chars = {"h": "t", "e": "q", "l": "x", "o": "a", "w": "t", "r": "l", "d": "i", "s": "y", "y": "x", "m": "y", "b": "n", "g": "j", "t": "e", "u": "f", "c": "b", "x": "c", "f": "d", "v": "g", "p": "h", "H": "T", "L": "X", "O": "A", "R": "L", "D": "I", "S": "Y", "W": "T"}
    output_str = ""
    for char in input_str:
        output_str += altered_chars.get(char, char.lower())
    return output_str
```

Please note that this approximation might not be perfect and might not cover all possible cases, but it represents the observed behavior in the tested inputs.