The function counts the number of letters in the input string and appends one of three characters based on the last letter of the input: U for uppercase letters or vowels, _ for digits and symbols, and ^ for lowercase consonants.

[CODE]:

```python
def approximate_function(input_string):
    letter_count = len(input_string)
    last_char = input_string[-1]
    
    if last_char.isupper() or last_char.lower() in ["a", "e", "i", "o", "u"]:
        return f"{letter_count}U"
    elif last_char.isdigit() or not last_char.isalnum():
        return f"{letter_count}_"
    else:
        return f"{letter_count}^"
```

This code approximates the observed behavior of the function. Note that the original function seems to have limitations in handling empty strings and may return non-standard numbers such as "foub" or "twelvU", but the provided approximation aims for a more generic and intuitive description.