The function takes an input string and replaces the first occurrence of 'e' with 'a' and the second occurrence of 'e' with 'u'. Then, it appends 'i' to the string.

[CODE]:
```python
def modified_function(input_string):
    count_e = 0
    new_string = ''
    for char in input_string:
        if char == 'e':
            count_e += 1
            if count_e == 1:
                new_string += 'a'
            elif count_e == 2:
                new_string += 'u'
            else:
                new_string += char
        else:
            new_string += char
    return new_string + 'i'
```
Please note that this is an approximation of the original function, and there may be small differences in the implementation.

Let's test this approximation with a few additional inputs to see if it produces the same results.

COMMAND: PYTHON(./temp/function.py "elephant" "deep" "needle")