The function replaces the first occurrence of the letter 'e' in the input string with an 'X' and then appends the length of the modified string in English words.

[CODE]:
```python
def approx_function(input_str):
    modified_str = input_str.replace('e', 'X', 1)
    length_in_english = {
        1: "oXne",
        2: "tXwo",
        3: "thXree",
        4: "foXur",
        5: "fiXve",
        6: "siXx",
        7: "seXven",
        8: "eiXght",
        9: "niXne",
        10: "tXn",
        11: "eleXven",
        12: "tweXlve",
        13: "thirtXn",
        14: "fourXteen",
        15: "fifXteen",
        16: "sixtXn",
        17: "sevXnteen",
        18: "eightXn",
        19: "ninXteen",
        20: "tweXnty"
    }
    return modified_str + ", " + length_in_english[len(modified_str)]
```
Keep in mind that this is an approximation, and there may be minor differences in the behavior of the actual function.