First of all, I understand this is an exercise in English, but there is nothing wrong with the sentence you are starting out with:
Anil is too fast a runner to not come first in the race.
This isn't really what we would call a "double negative". It is a perfectly coherent sentence, and similarly phrased sentences are often used, usually in response to a contrary opinion, ie if someone said that they didn't think Anil would come first.
The kind of double-negative that is unacceptable is where the statement is unbalanced and the meaning is obscured, for example "I never did nothing to nobody", which is plainly incorrect and redundant.
So, returning to your real question - the details of the original statement are that:
- Anil is a very fast runner
- You believe he will come first
So if you were looking to simply state this using as few words as possible and avoiding the so-called "double-negative" of the original statement, the most succinct statement would be the one written by your friends:
Anil is so fast a runner that he will come first in the race.
This makes the two points above, and is grammatically correct. However, some of the meaning of the original statement is lost. Originally there was a sense of belief that surely he will win, given his speed - but now in this shortened sentence it is being put across as an absolute certainty, but nobody can really say for sure that he will win, and so now the statement seems a little arrogant.
Your sentence does not scan correctly because technically you are repeating the word "not":
Anil is so fast a runner that he cannot not come first in the race.
If anything, this is more of a double negative that the original statement! Some English speakers may attempt to say such a sentence, pausing and emphasising the second "not", but you can't really get away with it in written English.
I would suggest:
Anil is so fast a runner that he cannot fail to come first in the race.
This reads better than your suggestion, and carries the same belief that he will win. I wouldn't call "cannot fail" a double negative, but technically the two words do cancel each other out because you could just say "win" instead of "didn't fail".
If the goal is really to eliminate all double-negatives, grammatical or otherwise, then you could say:
Anil is so fast a runner he surely must come first in the race.
Anil is so fast a runner I believe he will come first in the race
I believe these most accurately convey both the facts and the implied meaning of the original sentence.