Very nice. And 107 Bytes might even beat a Java-answer :)

@DenkerAffe So far it does, see my submission

While your algorithm is very interesting, a valid submission would have to accept user input and produce output of some type, as well as compute the square root of the sum of squares. In its current state, your answer is (sadly) invalid and subject to removal.

@Dennis I don't agree. The discussion about whether or not the brainfuck tape counts as arguments is not up to me, but the value at the pointer is definitly the return value of every brainfuck program. I already stated I might consider creating a square root algorithm, however, as DenkerAffe already stated, for the time being, its OK if I submit the squared distance instead, as long as I clearly state it in my answer.

1. According to point 3 of this meta post, a brainfuck code snippet does not count as a function, since you cannot execute it more than once. That leaves only a full program, with one of the defaults for I/O. 2. Unless the question is actually modified to include a "brainfuck clause", this answer does not meet the challenge's specification in its current state and is therefore invalid.

@Dennis Your second link conflict with what you said earlier in that turing machines CAN in fact accept input from tape and output CAN be written to the final tape position. Either way, I added a minimal complete definition, specially for you <3. This program no longer conflicts with the OP's rules (If we can get the discussion about precision out of the way). Please don't link random stackoverflow discussions without reading them, that way we won't get anywhere.

@YoYoYonnY Those links do not go to StackOverflow and are in no way random. The tape options for Turing machines (which doesn't support any of the "normal" ways of input) do not apply to brainfuck, because brainfuck is not a Turing machine.

Also, point 2 was meant to apply to the fact that your code doesn't calculate the square root. This is still an issue.

From this article on wikipedia (en.wikipedia.org/wiki/Turing_machine) it seems pretty clear to me that brainfuck is a turing machine, but I'm honestly wondering why we're still having this conversation. The answer is valid and the OP already stated that it's OK to return the squared distance as long as there doesn't exists a square root algorithm within the brainfuck environment. I'll post the square root algorithm somewhere today anyway, which would make the complete program valid.

If that's not enough, ZZ038 proved it in this esolang talk page

esolangs.org/wiki/Talk:Brainfuck#Turing_Machine

@YoYoYonnY That proves that brainfuck is equivalent to a Turing machine. We're still having this conversation because you accused me of linking random stackoverflow discussions without reading them, which are not random, not from Stack Overflow, and knowing what they say is kinda my job.

Wait. So you're telling me that brainfuck is equivalent to a Turing machine? brainfuck = turing machine? Then what's the problem? Do you want me to change my title to "Turing machine in brainfuck notation"?

By the way, I said your links were "random" because you sent two links, with conflicting statements, and quoted both of them as the same (that a brainfuck snippet does not count as a function (Which I never claimed it did), and that a turing machine may take input and put output from and to tape respectively (which my program does)). Oh, and "stack exchange", not "stack overflow"...

Equivalence does not imply equality. λ-calculus is also equivalent to a Turing machine. Anyway, there's no point in debating this if you've already edited the answer.