The first multiplication on the second line is the inverse of the function call on the first, the compiler shouldn't expect that, and the whole point of the multiplication is that it's not an inverse of the square root.

Caching repeated expressions is something every compiler is expected to know how to do, along with commuting additions and multiplications and inlining of small functions. Sometimes they'll even convert a cached value into a repeated expression, if they decide that register allocation is worth the extra CPU cycle. And modern CPUs won't even need that extra cycle.

By extra work, do you mean the ternary? I think it's better than spending five lines on the same.

It happens in Ruby at least

can I be given a gist of how this works?

That's a bug/feature with the output console. It interprets the stuff output as HTML rather than as plain text.

Cute. It wouldn't pass any code review, but it's still cute.

too few semicolons for my taste ;-)

Specification should not rely on Wikipedia to provide the definition of non-standard nomenclature. "constant R" is not standardised terminology in the context of arithmetic sequences.

Also, "elementh" is not a word. "Element" is (with the plural being "elements"). Also, "you" should not be capitalised. Only "I" gets the privilege because it's a single letter. Neither should "sum" be in all caps. Use proper formatting for emphasis - but this description is small enough to not need emphasis anyways.

Also, you should probably increase the range and count of your test cases.

Needs more test cases. Preferably a suite of random tests.

Upvoting just for the visuals ;-)

hmm... call method_missing manually? :-D