If abs(x) < 0.1, "sin(x)" is approximated really well by "x".
That's it. For small x, just return x.
(Obviously, there is some error involved, but for the speedup gained, it's a very good compromise)
That the sin(x) approximation still works well for 10^-1 (with an error of ~0.01%) is the really cool thing!
Whether it's appropriate in a numerical calculation obviously depends on the possible inputs and the acceptable error bars :)
What I really found amazing was that rather than reducing the number of multiplications to 2 (to calculate x^3), you can reduce it to 0, and it would still do surprisingly well.
N Error Limit
1 0.167% (1/6%)
2 8.35 x 10^-5% (1/11980%)
3 1.99 x 10^-8% (1/50316042%)
4 2.76 x 10^-12% (1/362275502328%)
For |x| up to Pi/2 (90°) the one term approximation falls apart. The guarantee from the Taylor series is within 65% (in reality it does better, but is still off by 36%). Two terms is guaranteed to be within 8%, three within 0.5%, and four within 0.02%.
Here's a quick and dirty little Python program to compute a table of error bounds for a given x [3].
[1] x - x^3/3! + x^5/5! - x^7/7! + ...
[2] In reality the error will usually be quite a bit below this upper limit. The Taylor series for a given x is a convergent alternating series. That is, it is of the form A0 - A1 + A2 - A3 + ... where all the A's have the same sign, |Ak| is a decreasing sequence past some particular k, and |Ak| has a limit of 0 as k goes to infinity. Such a series converges to some value, and if you approximate that by just taking the first N terms the error cannot exceed the first omitted term as long as N is large enough to take you to the point where the sequence from there on is decreasing. This is the upper bound I'm using above.
What is more interesting to me is that this can be one of rationales behind using radians.
And that tan(x)~x also holds for small angles, greatly easing estimations of distance to objects of known size (cf. mil-dot reticle).
The sin(x) = x approximation is actually exact (in terms of doubles) for |x| < 2^-26 = 1.4e-8. See also [1]. This happens to cover 48.6% of all doubles.
Similarly, cos(x) = 1 for |x| < 2^-27 = 7.45e-9 (see [2]).
Finally, sin(double(pi)) tells you the error in double(pi) - that is, how far the double representation of pi is away from the "real", mathematical pi [3].
[0]: https://randomascii.wordpress.com/2014/10/09/intel-underesti...
[1]: https://github.com/lattera/glibc/blob/master/sysdeps/ieee754...
[2]: https://github.com/lattera/glibc/blob/master/sysdeps/ieee754...
[3]: https://randomascii.wordpress.com/2012/02/25/comparing-float...
https://gist.github.com/mooman219/19b18ff07bb9d609a103ef0cd0...
> The term "memoization" was coined by Donald Michie in 1968[3] and is derived from the Latin word "memorandum" ("to be remembered"), usually truncated as "memo" in American English, and thus carries the meaning of "turning [the results of] a function into something to be remembered". While "memoization" might be confused with "memorization" (because they are etymological cognates), "memoization" has a specialized meaning in computing.
As wikipedia outlines, the r was dropped because of the memo. It's derived from the latin word memorandum that does contain the r, just like memory, but apparently it was more meant as an analogy to written memos.