[ThePadawan]

One of the things I recently learned that sounded the most "that can't possibly work well enough" is an optimization for sin(x):

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)

[tzs]

To put some numbers on it, using N terms of the Taylor series for sin(x) [1] with |x| <= 0.1, the maximum error percentage cannot exceed [2]:

  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%)

Even for |x| as large as 1 the sin(x) = x approximation is within 20%, which is not too bad when you are just doing a rough ballpark calculation, and a two term approximation gets you under 1%. Three terms is under 0.024% (1/43%).

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.

[3] https://pastebin.com/thN8B7Gf

[MauranKilom]

Some trivia, partly stolen from Bruce Dawson[0]:

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...

[quietbritishjim]

That is precisely the technique discussed in the article: it's the first term of the Taylor expansion. Except that the article used more terms of the expansion, and also used very slightly "wrong" coefficients to improve the overall accuracy within the small region.

[RicoElectrico]

Come on, at this point I've seen this "engineering approximation" memed so many times, even on Instagram ;)

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).

[lapetitejort]

tan(x)~x because cos(x)~1. It's approximations all the way down.

[chriswarbo]

This is a very common assumption in Physics, e.g. https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Small-a...

Whether it's appropriate in a numerical calculation obviously depends on the possible inputs and the acceptable error bars :)

[sharikone]

I think that you will find that for subnormal numbers any math library will use the identity function for sin(x) and 1 for cos(x)

[ThePadawan]

Right, but the largest subnormal number in single-precision floats is ~ 10^-38.

That the sin(x) approximation still works well for 10^-1 (with an error of ~0.01%) is the really cool thing!

[jvz01]

Another good hint is the classical half-angle formulas. You can often avoid calling sin() and cos() altogether!

[Bostonian]

Why wouldn't you at least include the x^3 term in the Taylor series for abs(x) < 0.1?

[ThePadawan]

That's what I assumed would have been a reasonable optimization!

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.