Fast implementation of trigonometric functions for c++

Question

Short version: I\'d like to know whether there are implementations of the standard trigonometric functions that are faster than the ones included in math.h.

Answer 1

For 2-3% gain, this is almost certainly not worth the risk of inaccuracy, error, assumptions no longer being true (e.g. never falling outside of [-1,-1]), etc., unless you are planning on running this on a huge number of machines (where 2-3% represents thousands or millions of dollars in electricity and amortized cost of the machine).

That said, if you have domain-specific knowledge about what you are trying to accomplish, you may be able to speed up your computations by a factor of two or more. For example, if you always need sin and cos of the same value, calculate them close to each other in the code and make sure that your compiler translates them into a FSINCOS assembly instruction (see this question). If you need only a small portion of the full range of the function, you can potentially use a set of low-order polynomials followed by an iteration of Newton's method to get full machine precision (or as much as you need). Again, this is much more powerful if you know that you only need some values--e.g. if you can use that sin(x) is close to x near zero, and you will only be needing values near zero, then you can dramatically decrease the number of terms you need.

But, again, my primary advice is: 2-3% is not worth it. Think harder about the algorithms used and other potential bottlenecks (e.g. is malloc eating too much time?) before you optimize this.

Answer 2

I've implemented a fast sine function on cpu side which is at least two times faster than math.h ' s sine function however I used a very small lookup table(20 floats). it's accuracy is also not bad at all; average relative error rate is 0.095%. you can check it out from http://www.hevi.info/tag/fast-sine-function/

Explanation of the method is quite simple and relies on the fact that for small a's sin(a) = a * pi / 180 (see the link above for the proof)

Some Trigonometry

Although it is possible to achieve relatively accurate results with the formula shown above for angles between 0 and 10, as the angle gets wider as it loses accuricy. Therefore we should use the formula for angles less than 10 but how?!

The answer comes from the trigonometric sine addition formula;

sin(a+b) = sin(a) cos(b) + sin(b) cos(a)

If we can keep the ‘b’ less than 10 then we will be able to use our formula in order to find the sine with a couple of aritchmetic operations.

Let’s say we are asked the sine value for 71.654, then;

a = 70

b = 1.654

and,

sin(71.654) = sin(70 + 1.654) = sin(70) cos(1.654) + sin(1.654) cos (70)

In this formula we are able to use the fast calculation for the sin(1.654) part and for the rest unfortunately we need to have sine and cosine tables. The good thing is we only need the multiply of tens for sine and natural number angles between 0 and 10 for cosine.

Answer 3

You can look at this. It talks about optimizing sin, cos.