Fast approximation to acos?

OpenGL
1

Anyone have any fast approximations to acos()?

Fast as in faster than doing a 1D texture lookup.

2

You could always use a partially expanded infinite series, but then again the built-in function may be doing just that.

http://functions.wolfram.com/ElementaryFunctions/ArcCos/06/01/01/0001/MainEq1.gif

3

Several Pade approximations are given to arcsin if you follow the link below. You could use one of those with arccos = pi/2 - arcsin.

http://math.fullerton.edu/mathews/n2003/PadeApproximationMod.html

4

I really doubt it will be faster then texture lookup…

5

The simplest way to find a fast approximation is to compile that shader with cgc:


varying float x;
void main (void){
        gl_FragColor = acos(x);
        }

cgc acos.frag -oglsl -profile glslf
(or use a profile like fp30 that is better readable) The algoritn requires round about 11 instructions and it looks like a polynom is used for the approximation.

6

Thanks for the links everyone.

I’m going to have to profile this to find out for sure, but I suspect you might be right on that…

7

Another option would be to store the coeffecients of 4 cubics in a 4x4 matrix. The cubics could be fit to acos on intervals spanning [0,1], and you could use acos(-x) = -acos(x)+pi for [-1,0]. You could also have the knots of the polynomials distributed cubically so the higher curvature near 1 would be fit to a polynomial with a smaller interval. The function that converts a point in the domain to an index could reuse the evaluation of xxx for the polynomial. Something like this where mat contains the polynomial coeffecients:

sgn = sign(x);
x = abs(x);
x2 = xx;
x3 = xx2;
t = {1, x, x2, x3};
i = floor(4*x3);
arccos = dot(mat(i), t);
return sgn * arccos - min(0, sgn * PI);

I don’t know how it would compare to texture lookup, but at least no division is required as in the Pade approximation.

8

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