Why atan2(y,x) is faster to compute than arcsin or arccos?

Question

I've read that when I know Y and X it's better to compute atan2(y,x) to get the angle instead of using a single value with asin and acos. I've tried to dig into math.h lib but I didn't found any formula.

Can someone explain why atan2 is better?

Answer 1

I guess you are comparing two pieces of code that look mostly like this:

angle = atan2(x, y);

and

angle = acos(x / sqrt(x * x + y * y));

(I assume C code)

The first part calculates what you need directly, while the second part does it in a roundabout way - it's natural to expect that the first one is faster (unless atan2 implementation contains some variant of the second code example).

In addition, atan is a pretty "primitive" function - it "feels" more general than "acos" or "asin". x87 has a command for it, so I expect old implementations of acos to use atan2 internally. Also, I don't know enough of SSE, but it would be reasonable to expect SSE to implement atan2, even just for compatibility with x87.

Answer 2

theta = atan2(y,x);

is more simple than:

float in = 1.0/sqrt(x*x+y*y);
theta = acos(x*in);
if(y<0)
    theta = -acos(x*in);
else
    theta = acos(y*in);

Don't you think? It's simpler, and maybe faster, depending on the implementation.

Now, if x and y are not arbitrary, but if you can assume for instance that x²+y²=1 and y>=0, then

theta = acos(x);

Is more simple, and maybe faster than atan2. But again, speed with vary with implementations. The atan2 might, or might not, be implemented with acos and asin, or using faster algorithms.