Approximating inverse trigonometric functions

Question

I have to implement asin, acos and atan in environment where I have only following math tools:

• sine
• cosine
• elementary fixed point arithmetic (floating point numbers are not available)

I also already have reasonably good square root function.

Can I use those to implement reasonably efficient inverse trigonometric functions?

I don't need too big precision (the floating point numbers have very limited precision anyways), basic approximation will do.

I'm already half decided to go with table lookup, but I would like to know if there is some neater option (that doesn't need several hundred lines of code just to implement basic math).

EDIT:

To clear things up: I need to run the function hundreds of times per frame at 35 frames per second.

Answer 1

In a fixed-point environment (S15.16) I successfully used the CORDIC algorithm (see Wikipedia for a general description) to compute atan2(y,x), then derived asin() and acos() from that using well-known functional identities that involve the square root:

asin(x) = atan2 (x, sqrt ((1.0 + x) * (1.0 - x)))
acos(x) = atan2 (sqrt ((1.0 + x) * (1.0 - x)), x)

It turns out that finding a useful description of the CORDIC iteration for atan2() on the double is harder than I thought. The following website appears to contain a sufficiently detailed description, and also discusses two alternative approaches, polynomial approximation and lookup tables:

http://ch.mathworks.com/examples/matlab-fixed-point-designer/615-calculate-fixed-point-arctangent