ArcCos(x)= Π √((1-x)/(4+x)).

This tradition came to its own as Madhava school of mathematics from Kerala. https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and...

Note the approximation is for 0 < x < 1. For the range [-1, 0] Bhaskara used symmetry.

If I remember correctly, Aryabhatta had derived a rational approximation about a hundred years before this.

EDIT https://doi.org/10.4169/math.mag.84.2.098

Let α represent a roll rotation, and β a pitch rotation.

Let R(α) be:

    ( cos α   sin α   0)
    (-sin α   cos α   0)
    (   0       0     1)

    (1     0       0   )
    (0   cos β   -sin β)
    (0   sin β    cos β)

    R(β).R(α) = (    cos α           sin α          0   )
                ((-sin α*cos β)   (cos α*cos β)   -sin β)
                ((-sin α*sin β)   (cos α*sin β)    cos β)

    ( 1   α   0)
    (-α   1  -β)
    ( 0   β   1)

    x' = x + αy
    y' = y - αx - βz
    z' = z + βy

  constexpr double a0 = 1.5707288;

That might explain why the original Cg (a GPU programming language) code did not use Estrin's, since at least the code in the post does add 1 extra op (squaring `abs_x`).

(AMD GPUs used to use VLIW (very long instruction word) which is "static" ILP).

That’s quite subjective. I happen to find trigonometry to be elegant and true.

I also agree that trigonometric functions lack efficiency in software.