Tricky arithmetic or sleight of hand?

Vincent answered Fast Arc Cos algorithm by suggesting this function.

float arccos(float x) 
{ 
    x = 1 - 开发者_JAVA技巧(x + 1); 
    return pi * x / 2; 
} 

The question is, why x = 1 - (x + 1) and not x = -x?

It returns a different result only when (x + 1) causes a loss of precision, that is, x is many orders of magnitude larger or smaller than one.

But I don't think this is tricky or sleight of hand, I think it's just plain wrong.

cos(0) = 1 but f(1) = -pi/2
cos(pi/2) = 0 but f(0) = 0
cos(pi) = -1 but f(-1) = pi/2

where f(x) is Vincent's arccos implementation. All of them are off by pi/2, a linear approximation that gets at least these three points correct would be

g(x) = (1 - x) * pi / 2

I don't see the details instantly, but think about what happens as x approaches 1 or -1 from either side, and consider roundoff error.

Addition causes that both numbers are normalized (in this case, relevant for x). IIRC, in Knuth's volume 2, in the chapter on floating-point arithmetic, you can even see an expression like x+0.