You pass an argument, let's call it a, to cos. Because the polynomial approximation is only valid for a relatively small value of the argument, the first step is to compute the remainder of a when divided by pi/2. The result, let's call it x, is as small as possible but will (or should) satisfy cos(x)=cos(a).

For example, let's say you're calculating cos(4). To reduce the argument you want to compute 4-n*pi/2 for some n. In this case n=2 so you want 4-pi, the smallest possible number whose cosine is equal to cos(4). In double-precision floating point, pi is not irrationall it's exactly 7074237752028440/2^51. Thus, x=4-pi will be 7731846010850208/2^53.

However, that's not enough to calculate cos(4) precisely, because pi is actually 1.224... * 10^-16 more than the fraction I gave before and thus cos(x) will not be exactly cos(4). To fix this, the remainder is computed in multiple precision, and that's where y comes in.

y is the result of computing (a-n*pi/2)-x with a value of pi that has a higher precision than just 7074237752028440/2^51. It is very small---so small that you don't need a polynomial to compute the remaining digits of cos(4), you can just use the linear approximation -x*y. But it's still very important in order to get the best possible approximation of cos(a), even if a is outside the interval -pi/2 to pi/2.