[dahart]

The input is a double-double where y << x by definition. So the first line is just using the standard cos identity cos(x+y) = cos(x)cos(y) - sin(x)sin(y). Since y is very small, we can approximate cos(y) ~= 1, and sin(y) ~= y. This yields the first line.

The second line, I believe, is not relying on x being small, but rather relying on sin(x)*y being small. The whole result of the sin(x)*y term is approximately on the order of the LSB of the result, and the relative error of approximating sin(x) using 1 is better than using 0, so this term will help produce more accurate rounding of the bottom bits.