Rotate and translate a 2D point N times
CONST is not a name; it doesn't convey any meaning. Call it something like rads_per_degree. If you feel you need to comment on the value, better do the calculation explicitly (take advantage of \$\pi\$ being defined for you in cmath):

const double rads_per_degree = M_PI / 180;

Each time you rotate, you recalculate radian measure, sin and cos. Within the test case these values do not change. You may safely precompute them:

double rads = rads_per_degree * angle;
double sin = std::sin(rads);
double cos = std::cos(rads);
while (v--) {
    do_math(x, y, sin, cos, L);
}

I am not sure why you special cased small values.
getDouble is somewhat dubious. It would've make sense to have such a function to sanitize input. As written, it is meaningless. Also, you should not read integer as double.