#include <math.h>

struct circle {
    double c;
    double s;
};

void approx_turn_update(double turn, struct circle* ret)
{
    static const double a = 8 * M_SQRT2 / 3 - 3;
    static const double b = 4 - 8 * M_SQRT2 / 3;

    double p = turn * (b * turn * turn + a);
    double q = p * p;
    double r = 1 + q;
    double c = (1 - q) / r, s = 2 * p / r;

    ret->c = c * c - s * s;
    ret->s = 2 * c * s;
}

void turn_update(double turn, struct circle* ret)
{
    double arg = M_PI * turn;
    ret->c = cos(arg);
    ret->s = sin(arg);
}