Endless sine generation in C

I am working on a project which incorporates computing a sine wave as input for a control loop.

The sine wave has a frequency of 280 Hz, and the control loop runs every 30 µs and everything is written in C for an Arm Cortex-M7.

At the moment we are simply doing:

double time;void control_loop() {    time += 30e-6;    double sine = sin(2 * M_PI * 280 * time);    ...}

Two problems/questions arise:

1. When running for a long time, time becomes bigger. Suddenly there is a point where the computation time for the sine function increases drastically (see image). Why is this? How are these functions usually implemented? Is there a way to circumvent this (without noticeable precision loss) as speed is a huge factor for us? We are using sin from math.h (Arm GCC).
2. How can I deal with time in general? When running for a long time, the variable time will inevitably reach the limits of double precision. Even using a counter time = counter++ * 30e-6; only improves this, but it does not solve it. As I am certainly not the first person who wants to generate a sine wave for a long time, there must be some ideas/papers/... on how to implement this fast and precise.