mystery wave: what to call it?

Question

I recently stumbled on a useful periodic waveform, but I don't know what to call it. It's drawn in blue in the enclosed image, with a sine wave drawn in red for reference. Do you know what to properly call this wave?

The most efficient way I've found to generate the wave is by offsetting each quadrant of a sine wave, as in the following C code. The even quadrants are displaced, while the odd quadrants are inverted and displaced.

double MysteryWave(double fPhase)
{
  double r = fmod(fPhase + 0.25, 1);
  double s = sin(r * PI * 2);
  if (r < 0.25)
    return s - 1;
  else if (r < 0.5)
    return 1 - s;
  else if (r < 0.75)
    return s + 1;
  else
    return -1 - s;
}

A wave that is very similar but NOT identical to my mystery wave can be generated via the arcsine of the cubed sine, rescaled to $[-1..1]$ by dividing by $\pi / 2$. In other words the following function is close but not quite the same:

2 * asin(pow(sin(fPhase * PI * 2), 3)) / PI

Answer 1

This is a more direct formula: there are two symmetries happening there, so you need to play with the values assigned to each $\frac{\pi}{2}$ interval in order to retrieve the correct "mapping" (probably it can be simplified more):

$$\sin{[x+\pi+[\frac{\pi}{2}\cdot(-1)^{\lfloor\frac{x}{\pi/2}\rfloor}]]}+(-1)^{\lfloor{x/\pi}\rfloor}$$

According to Wolfram Alpha if my calculations are correct it matches your waveform: