\sqrt{-\left(\frac{2\arcsin\left(\sin\left(\frac{\left(\pi x+\pi\right)}{2}\right)\right)}{\pi}\right)^{2}+1}\cdot\operatorname{sign}\left(\frac{2\arcsin\left(\sin\left(\frac{\left(\pi x+4\pi\right)}{2}\right)\right)}{\pi}\right)
update: f\left(x\right)=r\sqrt{-\left(\frac{2\arcsin\left(\sin\left(\frac{\pi x}{2r}+n\pi\right)\right)}{\pi}\right)^{2}+1}\cdot\operatorname{sign}\left(\cos\left(\frac{\pi x}{2r}+n\pi\right)\right)