The simplification rules as code

You will write three functions that give the values of the expressions of the lesson from their rules, without evaluating the expressions themselves. The checker compares each function with the direct value on forty-one numbers.

The idea. $\sin^{-1}\left(2x\sqrt{1 - x^2}\right)$ is $2\sin^{-1}x$ for $|x| \le \dfrac{1}{\sqrt{2}}$, then $\pi - 2\sin^{-1}x$ above that, and $-\pi - 2\sin^{-1}x$ below $-\dfrac{1}{\sqrt{2}}$. In the same way $\sin^{-1}(3x - 4x^3)$ has three regions with the border at $\pm\dfrac{1}{2}$ and the multiple $3$. And $\cos^{-1}\dfrac{1 - x^2}{1 + x^2}$ is $2\left|\tan^{-1}x\right|$.

Your tasks.

1. Finish fold(x) with the three regions.
2. Finish triple(x) with the three regions. The middle one is $|x| \le \dfrac{1}{2}$.
3. Finish cos_fraction(x). Use atan, and a modulus.