A function of two variables is given:
$$\chi(\xi, P) = -2 \left(-3 P \,\xi^{4}+3 P^{2} \xi^{2}-P^{3}\right)^{\frac{1}{6}} \cos \! \left(\frac{\pi}{3}+\frac{\arctan \! \left(\frac{\sqrt{-3 P \,\xi^{4}+3 P^{2} \xi^{2}-P^{3}}}{\xi^{3}-\xi}\right)}{3}\right)$$
I know the limits of changing each variable: $$P = -325 \ldots -3$$ $$\xi = 1 \ldots 3$$
I took two boundary values $P$ and plotted:
Now I am faced with the task of simplifying this expression. I want to replace it with a simpler one. I see that with increasing $\xi$, the dependence also changes from quadratic, or even from cubic to linear. But how can I make a new formula, which would also involve the variables $\xi$ and $P$, such that the graph repeats the original one as much as possible?
By the selection method , I derived such a function:
$$ \chi_{new} (\xi, P) = (\xi^2 - \xi) \cdot \frac{1}{P}$$
Now I have built a new function and an old one and compared them. In the area of small $\xi$ values, everything is fine, the functions are very close, but then the discrepancies begin:
(The dotted line indicates the new function, the solid line - the old one)
For the area of large hh values, I have compiled another function:
Conclusion:
The graph can be divided into several parts, into two or three, for each of them to choose its own function.
I do not know how to select the function correctly, I only followed the curvature, slope and linear vertical displacement
Question: By what method can I get the closest possible simple functions? Am I doing the right thing?