Let $0<x<1$ then define :
$$f(x)=x^{\operatorname{W}\left(x^{\frac{1}{x}}\right)}$$ $$g(x)=2-f(1-x)$$ $$h(x)=g'(0.5)(x-0.5)+g(0.5)$$ $$p(x)=g'(0.2)(x-0.2)+g(0.2)$$ Where we have the Lambert's function
Now denotes by $a$ the solution of $h(x)=p(x)$
We have $$a=0.378333357\cdots\simeq \frac{227}{600}=0.378333333\cdots$$
As you can see with the picture it's an attempt to evaluate an integral with tangent line wich conducts me to that .
Question :
How to explain that $a$ is so close to a rational number ?Is it a pure coincidence or not ?
Thanks
The analytical solution of $h(x)=p(x)$ is $$x=\frac 1 2+\frac{3 g'\left(\frac{1}{5}\right)+10 \left(g\left(\frac{1}{5}\right)-g\left(\frac{1}{2}\right)\right)}{10 \left(g'\left(\frac{1}{2}\right)-g'\left(\frac{1}{5}\right)\right)}$$ Rigorously computed, $$x=0.37833335765982278759994763708813774613464205787232352995953078149548\cdots$$
Rationalized it could be $$\left\{\frac{1}{3},\frac{3}{8},\frac{11}{29},\frac{14}{37},\frac{199}{526},\color{red}{\frac{22 7}{600}},\frac{18401}{48637},\frac{24984}{66037},\frac{25892}{684 37},\frac{155579}{411222},\frac{207363}{548096},\frac{2877190}{7604907},\cdots\right\}$$ Then why not $$ \sin \left(\frac{2557 }{20704}\pi\right)\quad \text{or} \quad \tan \left(\frac{538 }{4673}\pi\right)\quad \text{or} \quad W\left(\frac{16033}{29029}\right)$$