The blancmange curve is the set of points $(x,y)$ such that $0\le x\le 1$ and $y=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{s(2^{n}x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.

The area under the blancmange curve is equal to ½, shown in pink in the diagram below.

Let $C$ be the circle with centre $(\frac{1}{4},\frac{1}{2})$ and radius $\frac{1}{4}$, shown in black in the diagram.

What area under the blancmange curve is enclosed by $C$?
Give your answer rounded to eight decimal places in the form 0.abcdefgh.