For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor $f.$ The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which can all be represented by another dimensionless quantity, the Reynolds number $Re.$ A formula that predicts $f$ given $Re$ is the von Karman equation:

$\frac{1}{\sqrt[\phantom{2}]{f}}=4lo{g}_{10}(Re\sqrt[\phantom{2}]{f})-0\mathrm{.}4$

Typical values for the Reynolds number for turbulent flow are $10,000$ to $500,000$ and for the Fanning friction factor are $0\mathrm{.}001$ to $0\mathrm{.}01.$ Develop a function that uses regula$-$falsi to solve for $f$ given a user$-$input value of Re between $2500$ and $1,000,000.$ Design the function so that it ensures that the absolute error in the result is less than $0\mathrm{.}000005.$