  In this work, we develop a new iterative method for computing the digits of
$\pi$ by argument reduction of the tangent function. This method combines a
modified version of the iterative formula for $\pi$ with squared convergence
that we proposed in a previous work and a leading arctangent term from the
Machin-like formula. The computational test we performed shows that algorithmic
implementation can provide more than $17$ digits of $\pi$ per increment.
Mathematica codes, showing the convergence rate for computing the digits of
$\pi$, are presented.
