(It was Pi Day, at least in some parts of the world.)
Hwang's remarkable identities are Machin-like formulas (correcting two typos in Mathworld's eqns 33 and 34). The $3$-term, $$\frac{\pi}4=(n+2)\arctan\frac12-n\arctan\frac13-(n+1)\arctan\frac17\tag1$$ and the $8$-term, $$\frac{\pi}4=a\arctan\frac12-b\arctan\frac13+c\arctan\frac15+d\arctan\frac17-e\arctan\frac1{239}\\ -f_1\arctan\frac18-f_2\arctan\frac1{18}+f_3\arctan\frac1{57}\tag2$$ where, $$\small\begin{aligned} f_1&=3 a - 2 b + c + d + e-5\\ f_2&=a - b + c - 2 e-2\\ f_3&=2 a - b + c + d - e-3 \end{aligned}$$
Q: Are there other parametric Hwang-type identities with at least one free parameter?
P.S. It should not be derived by using the identity: $\cot^{-1}x = 2\cot^{-1}(2x)-\cot^{-1}(4x^3+3x)$.