Company D recently changed the policy of annual leave days, which confused many people. So I make up these mathematical formulas to make it more clear.

Assume Amy’s entry date is year \(x_0\), month \(y_0\), day \(z_0\), then \(F(x)\), the annual leave days for year \(x\) is \(F(x) = \dfrac{\left\lfloor{2\left(f(x)+\dfrac{1}{4}\right)}\right\rfloor}{2} f(x) = \begin{cases} \dfrac{(x - x_0 + 9)(y_0 - 1) + (x - x_0 + 10)(12 - y_0 + 1)}{12}, &\text{if $x > x_0$;}\\ \dfrac{10(13 - y_0)}{12}, &\text{if $x = x_0 , z_0 <= 15$;}\\ \dfrac{10(12 - y_0)}{12}, &\text{if $x = x_0, z_0 > 15$;}\\ \end{cases}\)

The current available annual leave days \(G(x, y, z)\), for year \(x\), month \(y\), day \(z\), is \(G(x, y, z) = \dfrac{\left\lfloor2\left(g(x, y, z) + \dfrac{1}{4}\right)\right\rfloor}{2} + M(x, y, z) - N g(x, y, z) = \begin{cases} \dfrac{f(x)(y - 1)}{12}, &\text{if $x > x_0$, $z$ is not the last day of month $y$;}\\ \dfrac{f(x)y}{12}, &\text{if $x > x_0$, $z$ is the last day of month $y$;}\\ \max(0, \dfrac{10(y - y_0 - 1)}{12}), &\text{if $x = x_0$, $z_0 > 15$, $z$ is not the last day of month $y$;}\\ \dfrac{10(y - y_0)}{12}, &\text{if $x = x_0$, $z_0 <= 15$, $z$ is not the last day of month $y$;}\\ \dfrac{10(y - y_0)}{12}, &\text{if $x = x_0$, $z_0 > 15$, $z$ is the last day of month $y$;}\\ \dfrac{10(y - y_0 + 1)}{12}, &\text{if $x = x_0$, $z_0 <= 15$, $z$ is the last day of month $y$;}\\ \end{cases} M(x, y, z) = \begin{cases} G(x - 1, 12, 31), &\text{if $(y, z) <= (6, 30)$;}\\ 0, &\text{if $(y, z) > (6, 30)$;}\\ \end{cases} N = \text{Annual leave days have been used for this year}\)