Let $\iota(z)\in[0,1)$ denote the fractional part of $z\in\mathbb R$. Can we derive a formula for the toroidal distance of $\iota(z_1)-\iota(z_2)$?

Question

Let $$\left|x\right|_{(-1,\:1)}:=\left.\begin{cases}|x|&\text{, if }|x|\le\frac12;\\1-|x|&\text{, otherwise}\end{cases}\right\}\;\;\;\text{for }x\in(-1,1).$$ Now, let $$\iota:\mathbb R\to[0,1)\;,\;\;\;z\mapsto z-\lfloor z\rfloor.$$ Can we derive a formula for $|\iota(z_1)-\iota(z_2)|_{(-1,\:1)}$ for $z_1,z_2\in\mathbb R$?

We can clearly write $$z_i=k_i+y_i$$ for some $k_i\in\mathbb Z$ and $r_i\in[0,1)$. By construction $$\iota(z_i)=y_i\tag2.$$