A12019 | Barcelonian Distance

In this problem we consider a very simplified model of Barcelona city.

Barcelona can be represented as a plane with streets of kind $x = c$ and $y = c$ for every integer $c$ (that is, the rectangular grid). However, there is a detail which makes Barcelona different from Manhattan. There is an avenue called Avinguda Diagonal which can be represented as a the set of points $(x, y)$ for which $ax + by + c = 0$ .

One can walk along streets, including the avenue. You are given two integer points $A$ and $B$ somewhere in Barcelona. Find the minimal possible distance one needs to travel to get to $B$ from $A$ .

The first line contains three integers $a$ , $b$ and $c$ ( $-10^9\leq a, b, c\leq 10^9$ , at least one of $a$ and $b$ is not zero) representing the Diagonal Avenue.

The next line contains four integers $x_1$ , $y_1$ , $x_2$ and $y_2$ ( $-10^9\leq x_1, y_1, x_2, y_2\leq 10^9$ ) denoting the points $A = (x_1, y_1)$ and $B = (x_2, y_2)$ .

Find the minimum possible travel distance between $A$ and $B$ . Your answer is considered correct if its absolute or relative error does not exceed $10^{-6}$ .

Formally, let your answer be $a$ , and the jury's answer be $b$ . Your answer is accepted if and only if $\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$ .