Delivering Carcinogen

题目描述

Qwerty the Ranger arrived to the Diatar system with a very important task. He should deliver a special carcinogen for scientific research to planet Persephone. This is urgent, so Qwerty has to get to the planet as soon as possible. A lost day may fail negotiations as nobody is going to pay for an overdue carcinogen. You can consider Qwerty's ship, the planet Persephone and the star Diatar points on a plane. Diatar is located in the origin of coordinate axes — at point $ (0,0) $ . Persephone goes round Diatar along a circular orbit with radius $ R $ in the counter-clockwise direction at constant linear speed $ v_{p} $ (thus, for instance, a full circle around the star takes  of time). At the initial moment of time Persephone is located at point $ (x_{p},y_{p}) $ . At the initial moment of time Qwerty's ship is at point $ (x,y) $ . Qwerty can move in any direction at speed of at most $ v $ ( $ v>v_{p} $ ). The star Diatar is hot (as all stars), so Qwerty can't get too close to it. The ship's metal sheathing melts at distance $ r $ ( $ r<R $ ) from the star. Find the minimum time Qwerty needs to get the carcinogen to planet Persephone.

输入输出格式

输入格式

The first line contains space-separated integers $ x_{p} $ , $ y_{p} $ and $ v_{p} $ ( $ -10^{4}<=x_{p},y_{p}<=10^{4} $ , $ 1<=v_{p}&lt;10^{4} $ ) — Persephone's initial position and the speed at which it goes round Diatar. The second line contains space-separated integers $ x $ , $ y $ , $ v $ and $ r $ ( $ -10^{4}<=x,y<=10^{4} $ , $ 1&lt;v<=10^{4} $ , $ 1<=r<=10^{4} $ ) — The intial position of Qwerty's ship, its maximum speed and the minimum safe distance to star Diatar. It is guaranteed that $ r^{2}&lt;x^{2}+y^{2} $ , $ r^{2}&lt;x_{p}^{2}+y_{p}^{2} $ and $ v_{p}&lt;v $ .

输出格式

Print a single real number — the minimum possible delivery time. The answer will be considered valid if its absolute or relative error does not exceed $ 10^{-6} $ .

输入输出样例

输入样例 #1

10 0 1
-10 0 2 8

输出样例 #1

9.584544103

输入样例 #2

50 60 10
50 60 20 40

输出样例 #2

0.000000000