Codeforces Round #499 (Div. 2) C题二分

最新推荐文章于 2022-10-24 20:22:18 发布

原创最新推荐文章于 2022-10-24 20:22:18 发布 · 535 阅读

0 ·

CC 4.0 BY-SA版权

二分，三分专栏收录该内容

4 篇文章

订阅专栏

本文介绍了一道编程竞赛题，涉及到Natasha驾驶火箭从地球出发，途经多个星球，最终返回地球的问题。关键在于计算所需的最小燃料量。通过理解题意，可以发现需要解一个方程，并利用二分搜索在1到1e9的范围内找到答案。题目中给出了行星数量、有效载荷重量、起飞和降落的燃料效率等参数，要求在不超过1e9吨燃料的情况下找到解决方案。如果无法完成飞行，则输出-1。

Fly

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

Natasha is going to fly on a rocket to Mars and return to Earth. Also, on the way to Mars, she will land on n−2n−2 intermediate planets. Formally: we number all the planets from 11 to nn. 11 is Earth, nn is Mars. Natasha will make exactly nn flights: 1→2→…n→11→2→…n→1.

Flight from xx to yy consists of two phases: take-off from planet xx and landing to planet yy. This way, the overall itinerary of the trip will be: the 11-st planet →→ take-off from the 11-st planet →→ landing to the 22-nd planet →→ 22-nd planet →→ take-off from the 22-nd planet →→ …… →→ landing to the nn-th planet →→ the nn-th planet →→ take-off from the nn-th planet →→ landing to the 11-st planet →→ the 11-st planet.

The mass of the rocket together with all the useful cargo (but without fuel) is mm tons. However, Natasha does not know how much fuel to load into the rocket. Unfortunately, fuel can only be loaded on Earth, so if the rocket runs out of fuel on some other planet, Natasha will not be able to return home. Fuel is needed to take-off from each planet and to land to each planet. It is known that 11 ton of fuel can lift off aiaitons of rocket from the ii-th planet or to land bibi tons of rocket onto the ii-th planet.

For example, if the weight of rocket is 99 tons, weight of fuel is 33 tons and take-off coefficient is 88 (ai=8ai=8), then 1.51.5 tons of fuel will be burnt (since 1.5⋅8=9+31.5⋅8=9+3). The new weight of fuel after take-off will be 1.51.5 tons.

Please note, that it is allowed to burn non-integral amount of fuel during take-off or landing, and the amount of initial fuel can be non-integral as well.

Help Natasha to calculate the minimum mass of fuel to load into the rocket. Note, that the rocket must spend fuel to carry both useful cargo and the fuel itself. However, it doesn't need to carry the fuel which has already been burnt. Assume, that the rocket takes off and lands instantly.

Input

The first line contains a single integer n (2≤n≤1000) — number of planets.

The second line contains the only integer m (1≤m≤1000) — weight of the payload.

The third line contains nn integers a1,a2,…,an (1≤ai≤1000), where aiai is the number of tons, which can be lifted off by one ton of fuel.

The fourth line contains nn integers b1,b2,…,bn (1≤bi≤1000), where bibi is the number of tons, which can be landed by one ton of fuel.

It is guaranteed, that if Natasha can make a flight, then it takes no more than 109109 tons of fuel.

Output

If Natasha can fly to Mars through (n−2)planets and return to Earth, print the minimum mass of fuel (in tons) that Natasha should take. Otherwise, print a single number −1.

It is guaranteed, that if Natasha can make a flight, then it takes no more than 109109 tons of fuel.

The answer will be considered correct if its absolute or relative error doesn't exceed 10−6. Formally, let your answer be pp, and the jury's answer be qq. Your answer is considered correct if |p−q|max(1,|q|)≤10−6.

Examples

input

2
12
11 8
7 5

output

10.0000000000

input

3 1 1 4 1 2 5 3

output
-1

input
6 2 4 6 3 3 5 6 2 6 3 6 5 3

output
85.4800000000

读懂题意之后可以发现要求出答案实际上是要解一个方程

很自然的想到二分求这个答案；

答案一定在1e9到1之间所以可以采用二分

坑点：题目保证答案的范围在1e9到1 所以二分的右端点应该是1e9+5；否则wa56（这个数据的答案是1e9，如果不这样会判断输出-1）

#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cstring>
using namespace std;
const int maxn=2e5+7;
const int INF=0x3f3f3f3f;
const double ESP=1e-8;
int a[maxn];
int b[maxn];
int n,m;
bool check(double x)
{
    //printf("%.10f\n",x);
    int i=1;//出发
    while(i<n)
    {
        x-=(x+m)/a[i];//起飞
        if(x<0) return false;
        x-=(x+m)/b[i+1];//降落
        if(x<0) return false;
        i++;
    }
    x-=(x+m)/a[n];
    if(x<0) return false;
    x-=(x+m)/b[1];
    if(x<0) return false;
    //cout<<x<<endl;
   // cout<<endl;
    return true;
}
int main()
{
    scanf("%d%d",&n,&m);
    for(int i=1;i<=n;i++)
        scanf("%d",&a[i]);
    for(int i=1;i<=n;++i)
        scanf("%d",&b[i]);
    double r=1e9+5;
    double l=0;
    double ans=-1;
    for(int i=0;i<=100;++i)
    {
        double mid=(l+r)/2;
        if(check(mid)) r=mid,ans=mid;
        else l=mid;
        //printf("%.10f %.10f\n",l,r);
    }
    printf("%.10f\n",ans);
}