// #pragma GCC target("avx")  // CPU 処理並列化
// #pragma GCC optimize("O3")  // CPU 処理並列化
// #pragma GCC optimize("unroll-loops")  // 条件処理の呼び出しを減らす
// #define BEGIN_STACK_EXTEND(size) void * stack_extend_memory_ = malloc(size);void * stack_extend_origin_memory_;char * stack_extend_dummy_memory_ = (char*)alloca((1+(int)(((long long)stack_extend_memory_)&127))*16);*stack_extend_dummy_memory_ = 0;asm volatile("mov %%rsp, %%rbx\nmov %%rax, %%rsp":"=b"(stack_extend_origin_memory_):"a"((char*)stack_extend_memory_+(size)-1024));
// #define END_STACK_EXTEND asm volatile("mov %%rax, %%rsp"::"a"(stack_extend_origin_memory_));free(stack_extend_memory_);

#include<stdio.h>
#include<math.h>
#include<algorithm>
#include<queue>
#include<deque>
#include<stack>
#include<string>
#include<string.h>
#include<vector>
#include<set>
#include<map>
#include<bitset>
#include<stdlib.h>
#include<cassert>
#include<time.h>
#include<bitset>
#include<numeric>
#include<unordered_set>
#include<unordered_map>
#include<complex>
using namespace std;
const long long mod=1000000009;
const long long inf=mod*mod;
const long long d2=(mod+1)/2;
const double EPS=1e-11;
const double INF=1e+10;
const double PI=acos(-1.0);
const int C_SIZE = 3100000;
const int UF_SIZE = 3100000;
namespace{
	long long fact[C_SIZE];
	long long finv[C_SIZE];
	long long inv[C_SIZE];
	inline long long Comb(int a,int b){
	 	if(a<b||b<0)return 0;
	 	return fact[a]*finv[b]%mod*finv[a-b]%mod;
	}
	void init_C(int n){
		fact[0]=finv[0]=inv[1]=1;
		for(int i=2;i<n;i++){
			inv[i]=(mod-(mod/i)*inv[mod%i]%mod)%mod;
		}
		for(int i=1;i<n;i++){
			fact[i]=fact[i-1]*i%mod;
			finv[i]=finv[i-1]*inv[i]%mod;
		}
	}
	long long pw(long long a,long long b){
		if(a<0LL)return 0;
		if(b<0LL)return 0;
		long long ret=1;
		while(b){
			if(b%2)ret=ret*a%mod;
			a=a*a%mod;
			b/=2;
		}
		return ret;
	}
	long long pw_mod(long long a,long long b,long long M){
		if(a<0LL)return 0;
		if(b<0LL)return 0;
		long long ret=1;
		while(b){
			if(b%2)ret=ret*a%M;
			a=a*a%M;
			b/=2;
		}
		return ret;
	}
	int pw_mod_int(int a,int b,int M){
		if(a<0)return 0;
		if(b<0)return 0;
		int ret=1;
		while(b){
			if(b%2)ret=(long long)ret*a%M;
			a=(long long)a*a%M;
			b/=2;
		}
		return ret;
	}
	int ABS(int a){return max(a,-a);}
	long long ABS(long long a){return max(a,-a);}
	double ABS(double a){return max(a,-a);}
	int sig(double r) { return (r < -EPS) ? -1 : (r > +EPS) ? +1 : 0; }
	int UF[UF_SIZE];
	void init_UF(int n){
		for(int i=0;i<n;i++)UF[i]=-1;
	}
	int FIND(int a){
		if(UF[a]<0)return a;
		return UF[a]=FIND(UF[a]);
	}
	void UNION(int a,int b){
		a=FIND(a);b=FIND(b);if(a==b)return;
		if(UF[a]>UF[b])swap(a,b);
		UF[a]+=UF[b];UF[b]=a;
	}
}
// ここから編集しろ
long long p[210000];
long long q[210000];

long long A[210000];
long long B[210000];
long long C[210000];
long long F[210000];
int g[210000];
int main(){
	int a;
	scanf("%d",&a);
	for(int i=0;i<a;i++){
		scanf("%lld",p+i);
	}
	for(int i=0;i<a;i++){
		scanf("%lld",q+i);
	}
	int at=0;
	for(int i=0;i<a;i++){
		if(p[at]<p[i])at=i;
	}
	int n=a+1;
	for(int i=0;i<=a;i++){
		A[i]=p[(at+i)%a];
		B[i]=q[(at+i)%a];
	}
	for(int i=2;i<n;i++){
		C[i]=2*C[i-1]-C[i-2]+2*B[i-1];
	}
	for(int i=0;i<n;i++){
		F[i]=A[i]-C[i];
	}
	int sz=0;
	g[0]=0;
	g[1]=1;
	sz=2;
	for(int i=2;i<n;i++){
		while(sz>1){
			if((F[g[sz-1]]-F[g[sz-2]])*(i-g[sz-1])>(F[i]-F[g[sz-1]])*(g[sz-1]-g[sz-2]))break;
			sz--;
		}
		g[sz++]=i;
	}
	// for(int i=0;i<sz;i++)printf("%d ",g[i]);
	double ret=0;
	for(int i=0;i<n-1;i++){
		int R=g[lower_bound(g,g+sz,i)-g];
		if(R==i){
			// printf("%d: %f\n",i,(double)F[i]+C[i]);
			ret+=F[i]+C[i];
			continue;
		}
		int L=g[lower_bound(g,g+sz,i)-g-1];
		double f=(double)(F[R]*(i-L)+F[L]*(R-i))/(R-L);
		// printf("%d: %f\n",i,f+C[i]);
		ret+=f+C[i];
	}
	printf("%.12f\n",ret/a);
}

Measured memory_bytes: 11824