import std.conv, std.functional, std.range, std.stdio, std.string;
import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons;
import core.bitop;

class EOFException : Throwable { this() { super("EOF"); } }
string[] tokens;
string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; }
int readInt() { return readToken.to!int; }
long readLong() { return readToken.to!long; }
real readReal() { return readToken.to!real; }

bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } }
bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } }

int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; }
int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); }
int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); }




void main() {
  try {
    for (; ; ) {
      const N = readInt();
      auto A = new long[N];
      foreach (i; 0 .. N) {
        A[i] = readLong();
      }
      auto B = new long[N];
      foreach (i; 0 .. N) {
        B[i] = readLong();
      }
      
      const im = A.maxIndex;
      A = A[im .. N] ~ A[0 .. im + 1];
      B = B[im .. N] ~ B[0 .. im];
      debug {
        writeln("A = ", A);
        writeln("B = ", B);
      }
      
      // fs[i] = -B[i] + (fs[i - 1] + fs[i + 1]) / 2
      auto fs = new long[N + 1];
      foreach (i; 1 .. N) {
        fs[i + 1] = 2 * (fs[i] + B[i]) - fs[i - 1];
      }
      
      auto ys = new long[N + 1];
      foreach (i; 0 .. N + 1) {
        ys[i] = A[i] - fs[i];
      }
      debug {
        writeln("fs = ", fs);
        writeln("ys = ", ys);
      }
      
      long tri(int i0, int i1, int i2) {
        return (i1 - i0) * (ys[i2] - ys[i0]) - (i2 - i0) * (ys[i1] - ys[i0]);
      }
      
      int jsLen;
      auto js = new int[N + 1];
      foreach (i; 0 .. N + 1) {
        for (; jsLen >= 2 && tri(js[jsLen - 2], js[jsLen - 1], i) >= 0; --jsLen) {}
        js[jsLen++] = i;
      }
      debug {
        writeln("js = ", js[0 .. jsLen]);
      }
      
      long ans;
      foreach (k; 0 .. jsLen - 1) {
        const i0 = js[k];
        const i1 = js[k + 1];
        ans += (i1 - i0) * (ys[i0] + ys[i1]);
      }
      foreach (i; 0 .. N) {
        ans += fs[i];
        ans += fs[i + 1];
      }
      writefln("%.20f", ans / 2.0L / N);
    }
  } catch (EOFException e) {
  }
}
