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)); }

// class Func = TX -> TY
//   for any f, g: Func, x -> sgn(g(x) - f(x)) must be monotone
class LiChaoTree(Func) {
  import std.algorithm : swap;
  alias TX = Func.TX;
  alias TY = Func.TY;
  const(TX) L, R;
  class Tree {
    Tree l, r;
    Func f;
    this(Func f) {
      this.f = f;
    }
  }
  Tree root;

  // [L, R)
  this(TX L, TX R) {
    assert(L < R, "LiChaoTree: L < R must hold");
    this.L = L;
    this.R = R;
  }
  // Add g to the whole [L, R)
  void add(Func g) {
    root = add(root, L, R, g);
  }
  // Add g to [a, b)
  void add(TX a, TX b, Func g) {
    root = add(root, L, R, a, b, g);
  }
  // Get max at a
  TY opCall(TX a) {
    TY mx = TY.min;
    TX l = L, r = R;
    for (Tree u = root; u; ) {
      if (u.f) {
        const fX = u.f(a);
        if (mx < fX) mx = fX;
      }
      const mid = l + (r - l) / 2;
      if (a < mid) {
        u = u.l; r = mid;
      } else {
        u = u.r; l = mid;
      }
    }
    return mx;
  }

 private:
  Tree add(Tree u0, TX l, TX r, Func g) {
    if (!u0) return new Tree(g);
    TY gL = g(l), gR = g(r);
    for (Tree u = u0; ; ) {
      TY fL = u.f ? u.f(l) : TY.min, fR = u.f ? u.f(r) : TY.min;
      if (fL >= gL && fR >= gR) return u0;
      if (fL <= gL && fR <= gR) { u.f = g; return u0; }
      if (r - l == 1) { if (fL <= gL) { u.f = g; } return u0; }
      const mid = l + (r - l) / 2;
      TY fMid = u.f(mid), gMid = g(mid);
      if (fMid < gMid) {
        swap(u.f, g);
        if (!g) return u0;
        if (gL < fL) {
          if (!u.l) { u.l = new Tree(g); return u0; }
          u = u.l; r = mid; gL = fL; gR = fMid;
        } else {
          if (!u.r) { u.r = new Tree(g); return u0; }
          u = u.r; l = mid; gL = fMid; gR = fR;
        }
      } else {
        if (fL < gL) {
          if (!u.l) { u.l = new Tree(g); return u0; }
          u = u.l; r = mid; gR = gMid;
        } else {
          if (!u.r) { u.r = new Tree(g); return u0; }
          u = u.r; l = mid; gL = gMid;
        }
      }
    }
  }
  Tree add(Tree u, TX l, TX r, TX a, TX b, Func g) {
    if (b <= l || r <= a) return u;
    if (a <= l && r <= b) return add(u, l, r, g);
    if (u && u.f && u.f(l) >= g(l) && u.f(r) >= g(r)) return u;
    if (!u) u = new Tree(null);
    const mid = l + (r - l) / 2;
    u.l = add(u.l, l, mid, a, b, g);
    u.r = add(u.r, mid, r, a, b, g);
    return u;
  }
}

// y = p x + q
class Line {
  alias TX = long;
  alias TY = long;
  long p, q;
  this(long p, long q) {
    this.p = p;
    this.q = q;
  }
  TY opCall(TX x) {
    return p * x + q;
  }
}


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);
      }
      
      auto sumB0 = new long[N + 1];
      auto sumB1 = new long[N + 1];
      auto sumB2 = new long[N + 1];
      foreach (i; 0 .. N) {
        sumB0[i + 1] = sumB0[i] + B[i];
        sumB1[i + 1] = sumB1[i] + B[i] * i;
        sumB2[i + 1] = sumB2[i] + B[i] * i * i;
      }
      
      long f(int j, int i) {
        return (i - j) * (A[j] + A[i])
            + 2 * (sumB0[i] - sumB0[j + 1]) * j * i
            - 2 * (sumB1[i] - sumB1[j + 1]) * (j + i)
            + 2 * (sumB2[i] - sumB2[j + 1]);
      }
      
      auto dp = new long[N + 1];
      dp[] = long.min;
      dp[0] = 0;
      
      auto que = new int[N + 1];
      int qb, qe;
      foreach (i; 1 .. N + 1) {
        long g(int j) {
          return dp[j] + f(j, i);
        }
        que[qe++] = i - 1;
        for (; qe - qb >= 2 && g(que[qe - 2]) >= g(que[qe - 1]); --qe) {}
        dp[i] = g(que[qe - 1]);
      }
      const ans = cast(real)(dp[N]) / 2.0L / N;
      writefln("%.20f", ans);
      
      debug {
        auto brt = new long[N + 1];
        brt[] = long.min;
        brt[0] = 0;
        auto cut = new int[N + 1];
        cut[] = -1;
        foreach (i; 1 .. N + 1) {
          foreach (j; 0 .. i) {
            if (chmax(brt[i], brt[j] + f(j, i))) {
              cut[i] = j;
            }
          }
        }
        writeln("dp = ", dp);
        writeln("brt = ", brt);
        writeln("cut = ", cut);
      }
    }
  } catch (EOFException e) {
  }
}


Measured memory_bytes: 27204