#include <bits/stdc++.h>
using namespace std;
template <class T>
void splitstr(const string &s, vector<T> &out) {
  istringstream in(s);
  out.clear();
  copy(istream_iterator<T>(in), istream_iterator<T>(), back_inserter(out));
}
template <class T>
T gcd(T a, T b) {
  return b ? gcd(b, a % b) : a;
}
static void redirect(int argc, const char **argv) {
  ios::sync_with_stdio(false);
  if (argc > 1) {
    static filebuf f;
    f.open(argv[1], ios::in);
    cin.rdbuf(&f);
    if (!cin) {
      cerr << "Failed to open '" << argv[1] << "'" << endl;
      exit(1);
    }
  }
  if (argc > 2) {
    static filebuf f;
    f.open(argv[2], ios::out | ios::trunc);
    cout.rdbuf(&f);
    if (!cout) {
      cerr << "Failed to open '" << argv[2] << "'" << endl;
    }
  }
}
static long long cross(const complex<long long> &a,
                       const complex<long long> &b) {
  return imag(conj(a) * b);
}
static int next(int x, int N) {
  if (++x == N) x = 0;
  return x;
}
int main(int argc, const char **argv) {
  redirect(argc, argv);
  long long N;
  cin >> N;
  vector<complex<long long> > pnts;
  for (int i = 0; i < N; i++) {
    int x, y;
    cin >> x >> y;
    pnts.emplace_back(x, y);
  }
  double den = 1.0 - ldexp(1.0, -N) * (N * (N - 1) / 2 + N + 1);
  double A = 0.0;
  for (int i = 0; i < N; i++) {
    int j = i;
    for (int d = 1; d < min(40LL, N - 1); d++) {
      j = next(j, N);
      double p = (ldexp(1.0, -1 - d) - ldexp(1.0, -N)) / den;
      double a = 0.5 * cross(pnts[i], pnts[j]);
      complex<long long> delta = pnts[j] - pnts[i];
      long long b = gcd(llabs(delta.real()), llabs(delta.imag()));
      A += p * (a - 0.5 * b);
    }
  }
  cout << setprecision(9) << fixed << A + 1.0 << '\n';
  return 0;
}
