#include <bits/stdc++.h>
int n;
int d[100000][2];
double pw[100000];
double ra, rb;
int GCD(int a, int b) {
  if (b == 0) return a;
  return a % b ? GCD(b, a % b) : b;
}
int main() {
  scanf("%d", &n);
  for (int i = 0; i < n; i++) scanf("%d%d", &d[i][0], &d[i][1]);
  pw[0] = 1;
  for (int i = 1; i < 100000; i++) pw[i] = pw[i - 1] / 2;
  for (int k = 1; k < 150 && k <= n - 2; k++) {
    double rra = 0, rrb = 0;
    for (int i = 0; i < n; i++) {
      int j = i + k;
      if (j >= n) j -= n;
      rra += ((double)d[i][0] * d[j][1] - (double)d[i][1] * d[j][0]) / 2;
      int dx = d[i][0] - d[j][0];
      int dy = d[i][1] - d[j][1];
      if (dx < 0) dx = -dx;
      if (dy < 0) dy = -dy;
      rrb += GCD(dx, dy);
    }
    ra += rra * pw[k + 1] * (1 - pw[n - k - 1]);
    rb += rrb * pw[k + 1] * (1 - pw[n - k - 1]);
  }
  double r = ra - rb / 2;
  if (n < 63) {
    long long p = 1;
    for (int i = 0; i < n; i++) p *= 2;
    long long q = p - 1 - n - n * (n - 1) / 2;
    r *= (double)p / q;
  }
  printf("%.15f", r + 1);
  return 0;
}
