#include <bits/stdc++.h>
using namespace std;
const int MAXN = 100005;
struct Point {
  long long x, y;
  Point() {}
  Point(long long _x, long long _y) : x(_x), y(_y) {}
  Point operator-(const Point &t) const { return Point(x - t.x, y - t.y); }
  long long operator*(const Point &t) const { return x * t.y - y * t.x; }
} p[MAXN];
long long gcd(long long a, long long b) { return b ? gcd(b, a % b) : a; }
long long pointOnLine(Point p, Point q) {
  long long tx = abs(p.x - q.x), ty = abs(p.y - q.y);
  if (tx == 0) return ty;
  if (ty == 0) return tx;
  return gcd(tx, ty);
}
int main() {
  int n;
  scanf("%d", &n);
  for (int i = 0; i < n; i++) scanf("%I64d%I64d", &p[i].x, &p[i].y);
  long long tmp = 0;
  for (int i = 0; i < n; i++) {
    tmp += p[i] * p[(i + 1) % n];
    tmp -= pointOnLine(p[i], p[(i + 1) % n]);
  }
  tmp = (tmp + 2) / 2;
  double res = 0, cut = pow(2.0, -n), buf = 1 - (1LL * n * n + n + 2) / 2 * cut;
  for (int i = 0; i < n; i++) {
    long long cnt = 0;
    double q = 0.5;
    for (int j = (i + 1) % n, k = 2; k <= min(n - 1, 40);
         j = (j + 1) % n, k++) {
      res += cnt * ((q *= 0.5) - cut) / buf;
      long long now = (p[j] - p[i]) * (p[(j + 1) % n] - p[i]);
      now -= pointOnLine(p[i], p[j]);
      now -= pointOnLine(p[j], p[(j + 1) % n]);
      now -= pointOnLine(p[(j + 1) % n], p[i]);
      cnt += (now + 2) / 2;
      cnt += pointOnLine(p[i], p[(j + 1) % n]) - 1;
    }
  }
  res = tmp - res;
  printf("%.10f", res);
  return 0;
}
