Problem p00805 (C++) — buggy submission, verdict: Memory Limit Exceeded

#include "bits/stdc++.h"
#include <unordered_map>
#include <unordered_set>
#pragma warning(disable : 4996)
using namespace std;
using ld = long double;
const ld eps = 1e-9;

/* 幾何の基本 */

#include <complex>

typedef complex<ld> Point;

const ld pi = acos(-1.0);
const ld dtop = pi / 180.;
const ld ptod = 1. / dtop;

namespace std {
bool operator<(const Point &lhs, const Point &rhs) {
  if (lhs.real() < rhs.real() - eps)
    return true;
  if (lhs.real() > rhs.real() + eps)
    return false;
  return lhs.imag() < rhs.imag();
}
} // namespace std

// 点の入力
Point input_Point() {
  ld x, y;
  cin >> x >> y;
  return Point(x, y);
}

// 誤差つき等号判定
bool eq(const ld a, const ld b) { return (abs(a - b) < eps); }

// 内積
ld dot(const Point &a, const Point &b) { return real(conj(a) * b); }

// 外積
ld cross(const Point &a, const Point &b) { return imag(conj(a) * b); }

// 直線の定義
class Line {
public:
  Point a, b;
  Line() : a(Point(0, 0)), b(Point(0, 0)) {}
  Line(Point a, Point b) : a(a), b(b) {}
  Point operator[](const int _num) const {
    if (_num == 0)
      return a;
    else if (_num == 1)
      return b;
    else {
      assert(false);
      return Point();
    }
  }
};

// 円の定義
class Circle {
public:
  Point p;
  ld r;
  Circle() : p(Point(0, 0)), r(0) {}
  Circle(Point p, ld r) : p(p), r(r) {}
};

// ccw
// 1: a,b,cが反時計周りの順に並ぶ
//-1: a,b,cが時計周りの順に並ぶ
// 2: c,a,bの順に直線に並ぶ
//-2: a,b,cの順に直線に並ぶ
// 0: a,c,bの順に直線に並ぶ
int ccw(const Point &a, const Point &b, const Point &c) {
  const Point nb(b - a);
  const Point nc(c - a);
  if (cross(nb, nc) > eps)
    return 1; // a,b,cが反時計周りの順に並ぶ
  if (cross(nb, nc) < -eps)
    return -1; // a,b,cが時計周りの順に並ぶ
  if (dot(nb, nc) < 0)
    return 2; // c,a,bの順に直線に並ぶ
  if (norm(nb) < norm(nc))
    return -2; // a,b,cの順に直線に並ぶ
  return 0;    // a,c,bの順に直線に並ぶ
}

/* 交差判定 */

// 直線と直線の交差判定
bool isis_ll(const Line &l, const Line &m) {
  return !eq(cross(l.b - l.a, m.b - m.a), 0);
}

// 直線と線分の交差判定
bool isis_ls(const Line &l, const Line &s) {
  return isis_ll(l, s) &&
         (cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < eps);
}

// 線分と線分の交差判定
bool isis_ss(const Line &s, const Line &t) {
  return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
         ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}

// 点の直線上判定
bool isis_lp(const Line &l, const Point &p) {
  return (abs(cross(l.b - p, l.a - p)) < eps);
}

// 点の線分上判定
bool isis_sp(const Line &s, const Point &p) {
  return (abs(s.a - p) + abs(s.b - p) - abs(s.b - s.a) < eps);
}

// 垂線の足
Point proj(const Line &l, const Point &p) {
  ld t = dot(p - l.a, l.b - l.a) / norm(l.a - l.b);
  return l.a + t * (l.b - l.a);
}

// 線対象の位置にある点
Point reflect(const Line &l, const Point &p) {
  Point pr = proj(l, p);
  return pr * 2.l - p;
}

// 直線と直線の交点
Point is_ll(const Line &s, const Line &t) {
  Point sv = s.b - s.a, tv = t.b - t.a;
  assert(cross(sv, tv) != 0);
  return s.a + sv * cross(tv, t.a - s.a) / cross(tv, sv);
}
// 直線と直線の交点
vector<Point> is_ll2(const Line &s, const Line &t) {
  Point sv = s.b - s.a, tv = t.b - t.a;
  if (cross(sv, tv) != 0)
    return vector<Point>(1, is_ll(s, t));
  else {
    vector<Point> ans;
    for (int k = 0; k < 2; ++k) {
      if (isis_sp(s, t[k]) && find(ans.begin(), ans.end(), t[k]) == ans.end())
        ans.push_back(t[k]);
      if (isis_sp(t, s[k]) && find(ans.begin(), ans.end(), s[k]) == ans.end())
        ans.push_back(s[k]);
    }
    return ans;
  }
}
// 線分と線分の交点
// 　重なってる部分あるとassert(false)
Point is_ss(const Line &s, const Line &t) {
  if (isis_ss(s, t)) {
    for (int k = 0; k < 2; ++k) {
      for (int l = 0; l < 2; ++l) {
        if (s[k] == t[l])
          return s[k];
      }
    }
    return is_ll(s, t);
  } else {
    // 先にisis_ssしてね
    assert(false);
    return Point(0, 0);
  }
}
// 線分と線分の交点
vector<Point> is_ss2(const Line &s, const Line &t) {
  vector<Point> kouho(is_ll2(s, t));
  vector<Point> ans;
  for (auto p : kouho) {
    if (isis_sp(s, p) && isis_sp(t, p))
      ans.emplace_back(p);
  }
  return ans;
}
// 直線と点の距離
ld dist_lp(const Line &l, const Point &p) { return abs(p - proj(l, p)); }

// 直線と直線の距離
ld dist_ll(const Line &l, const Line &m) {
  return isis_ll(l, m) ? 0 : dist_lp(l, m.a);
}

// 直線と線分の距離
ld dist_ls(const Line &l, const Line &s) {
  return isis_ls(l, s) ? 0 : min(dist_lp(l, s.a), dist_lp(l, s.b));
}

// 線分と点の距離
ld dist_sp(const Line &s, const Point &p) {
  Point r = proj(s, p);
  return isis_sp(s, r) ? abs(r - p) : min(abs(s.a - p), abs(s.b - p));
}

// 線分と線分の距離
ld dist_ss(const Line &s, const Line &t) {
  if (isis_ss(s, t))
    return 0;
  return min(
      {dist_sp(s, t.a), dist_sp(s, t.b), dist_sp(t, s.a), dist_sp(t, s.b)});
}

// 直線と直線の二等分線のベクトル
Line line_bisection(const Line &s, const Line &t) {
  const Point laglanju(is_ll(s, t));
  const Point avec =
      !(abs(laglanju - s[0]) < eps) ? s[0] - laglanju : s[1] - laglanju;
  const Point bvec =
      !(abs(laglanju - t[0]) < eps) ? t[0] - laglanju : t[1] - laglanju;

  return Line(laglanju, laglanju + (abs(bvec) * avec + abs(avec) * bvec) /
                                       (abs(avec) + abs(bvec)));
}
// 点と点の垂直二等分線　aを左に見ながら
Line point_bisection(const Point &a, const Point &b) {
  const Point cen((a + b) / 2.l);
  const Point vec = (b - a) * Point(0, 1);
  return Line(cen, cen + vec);
}

// 三つの点からなる外心
Point outer_center(const vector<Point> &ps) {
  assert(ps.size() == 3);
  Line l1 = point_bisection(ps[0], ps[1]);
  Line l2 = point_bisection(ps[1], ps[2]);
  return is_ll(l1, l2);
}

// 三つの直線からなる内心
// 三つの直線が並行でないことは確かめといてね
Point inner_center(const vector<Line> &ls) {
  vector<Point> vertics;
  for (int i = 0; i < static_cast<int>(ls.size()); ++i) {
    vertics.push_back(is_ll(ls[i], ls[(i + 1) % 3]));
  }
  if (vertics[0] == vertics[1] || vertics[1] == vertics[2] ||
      vertics[2] == vertics[0])
    return vertics[0];
  Line bi1(line_bisection(Line(vertics[0], vertics[1]),
                          Line(vertics[0], vertics[2])));
  Line bi2(line_bisection(Line(vertics[1], vertics[2]),
                          Line(vertics[1], vertics[0])));
  if (bi1[0] == bi2[0])
    return bi1[0];
  else {
    return is_ll(bi1, bi2);
  }
}

// 三つの直線からなる傍心
// 三つの直線が並行でないことは確かめといてね
vector<Point> ex_center(const vector<Line> &ls) {
  vector<Point> vertics;
  for (int i = 0; i < static_cast<int>(ls.size()); ++i) {
    vertics.push_back(is_ll(ls[i], ls[(i + 1) % 3]));
  }
  if (abs(vertics[0] - vertics[1]) < eps ||
      abs(vertics[1] - vertics[2]) < eps ||
      (abs(vertics[2] - vertics[0]) < eps))
    return vector<Point>();
  vector<Point> ecs;
  for (int i = 0; i < 3; ++i) {
    Line bi1(line_bisection(
        Line(vertics[i], vertics[i] * 2.0l - vertics[(i + 2) % 3]),
        Line(vertics[i], vertics[(i + 1) % 3])));
    Line bi2(
        line_bisection(Line(vertics[(i + 1) % 3],
                            vertics[(i + 1) % 3] * 2.0l - vertics[(i + 2) % 3]),
                       Line(vertics[(i + 1) % 3], vertics[i])));
    ecs.push_back(is_ll(bi1, bi2));
  }
  return ecs;
}

// a,b:並行
// c:並行でない
// 三つの直線から同距離の位置を求める。
vector<Point> same_dis(const vector<Line> &ls) {
  vector<Point> vertics;
  vertics.push_back(is_ll(ls[0], ls[2]));
  vertics.push_back(is_ll(ls[1], ls[2]));

  if (abs(vertics[0] - vertics[1]) < eps)
    return vector<Point>{vertics[0]};
  Line bis(line_bisection(ls[0], ls[1]));
  vector<Point> ecs;

  Line abi(line_bisection(Line(vertics[0], vertics[1]), ls[0]));
  ecs.push_back(is_ll(bis, abi));

  Line bbi(
      line_bisection(Line(vertics[0], 2.l * vertics[0] - vertics[1]), ls[0]));
  ecs.push_back(is_ll(bis, bbi));

  return ecs;
}
/* 円 */

// 円と円の交点
vector<Point> is_cc(const Circle &c1, const Circle &c2) {
  vector<Point> res;
  ld d = abs(c1.p - c2.p);
  ld rc = (d * d + c1.r * c1.r - c2.r * c2.r) / (2 * d);
  ld dfr = c1.r * c1.r - rc * rc;
  if (abs(dfr) < eps)
    dfr = 0.0;
  else if (dfr < 0.0)
    return res; // no intersection
  ld rs = sqrt(dfr);
  Point diff = (c2.p - c1.p) / d;
  res.push_back(c1.p + diff * Point(rc, rs));
  if (dfr != 0.0)
    res.push_back(c1.p + diff * Point(rc, -rs));
  return res;
}

/*
点が円の中にいるか
0 => out
1 => on
2 => in*/
int is_in_Circle(const Point &p, const Circle &cir) {
  ld dis = abs(cir.p - p);
  if (dis > cir.r + eps)
    return 0;
  else if (dis < cir.r - eps)
    return 2;
  else
    return 1;
}
int is_in_Circle(const Circle &cir, const Point &p) {
  ld dis = abs(cir.p - p);
  if (dis > cir.r + eps)
    return 0;
  else if (dis < cir.r - eps)
    return 2;
  else
    return 1;
}
/*
円lcが円rcの中にいるか
0 => out
1 => on
2 => in*/
int Circle_in_Circle(const Circle &lc, const Circle &rc) {
  ld dis = abs(lc.p - rc.p);
  if (dis < rc.r - lc.r - eps)
    return 2;
  else if (dis > rc.r - lc.r + eps)
    return 0;
  else
    return 1;
}

// 円と直線の交点
vector<Point> is_lc(const Circle &c, const Line &l) {
  vector<Point> res;
  ld d = dist_lp(l, c.p);
  if (d < c.r + eps) {
    ld len = (d > c.r) ? 0.0 : sqrt(c.r * c.r - d * d); // safety;
    Point nor = (l.a - l.b) / abs(l.a - l.b);
    res.push_back(proj(l, c.p) + len * nor);
    res.push_back(proj(l, c.p) - len * nor);
  }
  return res;
}

// 円と線分の距離
vector<Point> is_sc(const Circle &c, const Line &l) {
  vector<Point> v = is_lc(c, l), res;
  for (Point p : v)
    if (isis_sp(l, p))
      res.push_back(p);
  return res;
}

// 円と点の接線
vector<Line> tangent_cp(const Circle &c, const Point &p) {
  vector<Line> ret;
  Point v = c.p - p;
  ld d = abs(v);
  ld l = sqrt(norm(v) - c.r * c.r);
  if (isnan(l)) {
    return ret;
  }
  Point v1 = v * Point(l / d, c.r / d);
  Point v2 = v * Point(l / d, -c.r / d);
  ret.push_back(Line(p, p + v1));
  if (l < eps)
    return ret;
  ret.push_back(Line(p, p + v2));
  return ret;
}

// 円と円の接線
vector<Line> tangent_cc(const Circle &c1, const Circle &c2) {
  vector<Line> ret;
  if (abs(c1.p - c2.p) - (c1.r + c2.r) > -eps) {
    Point center = (c1.p * c2.r + c2.p * c1.r) / (c1.r + c2.r);
    ret = tangent_cp(c1, center);
  }
  if (abs(c1.r - c2.r) > eps) {
    Point out = (-c1.p * c2.r + c2.p * c1.r) / (c1.r - c2.r);
    vector<Line> nret = tangent_cp(c1, out);
    ret.insert(ret.end(), nret.begin(), nret.end());
  } else {
    Point v = c2.p - c1.p;
    v /= abs(v);
    Point q1 = c1.p + v * Point(0, 1) * c1.r;
    Point q2 = c1.p + v * Point(0, -1) * c1.r;
    ret.push_back(Line(q1, q1 + v));
    ret.push_back(Line(q2, q2 + v));
  }
  return ret;
}
// 二つの円の重なり面積
ld two_Circle_area(const Circle &l, const Circle &r) {
  ld dis = abs(l.p - r.p);
  if (dis > l.r + r.r)
    return 0;
  else if (dis + r.r < l.r) {
    return r.r * r.r * pi;
  } else if (dis + l.r < r.r) {
    return l.r * l.r * pi;
  } else {
    ld ans = (l.r) * (l.r) *
                 acos((dis * dis + l.r * l.r - r.r * r.r) / (2 * dis * l.r)) +
             (r.r) * (r.r) *
                 acos((dis * dis + r.r * r.r - l.r * l.r) / (2 * dis * r.r)) -
             sqrt(4 * dis * dis * l.r * l.r -
                  (dis * dis + l.r * l.r - r.r * r.r) *
                      (dis * dis + l.r * l.r - r.r * r.r)) /
                 2;
    return ans;
  }
}

/* 多角形 */

typedef vector<Point> Polygon;

// 多角形(複数の点)の最小包含円をO(N=頂点数)で求めるアルゴリズム
// 同一直線上に三つの点がないこと
#include <random>
Circle welzl(vector<Point> ps) {
  struct solver {
    Circle solve(vector<Point> &ps, vector<Point> &rs) {
      if (ps.empty() || rs.size() == 3) {
        if (rs.size() == 1) {
          return Circle(Point(rs[0]), 0);
        } else if (rs.size() == 2) {
          return Circle((rs[0] + rs[1]) / 2.0l, abs(rs[1] - rs[0]) / 2);
        } else if (rs.size() == 3) {
          vector<Line> ls(3);
          for (int i = 0; i < 3; ++i) {
            ls[i] = Line(rs[i], rs[(i + 1) % 3]);
          }
          Point center = outer_center(rs);
          return Circle(center, abs(center - rs[0]));
        } else {
          return Circle(Point(), 0);
        }
      } else {

        Point p_ba = ps.back();
        ps.pop_back();
        Circle d = solve(ps, rs);
        ps.push_back(p_ba);
        if (is_in_Circle(d, p_ba)) {
          return d;
        } else {
          rs.push_back(p_ba);
          ps.pop_back();
          auto ans = solve(ps, rs);
          ps.push_back(p_ba);
          rs.pop_back();
          return ans;
        }
      }
    }
  } so;
  std::random_device rd;
  std::mt19937 mt(rd());
  shuffle(ps.begin(), ps.end(), mt);
  vector<Point> rs;
  Circle ans = so.solve(ps, rs);
  return ans;
}
// 面積
ld get_area(const Polygon &p) {
  ld res = 0;
  int n = p.size();
  for (int j = 0; j < n; ++j)
    res += cross(p[j], p[(j + 1) % n]);
  return res / 2;
}

// 多角形の回転方向
bool is_counter_clockwise(const Polygon &poly) {
  ld angle = 0;
  int n = poly.size();
  for (int i = 0; i < n; ++i) {
    Point a = poly[i], b = poly[(i + 1) % n], c = poly[(i + 2) % n];
    angle += arg((c - b) / (b - a));
  }
  return angle > eps;
}

// 円の内外判定
/*0 => out
1 => on
2 => in*/
int is_in_Polygon(const Polygon &poly, const Point &p) {
  ld angle = 0;
  int n = poly.size();
  for (int i = 0; i < n; ++i) {
    Point a = poly[i], b = poly[(i + 1) % n];
    if (isis_sp(Line(a, b), p))
      return 1;
    angle += arg((b - p) / (a - p));
  }
  return eq(angle, 0) ? 0 : 2;
}
// 円の内外判定2　高速
enum { out, on, in };
int convex_contains(const Polygon &P, const Point &p) {
  const int n = P.size();
  Point g = (P[0] + P[n / 3] + P[2 * n / 3]) / 3.0l; // inner-point
  int a = 0, b = n;
  while (a + 1 < b) { // invariant: c is in fan g-P[a]-P[b]
    int c = (a + b) / 2;
    if (cross(P[a] - g, P[c] - g) > 0) { // angle < 180 deg
      if (cross(P[a] - g, p - g) > 0 && cross(P[c] - g, p - g) < 0)
        b = c;
      else
        a = c;
    } else {
      if (cross(P[a] - g, p - g) < 0 && cross(P[c] - g, p - g) > 0)
        a = c;
      else
        b = c;
    }
  }
  b %= n;
  if (cross(P[a] - p, P[b] - p) < 0)
    return 0;
  if (cross(P[a] - p, P[b] - p) > 0)
    return 2;
  return 1;
}

// 凸包
// 点や線を返すことも有り得るので注意
Polygon convex_hull(vector<Point> ps) {
  int n = ps.size();
  int k = 0;
  sort(ps.begin(), ps.end());
  Polygon ch(2 * n);
  for (int i = 0; i < n; ch[k++] = ps[i++])
    while (k >= 2 && ccw(ch[k - 2], ch[k - 1], ps[i]) <= 0)
      --k;
  for (int i = n - 2, t = k + 1; i >= 0; ch[k++] = ps[i--])
    while (k >= t && ccw(ch[k - 2], ch[k - 1], ps[i]) <= 0)
      --k;
  ch.resize(k - 1);
  return ch;
}

// 凸カット
vector<Polygon> convex_cut(const Polygon &ps, const Line &l) {
  int n = ps.size();
  Polygon q;
  Polygon r;
  for (int i = 0; i < n; ++i) {
    Point a = ps[i], b = ps[(i + 1) % n];
    Line m = Line(a, b);
    if (ccw(l.a, l.b, a) != -1)
      q.push_back(a);
    if (ccw(l.a, l.b, a) != 1)
      r.push_back(a);
    if (ccw(l.a, l.b, a) * ccw(l.a, l.b, b) < 0 && isis_ll(l, m)) {
      q.push_back(is_ll(l, m));
      r.push_back(is_ll(l, m));
    }
  }
  const vector<Polygon> polys{q, r};
  return polys;
}

int main() {
  while (true) {
    int N;
    cin >> N;
    if (!N)
      break;
    vector<ld> us, ds, ls, rs;
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      ds.push_back(n);
    }
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      us.push_back(n);
    }
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      ls.push_back(n);
    }
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      rs.push_back(n);
    }

    Polygon start_poly{Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1)};

    vector<Polygon> polys{start_poly};
    for (int i = 0; i < N; ++i) {
      vector<Polygon> nextpolys;
      for (auto po : polys) {
        Line l(Point(ds[i], 0), Point(us[i], 1));
        auto pos = convex_cut(po, l);

        for (auto aa : pos) {
          if (abs(get_area(aa)) > eps) {

            nextpolys.push_back(aa);
          }
        }
      }
      polys = nextpolys;
    }
    for (int i = 0; i < N; ++i) {
      vector<Polygon> nextpolys;
      for (auto po : polys) {
        Line l(Point(0, ls[i]), Point(1, rs[i]));
        auto pos = convex_cut(po, l);

        for (auto aa : pos) {
          if (abs(get_area(aa)) > eps) {

            nextpolys.push_back(aa);
          }
        }
      }
      polys = nextpolys;
    }
    ld ans = 0;
    for (auto poly : polys) {
      ans = max(ans, abs(get_area(poly)));
    }
    cout << setprecision(6) << fixed << ans << endl;
  }
  return 0;
}


Error: MLE

Fixed submission:

#include "bits/stdc++.h"
#include <unordered_map>
#include <unordered_set>
#pragma warning(disable : 4996)
using namespace std;
using ld = long double;
const ld eps = 1e-9;

/* 幾何の基本 */

#include <complex>

typedef complex<ld> Point;

const ld pi = acos(-1.0);
const ld dtop = pi / 180.;
const ld ptod = 1. / dtop;

namespace std {
bool operator<(const Point &lhs, const Point &rhs) {
  if (lhs.real() < rhs.real() - eps)
    return true;
  if (lhs.real() > rhs.real() + eps)
    return false;
  return lhs.imag() < rhs.imag();
}
} // namespace std

// 点の入力
Point input_Point() {
  ld x, y;
  cin >> x >> y;
  return Point(x, y);
}

// 誤差つき等号判定
bool eq(const ld a, const ld b) { return (abs(a - b) < eps); }

// 内積
ld dot(const Point &a, const Point &b) { return real(conj(a) * b); }

// 外積
ld cross(const Point &a, const Point &b) { return imag(conj(a) * b); }

// 直線の定義
class Line {
public:
  Point a, b;
  Line() : a(Point(0, 0)), b(Point(0, 0)) {}
  Line(Point a, Point b) : a(a), b(b) {}
  Point operator[](const int _num) const {
    if (_num == 0)
      return a;
    else if (_num == 1)
      return b;
    else {
      assert(false);
      return Point();
    }
  }
};

// 円の定義
class Circle {
public:
  Point p;
  ld r;
  Circle() : p(Point(0, 0)), r(0) {}
  Circle(Point p, ld r) : p(p), r(r) {}
};

// ccw
// 1: a,b,cが反時計周りの順に並ぶ
//-1: a,b,cが時計周りの順に並ぶ
// 2: c,a,bの順に直線に並ぶ
//-2: a,b,cの順に直線に並ぶ
// 0: a,c,bの順に直線に並ぶ
int ccw(const Point &a, const Point &b, const Point &c) {
  const Point nb(b - a);
  const Point nc(c - a);
  if (cross(nb, nc) > eps)
    return 1; // a,b,cが反時計周りの順に並ぶ
  if (cross(nb, nc) < -eps)
    return -1; // a,b,cが時計周りの順に並ぶ
  if (dot(nb, nc) < 0)
    return 2; // c,a,bの順に直線に並ぶ
  if (norm(nb) < norm(nc))
    return -2; // a,b,cの順に直線に並ぶ
  return 0;    // a,c,bの順に直線に並ぶ
}

/* 交差判定 */

// 直線と直線の交差判定
bool isis_ll(const Line &l, const Line &m) {
  return !eq(cross(l.b - l.a, m.b - m.a), 0);
}

// 直線と線分の交差判定
bool isis_ls(const Line &l, const Line &s) {
  return isis_ll(l, s) &&
         (cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < eps);
}

// 線分と線分の交差判定
bool isis_ss(const Line &s, const Line &t) {
  return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
         ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}

// 点の直線上判定
bool isis_lp(const Line &l, const Point &p) {
  return (abs(cross(l.b - p, l.a - p)) < eps);
}

// 点の線分上判定
bool isis_sp(const Line &s, const Point &p) {
  return (abs(s.a - p) + abs(s.b - p) - abs(s.b - s.a) < eps);
}

// 垂線の足
Point proj(const Line &l, const Point &p) {
  ld t = dot(p - l.a, l.b - l.a) / norm(l.a - l.b);
  return l.a + t * (l.b - l.a);
}

// 線対象の位置にある点
Point reflect(const Line &l, const Point &p) {
  Point pr = proj(l, p);
  return pr * 2.l - p;
}

// 直線と直線の交点
Point is_ll(const Line &s, const Line &t) {
  Point sv = s.b - s.a, tv = t.b - t.a;
  assert(cross(sv, tv) != 0);
  return s.a + sv * cross(tv, t.a - s.a) / cross(tv, sv);
}
// 直線と直線の交点
vector<Point> is_ll2(const Line &s, const Line &t) {
  Point sv = s.b - s.a, tv = t.b - t.a;
  if (cross(sv, tv) != 0)
    return vector<Point>(1, is_ll(s, t));
  else {
    vector<Point> ans;
    for (int k = 0; k < 2; ++k) {
      if (isis_sp(s, t[k]) && find(ans.begin(), ans.end(), t[k]) == ans.end())
        ans.push_back(t[k]);
      if (isis_sp(t, s[k]) && find(ans.begin(), ans.end(), s[k]) == ans.end())
        ans.push_back(s[k]);
    }
    return ans;
  }
}
// 線分と線分の交点
// 　重なってる部分あるとassert(false)
Point is_ss(const Line &s, const Line &t) {
  if (isis_ss(s, t)) {
    for (int k = 0; k < 2; ++k) {
      for (int l = 0; l < 2; ++l) {
        if (s[k] == t[l])
          return s[k];
      }
    }
    return is_ll(s, t);
  } else {
    // 先にisis_ssしてね
    assert(false);
    return Point(0, 0);
  }
}
// 線分と線分の交点
vector<Point> is_ss2(const Line &s, const Line &t) {
  vector<Point> kouho(is_ll2(s, t));
  vector<Point> ans;
  for (auto p : kouho) {
    if (isis_sp(s, p) && isis_sp(t, p))
      ans.emplace_back(p);
  }
  return ans;
}
// 直線と点の距離
ld dist_lp(const Line &l, const Point &p) { return abs(p - proj(l, p)); }

// 直線と直線の距離
ld dist_ll(const Line &l, const Line &m) {
  return isis_ll(l, m) ? 0 : dist_lp(l, m.a);
}

// 直線と線分の距離
ld dist_ls(const Line &l, const Line &s) {
  return isis_ls(l, s) ? 0 : min(dist_lp(l, s.a), dist_lp(l, s.b));
}

// 線分と点の距離
ld dist_sp(const Line &s, const Point &p) {
  Point r = proj(s, p);
  return isis_sp(s, r) ? abs(r - p) : min(abs(s.a - p), abs(s.b - p));
}

// 線分と線分の距離
ld dist_ss(const Line &s, const Line &t) {
  if (isis_ss(s, t))
    return 0;
  return min(
      {dist_sp(s, t.a), dist_sp(s, t.b), dist_sp(t, s.a), dist_sp(t, s.b)});
}

// 直線と直線の二等分線のベクトル
Line line_bisection(const Line &s, const Line &t) {
  const Point laglanju(is_ll(s, t));
  const Point avec =
      !(abs(laglanju - s[0]) < eps) ? s[0] - laglanju : s[1] - laglanju;
  const Point bvec =
      !(abs(laglanju - t[0]) < eps) ? t[0] - laglanju : t[1] - laglanju;

  return Line(laglanju, laglanju + (abs(bvec) * avec + abs(avec) * bvec) /
                                       (abs(avec) + abs(bvec)));
}
// 点と点の垂直二等分線　aを左に見ながら
Line point_bisection(const Point &a, const Point &b) {
  const Point cen((a + b) / 2.l);
  const Point vec = (b - a) * Point(0, 1);
  return Line(cen, cen + vec);
}

// 三つの点からなる外心
Point outer_center(const vector<Point> &ps) {
  assert(ps.size() == 3);
  Line l1 = point_bisection(ps[0], ps[1]);
  Line l2 = point_bisection(ps[1], ps[2]);
  return is_ll(l1, l2);
}

// 三つの直線からなる内心
// 三つの直線が並行でないことは確かめといてね
Point inner_center(const vector<Line> &ls) {
  vector<Point> vertics;
  for (int i = 0; i < static_cast<int>(ls.size()); ++i) {
    vertics.push_back(is_ll(ls[i], ls[(i + 1) % 3]));
  }
  if (vertics[0] == vertics[1] || vertics[1] == vertics[2] ||
      vertics[2] == vertics[0])
    return vertics[0];
  Line bi1(line_bisection(Line(vertics[0], vertics[1]),
                          Line(vertics[0], vertics[2])));
  Line bi2(line_bisection(Line(vertics[1], vertics[2]),
                          Line(vertics[1], vertics[0])));
  if (bi1[0] == bi2[0])
    return bi1[0];
  else {
    return is_ll(bi1, bi2);
  }
}

// 三つの直線からなる傍心
// 三つの直線が並行でないことは確かめといてね
vector<Point> ex_center(const vector<Line> &ls) {
  vector<Point> vertics;
  for (int i = 0; i < static_cast<int>(ls.size()); ++i) {
    vertics.push_back(is_ll(ls[i], ls[(i + 1) % 3]));
  }
  if (abs(vertics[0] - vertics[1]) < eps ||
      abs(vertics[1] - vertics[2]) < eps ||
      (abs(vertics[2] - vertics[0]) < eps))
    return vector<Point>();
  vector<Point> ecs;
  for (int i = 0; i < 3; ++i) {
    Line bi1(line_bisection(
        Line(vertics[i], vertics[i] * 2.0l - vertics[(i + 2) % 3]),
        Line(vertics[i], vertics[(i + 1) % 3])));
    Line bi2(
        line_bisection(Line(vertics[(i + 1) % 3],
                            vertics[(i + 1) % 3] * 2.0l - vertics[(i + 2) % 3]),
                       Line(vertics[(i + 1) % 3], vertics[i])));
    ecs.push_back(is_ll(bi1, bi2));
  }
  return ecs;
}

// a,b:並行
// c:並行でない
// 三つの直線から同距離の位置を求める。
vector<Point> same_dis(const vector<Line> &ls) {
  vector<Point> vertics;
  vertics.push_back(is_ll(ls[0], ls[2]));
  vertics.push_back(is_ll(ls[1], ls[2]));

  if (abs(vertics[0] - vertics[1]) < eps)
    return vector<Point>{vertics[0]};
  Line bis(line_bisection(ls[0], ls[1]));
  vector<Point> ecs;

  Line abi(line_bisection(Line(vertics[0], vertics[1]), ls[0]));
  ecs.push_back(is_ll(bis, abi));

  Line bbi(
      line_bisection(Line(vertics[0], 2.l * vertics[0] - vertics[1]), ls[0]));
  ecs.push_back(is_ll(bis, bbi));

  return ecs;
}
/* 円 */

// 円と円の交点
vector<Point> is_cc(const Circle &c1, const Circle &c2) {
  vector<Point> res;
  ld d = abs(c1.p - c2.p);
  ld rc = (d * d + c1.r * c1.r - c2.r * c2.r) / (2 * d);
  ld dfr = c1.r * c1.r - rc * rc;
  if (abs(dfr) < eps)
    dfr = 0.0;
  else if (dfr < 0.0)
    return res; // no intersection
  ld rs = sqrt(dfr);
  Point diff = (c2.p - c1.p) / d;
  res.push_back(c1.p + diff * Point(rc, rs));
  if (dfr != 0.0)
    res.push_back(c1.p + diff * Point(rc, -rs));
  return res;
}

/*
点が円の中にいるか
0 => out
1 => on
2 => in*/
int is_in_Circle(const Point &p, const Circle &cir) {
  ld dis = abs(cir.p - p);
  if (dis > cir.r + eps)
    return 0;
  else if (dis < cir.r - eps)
    return 2;
  else
    return 1;
}
int is_in_Circle(const Circle &cir, const Point &p) {
  ld dis = abs(cir.p - p);
  if (dis > cir.r + eps)
    return 0;
  else if (dis < cir.r - eps)
    return 2;
  else
    return 1;
}
/*
円lcが円rcの中にいるか
0 => out
1 => on
2 => in*/
int Circle_in_Circle(const Circle &lc, const Circle &rc) {
  ld dis = abs(lc.p - rc.p);
  if (dis < rc.r - lc.r - eps)
    return 2;
  else if (dis > rc.r - lc.r + eps)
    return 0;
  else
    return 1;
}

// 円と直線の交点
vector<Point> is_lc(const Circle &c, const Line &l) {
  vector<Point> res;
  ld d = dist_lp(l, c.p);
  if (d < c.r + eps) {
    ld len = (d > c.r) ? 0.0 : sqrt(c.r * c.r - d * d); // safety;
    Point nor = (l.a - l.b) / abs(l.a - l.b);
    res.push_back(proj(l, c.p) + len * nor);
    res.push_back(proj(l, c.p) - len * nor);
  }
  return res;
}

// 円と線分の距離
vector<Point> is_sc(const Circle &c, const Line &l) {
  vector<Point> v = is_lc(c, l), res;
  for (Point p : v)
    if (isis_sp(l, p))
      res.push_back(p);
  return res;
}

// 円と点の接線
vector<Line> tangent_cp(const Circle &c, const Point &p) {
  vector<Line> ret;
  Point v = c.p - p;
  ld d = abs(v);
  ld l = sqrt(norm(v) - c.r * c.r);
  if (isnan(l)) {
    return ret;
  }
  Point v1 = v * Point(l / d, c.r / d);
  Point v2 = v * Point(l / d, -c.r / d);
  ret.push_back(Line(p, p + v1));
  if (l < eps)
    return ret;
  ret.push_back(Line(p, p + v2));
  return ret;
}

// 円と円の接線
vector<Line> tangent_cc(const Circle &c1, const Circle &c2) {
  vector<Line> ret;
  if (abs(c1.p - c2.p) - (c1.r + c2.r) > -eps) {
    Point center = (c1.p * c2.r + c2.p * c1.r) / (c1.r + c2.r);
    ret = tangent_cp(c1, center);
  }
  if (abs(c1.r - c2.r) > eps) {
    Point out = (-c1.p * c2.r + c2.p * c1.r) / (c1.r - c2.r);
    vector<Line> nret = tangent_cp(c1, out);
    ret.insert(ret.end(), nret.begin(), nret.end());
  } else {
    Point v = c2.p - c1.p;
    v /= abs(v);
    Point q1 = c1.p + v * Point(0, 1) * c1.r;
    Point q2 = c1.p + v * Point(0, -1) * c1.r;
    ret.push_back(Line(q1, q1 + v));
    ret.push_back(Line(q2, q2 + v));
  }
  return ret;
}
// 二つの円の重なり面積
ld two_Circle_area(const Circle &l, const Circle &r) {
  ld dis = abs(l.p - r.p);
  if (dis > l.r + r.r)
    return 0;
  else if (dis + r.r < l.r) {
    return r.r * r.r * pi;
  } else if (dis + l.r < r.r) {
    return l.r * l.r * pi;
  } else {
    ld ans = (l.r) * (l.r) *
                 acos((dis * dis + l.r * l.r - r.r * r.r) / (2 * dis * l.r)) +
             (r.r) * (r.r) *
                 acos((dis * dis + r.r * r.r - l.r * l.r) / (2 * dis * r.r)) -
             sqrt(4 * dis * dis * l.r * l.r -
                  (dis * dis + l.r * l.r - r.r * r.r) *
                      (dis * dis + l.r * l.r - r.r * r.r)) /
                 2;
    return ans;
  }
}

/* 多角形 */

typedef vector<Point> Polygon;

// 多角形(複数の点)の最小包含円をO(N=頂点数)で求めるアルゴリズム
// 同一直線上に三つの点がないこと
#include <random>
Circle welzl(vector<Point> ps) {
  struct solver {
    Circle solve(vector<Point> &ps, vector<Point> &rs) {
      if (ps.empty() || rs.size() == 3) {
        if (rs.size() == 1) {
          return Circle(Point(rs[0]), 0);
        } else if (rs.size() == 2) {
          return Circle((rs[0] + rs[1]) / 2.0l, abs(rs[1] - rs[0]) / 2);
        } else if (rs.size() == 3) {
          vector<Line> ls(3);
          for (int i = 0; i < 3; ++i) {
            ls[i] = Line(rs[i], rs[(i + 1) % 3]);
          }
          Point center = outer_center(rs);
          return Circle(center, abs(center - rs[0]));
        } else {
          return Circle(Point(), 0);
        }
      } else {

        Point p_ba = ps.back();
        ps.pop_back();
        Circle d = solve(ps, rs);
        ps.push_back(p_ba);
        if (is_in_Circle(d, p_ba)) {
          return d;
        } else {
          rs.push_back(p_ba);
          ps.pop_back();
          auto ans = solve(ps, rs);
          ps.push_back(p_ba);
          rs.pop_back();
          return ans;
        }
      }
    }
  } so;
  std::random_device rd;
  std::mt19937 mt(rd());
  shuffle(ps.begin(), ps.end(), mt);
  vector<Point> rs;
  Circle ans = so.solve(ps, rs);
  return ans;
}
// 面積
ld get_area(const Polygon &p) {
  ld res = 0;
  int n = p.size();
  for (int j = 0; j < n; ++j)
    res += cross(p[j], p[(j + 1) % n]);
  return res / 2;
}

// 多角形の回転方向
bool is_counter_clockwise(const Polygon &poly) {
  ld angle = 0;
  int n = poly.size();
  for (int i = 0; i < n; ++i) {
    Point a = poly[i], b = poly[(i + 1) % n], c = poly[(i + 2) % n];
    angle += arg((c - b) / (b - a));
  }
  return angle > eps;
}

// 円の内外判定
/*0 => out
1 => on
2 => in*/
int is_in_Polygon(const Polygon &poly, const Point &p) {
  ld angle = 0;
  int n = poly.size();
  for (int i = 0; i < n; ++i) {
    Point a = poly[i], b = poly[(i + 1) % n];
    if (isis_sp(Line(a, b), p))
      return 1;
    angle += arg((b - p) / (a - p));
  }
  return eq(angle, 0) ? 0 : 2;
}
// 円の内外判定2　高速
enum { out, on, in };
int convex_contains(const Polygon &P, const Point &p) {
  const int n = P.size();
  Point g = (P[0] + P[n / 3] + P[2 * n / 3]) / 3.0l; // inner-point
  int a = 0, b = n;
  while (a + 1 < b) { // invariant: c is in fan g-P[a]-P[b]
    int c = (a + b) / 2;
    if (cross(P[a] - g, P[c] - g) > 0) { // angle < 180 deg
      if (cross(P[a] - g, p - g) > 0 && cross(P[c] - g, p - g) < 0)
        b = c;
      else
        a = c;
    } else {
      if (cross(P[a] - g, p - g) < 0 && cross(P[c] - g, p - g) > 0)
        a = c;
      else
        b = c;
    }
  }
  b %= n;
  if (cross(P[a] - p, P[b] - p) < 0)
    return 0;
  if (cross(P[a] - p, P[b] - p) > 0)
    return 2;
  return 1;
}

// 凸包
// 点や線を返すことも有り得るので注意
Polygon convex_hull(vector<Point> ps) {
  int n = ps.size();
  int k = 0;
  sort(ps.begin(), ps.end());
  Polygon ch(2 * n);
  for (int i = 0; i < n; ch[k++] = ps[i++])
    while (k >= 2 && ccw(ch[k - 2], ch[k - 1], ps[i]) <= 0)
      --k;
  for (int i = n - 2, t = k + 1; i >= 0; ch[k++] = ps[i--])
    while (k >= t && ccw(ch[k - 2], ch[k - 1], ps[i]) <= 0)
      --k;
  ch.resize(k - 1);
  return ch;
}

// 凸カット
vector<Polygon> convex_cut(const Polygon &ps, const Line &l) {
  int n = ps.size();
  Polygon q;
  Polygon r;
  for (int i = 0; i < n; ++i) {
    Point a = ps[i], b = ps[(i + 1) % n];
    Line m = Line(a, b);
    if (ccw(l.a, l.b, a) != -1)
      q.push_back(a);
    if (ccw(l.a, l.b, a) != 1)
      r.push_back(a);
    if (ccw(l.a, l.b, a) * ccw(l.a, l.b, b) < 0 && isis_ll(l, m)) {
      q.push_back(is_ll(l, m));
      r.push_back(is_ll(l, m));
    }
  }
  const vector<Polygon> polys{q, r};
  return polys;
}

int main() {
  while (true) {
    int N;
    cin >> N;
    if (!N)
      break;
    vector<ld> us, ds, ls, rs;
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      ds.push_back(n);
    }
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      us.push_back(n);
    }
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      ls.push_back(n);
    }
    for (int i = 0; i < N; ++i) {
      ld n;
      cin >> n;
      rs.push_back(n);
    }

    Polygon start_poly{Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1)};

    vector<Polygon> polys{start_poly};
    for (int i = 0; i < N; ++i) {
      vector<Polygon> nextpolys;
      for (auto po : polys) {
        Line l(Point(ds[i], 0), Point(us[i], 1));
        auto pos = convex_cut(po, l);

        for (auto aa : pos) {
          nextpolys.push_back(aa);
        }
      }
      polys = nextpolys;
    }
    for (int i = 0; i < N; ++i) {
      vector<Polygon> nextpolys;
      for (auto po : polys) {
        Line l(Point(0, ls[i]), Point(1, rs[i]));
        auto pos = convex_cut(po, l);

        for (auto aa : pos) {
          nextpolys.push_back(aa);
        }
      }
      polys = nextpolys;
    }
    ld ans = 0;
    for (auto poly : polys) {
      ans = max(ans, abs(get_area(poly)));
    }
    cout << setprecision(6) << fixed << ans << endl;
  }
  return 0;
}