Problem p02312 (C++) — buggy submission, verdict: Runtime Error

#include <bits/stdc++.h>
using namespace std;
typedef complex<double> P;
typedef complex<double> V;
typedef vector<P> vecP;
typedef pair<P, P> L;
typedef pair<P, P> S;
typedef pair<P, double> C;
const double eps = 1e-8;
const double PI = acos(-1);
const double PI2 = PI * 2.0;

namespace std {
bool operator<(const P &a, const P &b) {
  return (a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag());
}
}; // namespace std

V normal(V a) {
  assert(abs(a) > 0);
  return a / abs(a);
}

double Sqrt(double x) {
  if (x < 0)
    return 0;
  else
    return sqrt(x);
}

P Vector(L a) { return a.second - a.first; }

bool eq(double a, double b) { return (-eps < a - b && a - b < eps); }

bool eq(P a, P b) { return (eq(a.real(), b.real()) && eq(a.imag(), b.imag())); }

double dot(P a, P b) { return real(b * conj(a)); }

double cross(P a, P b) { return imag(b * conj(a)); }

double getArg(P a, P b) { return arg(b * conj(a)); }

double getTime(V a, V b) {
  assert(eq(cross(a, b), 0));
  return (dot(a, b) < 0 ? -1.0 : 1.0) * abs(b) / abs(a);
}

P project(P a, P b, P c) {
  b -= a, c -= a;
  return a + b * real(c / b);
}

P reflect(P a, P b, P c) {
  b -= a, c -= a;
  return a + b * conj(c / b);
}

int ccw(P a, P b, P c) {
  P ab = b - a, ac = c - a;
  P k = ac * conj(ab);
  if (k.imag() > 0)
    return 1;
  if (k.imag() < 0)
    return -1;
  if (k.real() < 0)
    return 2;
  if (abs(ab) < abs(ac))
    return -2;
  return 0;
}

bool isParallel(P a, P b) { return eq(0, cross(a, b)); }

bool isParallel(S a, S b) { return eq(0, cross(Vector(a), Vector(b))); }

bool onLP(L l, P p) {
  P a = l.first, b = l.second;
  return eq(0, cross(b - a, p - a));
}

bool onSP(S s, P p) {
  P a = s.first, b = s.second;
  return eq(abs(b - a), abs(a - p) + abs(b - p));
}

bool isCrossSS(S s0, S s1) {
  P a = s0.first, b = s0.second;
  P c = s1.first, d = s1.second;
  int f0 = ccw(a, b, c) * ccw(a, b, d);
  int f1 = ccw(c, d, a) * ccw(c, d, b);
  return (f0 <= 0 && f1 <= 0);
}

bool isCrossLS(L l, S s) {
  P a = l.first, b = l.second;
  P c = s.first, d = s.second;
  return (ccw(a, b, c) * ccw(a, b, d) <= 0);
}

double distLP(L l, P p) {
  P a = l.first, b = l.second;
  double res = cross(b - a, p - a) / abs(b - a);
  return abs(res);
}

double distSP(S s, P p) {
  P a = s.first, b = s.second;
  if (dot(b - a, p - a) < eps)
    return abs(p - a);
  if (dot(a - b, p - b) < eps)
    return abs(p - b);
  return distLP(s, p);
}

double distSS(S s0, S s1) {
  if (isCrossSS(s0, s1))
    return 0;
  double res0 = min(distSP(s0, s1.first), distSP(s0, s1.second));
  double res1 = min(distSP(s1, s0.first), distSP(s1, s0.second));
  return min(res0, res1);
}

P getCrossLL(L l0, L l1) {
  P a = l0.first, b = l0.second;
  P c = l1.first, d = l1.second;
  a -= d;
  b -= d;
  c -= d;
  return d + a + (b - a) * imag(a / c) / imag(a / c - b / c);
}

int inPolygon(vecP &t, P p) {
  int n = t.size();
  double sum = 0;
  for (int i = 0; i < n; i++) {
    P a = t[i], b = t[(i + 1 == n ? 0 : i + 1)];
    if (ccw(a, b, p) == 0)
      return 1;
    sum += getArg(a - p, b - p);
  }
  if (abs(sum) < eps)
    return 0;
  else
    return 2;
}

vecP andrewScan(vecP &t) {
  int N = t.size(), C = 0;
  vecP R(N);
  for (int i = 0; i < N; i++) {
    while (2 <= C && ccw(R[C - 2], R[C - 1], t[i]) == -1)
      C--;
    R[C++] = t[i];
  }
  vecP res(C);
  for (int i = 0; i < C; i++)
    res[i] = R[i];
  return res;
}

vecP convexHull(vecP &t) {
  sort(t.begin(), t.end());
  vecP u = andrewScan(t);
  reverse(t.begin(), t.end());
  vecP l = andrewScan(t);
  for (int i = 1; i + 1 < (int)l.size(); i++)
    u.push_back(l[i]);
  return u;
}

vecP cutConvex(vecP &t, L l) {
  P a = l.first, b = l.second;
  int N = t.size();
  vecP res;
  for (int i = 0; i < N; i++) {
    P c = t[i], d = t[(i + 1) % N];
    int C = ccw(a, b, c), D = ccw(a, b, d);
    if (C != -1)
      res.push_back(c);
    if (C == -D && abs(C) == 1)
      res.push_back(getCrossLL(l, L(c, d)));
  }
  return res;
}

P getVector(const vecP &t, int id) {
  int n = t.size();
  return t[(id + 1) % n] - t[id % n];
}

double convex_diameter(vecP &t) {
  int n = t.size();
  int is = 0, js = 0;
  for (int i = 1; i < n; ++i) {
    if (imag(t[i]) > imag(t[is]))
      is = i;
    if (imag(t[i]) < imag(t[js]))
      js = i;
  }
  double maxd = norm(t[is] - t[js]);

  int i, maxi, j, maxj;
  i = maxi = is;
  j = maxj = js;
  do {

    if (cross(getVector(t, i), getVector(t, j)) >= 0)
      j = (j + 1) % n;

    else
      i = (i + 1) % n;
    if (norm(t[i] - t[j]) > maxd) {
      maxd = norm(t[i] - t[j]);
      maxi = i;
      maxj = j;
    }
  } while (i != is || j != js);
  return sqrt(maxd); /* farthest pair is (maxi, maxj). */
}

bool compare_y(const P &a, const P &b) { return a.imag() < b.imag(); }

double closest_pair(P *a, int n) {
  if (n <= 1)
    return 1e30;
  int m = n / 2;
  double x = a[m].real();
  double d = min(closest_pair(a, m), closest_pair(a + m, n - m));
  inplace_merge(a, a + m, a + n, compare_y);
  vector<P> b;
  for (int i = 0; i < n; i++) {
    if (abs(a[i].real() - x) >= d)
      continue;
    for (int j = 0; j < (int)b.size(); j++) {
      double dx = real(a[i] - b[b.size() - j - 1]);
      double dy = imag(a[i] - b[b.size() - j - 1]);
      if (dy >= d)
        break;
      d = min(d, sqrt(dx * dx + dy * dy));
    }
    b.push_back(a[i]);
  }
  return d;
}

P _pool[200005];
double minDist(vecP &t) {
  int n = t.size();
  for (int i = 0; i < n; i++)
    _pool[i] = t[i];
  sort(_pool, _pool + n);
  return closest_pair(_pool, n);
}

int getStateCC(C a, C b) {
  double ar = a.second, br = b.second;
  double dist = abs(a.first - b.first);
  if (dist > ar + br + eps)
    return 4;
  if (dist > ar + br - eps)
    return 3;
  if (dist > abs(ar - br) + eps)
    return 2;
  if (dist > abs(ar - br) - eps)
    return 1;
  return 0;
}

P getCrossCC(C a, C b) {
  P p1 = a.first, p2 = b.first;
  double r1 = a.second, r2 = b.second;
  double cA = (r1 * r1 + norm(p1 - p2) - r2 * r2) / (2.0 * r1 * abs(p1 - p2));
  return p1 + (p2 - p1) / abs(p1 - p2) * r1 * P(cA, Sqrt(1.0 - cA * cA));
}

P getTangentCP_(C a, P p, int flg) {
  P base = a.first - p;
  double ar = a.second;
  double w = Sqrt(norm(base) - ar * ar);
  P s = p + base * P(w, ar * flg) / norm(base) * w;
  return s;
}

vector<S> getTangentCP(C a, P p) {
  vector<S> res;
  P s = getTangentCP_(a, p, 1);
  P t = getTangentCP_(a, p, -1);

  if (eq(s, t)) {
    res.push_back(S(s, s + (a.first - p) * P(0, 1)));
  } else {
    res.push_back(S(p, s));
    res.push_back(S(p, t));
  }
  return res;
}

S getInTangent(C a, C b, double flg = 1.0) {
  P ap = a.first, bp = b.first;
  double ar = a.second, br = b.second;

  P base = bp - ap;
  double w = ar + br;
  double h = Sqrt(norm(base) - w * w);
  P k = base * P(w, h * flg) / norm(base);
  return S(ap + k * ar, bp - k * br);
}

S getOutTangent(C a, C b, double flg = 1.0) {
  P ap = a.first, bp = b.first;
  double ar = a.second, br = b.second;

  P base = bp - ap;
  double h = br - ar;

  double w = Sqrt(norm(base) - h * h);
  P k = base * P(w, h * flg) / norm(base) * P(0, flg);
  return S(ap + k * ar, bp + k * br);
}

vector<S> getTangentCC(C a, C b) {
  P ap = a.first, bp = b.first;
  double ar = a.second, br = b.second;

  vector<S> res;
  double dist = abs(ap - bp);

  if (dist > ar + br + eps)
    res.push_back(getInTangent(a, b, 1));

  if (dist > ar + br - eps)
    res.push_back(getInTangent(a, b, -1));

  if (dist > abs(ar - br) + eps)
    res.push_back(getOutTangent(a, b, 1));

  if (dist > abs(ar - br) - eps)
    res.push_back(getOutTangent(a, b, -1));

  return res;
}

vecP getCrossCL(C cir, L l) {
  P a = l.first, b = l.second;
  double cr = cir.second;
  P cp = cir.first;

  vecP res;
  P base = b - a, target = project(a, b, cp);

  double length = abs(base), h = abs(cp - target);
  base /= length;

  if (cr + eps < h)
    return res;
  double w = Sqrt(cr * cr - h * h);
  double L = getTime(normal(b - a), target - a) - w, R = L + w * 2.0;

  res.push_back(a + base * L);
  if (eq(L, R))
    return res;
  res.push_back(a + base * R);

  return res;
}

vecP getCrossCS(C cir, S s) {
  vecP tmp = getCrossCL(cir, s);
  vecP res;
  for (int i = 0; i < (int)tmp.size(); i++)
    //  if( ccw(s.first,s.second, tmp[i] ) == 0)
    if (onSP(s, tmp[i]))
      res.push_back(tmp[i]);
  return res;
}

double getArea(C c, P a, P b) {
  P cp = c.first;
  double cr = c.second;

  P va = cp - a, vb = cp - b;
  double A = abs(va), B = abs(vb);
  double f = cross(va, vb), d = distSP(S(a, b), cp), res = 0;

  if (eq(0, f))
    return 0;
  if (A < cr + eps && B < cr + eps)
    return f * 0.5;
  if (d > cr - eps)
    return cr * cr * PI * getArg(va, vb) / PI2;

  vecP u = getCrossCS(c, S(a, b));

  assert(!u.empty());
  u.insert(u.begin(), a), u.push_back(b);

  for (int i = 0; i + 1 < (int)u.size(); i++)
    res += getArea(c, u[i], u[i + 1]);
  return res;
}

double getCrossArea(vecP t, C c) {
  int n = t.size();
  if (n < 3)
    return 0;
  double res = 0;
  for (int i = 0; i < n; i++) {
    P a = t[i], b = t[(i + 1) % n];
    res += getArea(c, a, b);
  }
  return res;
}

double calcPolygonArea(const vecP &t) {
  double res = 0;
  int n = t.size();
  for (int i = 0; i < n; i++) {
    res += cross(t[(i + 1) % n], t[i]);
  }
  return abs(res) * 0.5;
}

P input() {
  double x, y;
  cin >> x >> y;
  return P(x, y);
}

void pr(P p, string str = "\\n") {
  printf("%.10f %.10f", p.real(), p.imag());
  cout << str;
}

int main() {
  int n;
  double r;
  cin >> n >> r;
  C a = C(P(0, 0), r);

  vecP t(n);
  for (int i = 0; i < n; i++)
    t[i] = input();

  printf("%.10f\\n", getCrossArea(t, a));
  return 0;
}

Error: -6
9e720fcb-cca1-4361-b937-a1459990b32d.out: /home/alex/Documents/bug-detection/input/Project_CodeNet/data/p02312/C++/s412921055.cpp:380: double getArea(C, P, P): Assertion `!u.empty()' failed.


Fixed submission:

#include <bits/stdc++.h>
using namespace std;
typedef complex<double> P;
typedef complex<double> V;
typedef vector<P> vecP;
typedef pair<P, P> L;
typedef pair<P, P> S;
typedef pair<P, double> C;
const double eps = 1e-8;
const double PI = acos(-1);
const double PI2 = PI * 2.0;

namespace std {
bool operator<(const P &a, const P &b) {
  return (a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag());
}
}; // namespace std

V normal(V a) {
  assert(abs(a) > 0);
  return a / abs(a);
}

double Sqrt(double x) {
  if (x < 0)
    return 0;
  else
    return sqrt(x);
}

P Vector(L a) { return a.second - a.first; }

bool eq(double a, double b) { return (-eps < a - b && a - b < eps); }

bool eq(P a, P b) { return (eq(a.real(), b.real()) && eq(a.imag(), b.imag())); }

double dot(P a, P b) { return real(b * conj(a)); }

double cross(P a, P b) { return imag(b * conj(a)); }

double getArg(P a, P b) { return arg(b * conj(a)); }

double getTime(V a, V b) {
  assert(eq(cross(a, b), 0));
  return (dot(a, b) < 0 ? -1.0 : 1.0) * abs(b) / abs(a);
}

P project(P a, P b, P c) {
  b -= a, c -= a;
  return a + b * real(c / b);
}

P reflect(P a, P b, P c) {
  b -= a, c -= a;
  return a + b * conj(c / b);
}

int ccw(P a, P b, P c) {
  P ab = b - a, ac = c - a;
  P k = ac * conj(ab);
  if (k.imag() > 0)
    return 1;
  if (k.imag() < 0)
    return -1;
  if (k.real() < 0)
    return 2;
  if (abs(ab) < abs(ac))
    return -2;
  return 0;
}

bool isParallel(P a, P b) { return eq(0, cross(a, b)); }

bool isParallel(S a, S b) { return eq(0, cross(Vector(a), Vector(b))); }

bool onLP(L l, P p) {
  P a = l.first, b = l.second;
  return eq(0, cross(b - a, p - a));
}

bool onSP(S s, P p) {
  P a = s.first, b = s.second;
  return eq(abs(b - a), abs(a - p) + abs(b - p));
}

bool isCrossSS(S s0, S s1) {
  P a = s0.first, b = s0.second;
  P c = s1.first, d = s1.second;
  int f0 = ccw(a, b, c) * ccw(a, b, d);
  int f1 = ccw(c, d, a) * ccw(c, d, b);
  return (f0 <= 0 && f1 <= 0);
}

bool isCrossLS(L l, S s) {
  P a = l.first, b = l.second;
  P c = s.first, d = s.second;
  return (ccw(a, b, c) * ccw(a, b, d) <= 0);
}

double distLP(L l, P p) {
  P a = l.first, b = l.second;
  double res = cross(b - a, p - a) / abs(b - a);
  return abs(res);
}

double distSP(S s, P p) {
  P a = s.first, b = s.second;
  if (dot(b - a, p - a) < eps)
    return abs(p - a);
  if (dot(a - b, p - b) < eps)
    return abs(p - b);
  return distLP(s, p);
}

double distSS(S s0, S s1) {
  if (isCrossSS(s0, s1))
    return 0;
  double res0 = min(distSP(s0, s1.first), distSP(s0, s1.second));
  double res1 = min(distSP(s1, s0.first), distSP(s1, s0.second));
  return min(res0, res1);
}

P getCrossLL(L l0, L l1) {
  P a = l0.first, b = l0.second;
  P c = l1.first, d = l1.second;
  a -= d;
  b -= d;
  c -= d;
  return d + a + (b - a) * imag(a / c) / imag(a / c - b / c);
}

int inPolygon(vecP &t, P p) {
  int n = t.size();
  double sum = 0;
  for (int i = 0; i < n; i++) {
    P a = t[i], b = t[(i + 1 == n ? 0 : i + 1)];
    if (ccw(a, b, p) == 0)
      return 1;
    sum += getArg(a - p, b - p);
  }
  if (abs(sum) < eps)
    return 0;
  else
    return 2;
}

vecP andrewScan(vecP &t) {
  int N = t.size(), C = 0;
  vecP R(N);
  for (int i = 0; i < N; i++) {
    while (2 <= C && ccw(R[C - 2], R[C - 1], t[i]) == -1)
      C--;
    R[C++] = t[i];
  }
  vecP res(C);
  for (int i = 0; i < C; i++)
    res[i] = R[i];
  return res;
}

vecP convexHull(vecP &t) {
  sort(t.begin(), t.end());
  vecP u = andrewScan(t);
  reverse(t.begin(), t.end());
  vecP l = andrewScan(t);
  for (int i = 1; i + 1 < (int)l.size(); i++)
    u.push_back(l[i]);
  return u;
}

vecP cutConvex(vecP &t, L l) {
  P a = l.first, b = l.second;
  int N = t.size();
  vecP res;
  for (int i = 0; i < N; i++) {
    P c = t[i], d = t[(i + 1) % N];
    int C = ccw(a, b, c), D = ccw(a, b, d);
    if (C != -1)
      res.push_back(c);
    if (C == -D && abs(C) == 1)
      res.push_back(getCrossLL(l, L(c, d)));
  }
  return res;
}

P getVector(const vecP &t, int id) {
  int n = t.size();
  return t[(id + 1) % n] - t[id % n];
}

double convex_diameter(vecP &t) {
  int n = t.size();
  int is = 0, js = 0;
  for (int i = 1; i < n; ++i) {
    if (imag(t[i]) > imag(t[is]))
      is = i;
    if (imag(t[i]) < imag(t[js]))
      js = i;
  }
  double maxd = norm(t[is] - t[js]);

  int i, maxi, j, maxj;
  i = maxi = is;
  j = maxj = js;
  do {

    if (cross(getVector(t, i), getVector(t, j)) >= 0)
      j = (j + 1) % n;

    else
      i = (i + 1) % n;
    if (norm(t[i] - t[j]) > maxd) {
      maxd = norm(t[i] - t[j]);
      maxi = i;
      maxj = j;
    }
  } while (i != is || j != js);
  return sqrt(maxd); /* farthest pair is (maxi, maxj). */
}

bool compare_y(const P &a, const P &b) { return a.imag() < b.imag(); }

double closest_pair(P *a, int n) {
  if (n <= 1)
    return 1e30;
  int m = n / 2;
  double x = a[m].real();
  double d = min(closest_pair(a, m), closest_pair(a + m, n - m));
  inplace_merge(a, a + m, a + n, compare_y);
  vector<P> b;
  for (int i = 0; i < n; i++) {
    if (abs(a[i].real() - x) >= d)
      continue;
    for (int j = 0; j < (int)b.size(); j++) {
      double dx = real(a[i] - b[b.size() - j - 1]);
      double dy = imag(a[i] - b[b.size() - j - 1]);
      if (dy >= d)
        break;
      d = min(d, sqrt(dx * dx + dy * dy));
    }
    b.push_back(a[i]);
  }
  return d;
}

P _pool[200005];
double minDist(vecP &t) {
  int n = t.size();
  for (int i = 0; i < n; i++)
    _pool[i] = t[i];
  sort(_pool, _pool + n);
  return closest_pair(_pool, n);
}

int getStateCC(C a, C b) {
  double ar = a.second, br = b.second;
  double dist = abs(a.first - b.first);
  if (dist > ar + br + eps)
    return 4;
  if (dist > ar + br - eps)
    return 3;
  if (dist > abs(ar - br) + eps)
    return 2;
  if (dist > abs(ar - br) - eps)
    return 1;
  return 0;
}

P getCrossCC(C a, C b) {
  P p1 = a.first, p2 = b.first;
  double r1 = a.second, r2 = b.second;
  double cA = (r1 * r1 + norm(p1 - p2) - r2 * r2) / (2.0 * r1 * abs(p1 - p2));
  return p1 + (p2 - p1) / abs(p1 - p2) * r1 * P(cA, Sqrt(1.0 - cA * cA));
}

P getTangentCP_(C a, P p, int flg) {
  P base = a.first - p;
  double ar = a.second;
  double w = Sqrt(norm(base) - ar * ar);
  P s = p + base * P(w, ar * flg) / norm(base) * w;
  return s;
}

vector<S> getTangentCP(C a, P p) {
  vector<S> res;
  P s = getTangentCP_(a, p, 1);
  P t = getTangentCP_(a, p, -1);

  if (eq(s, t)) {
    res.push_back(S(s, s + (a.first - p) * P(0, 1)));
  } else {
    res.push_back(S(p, s));
    res.push_back(S(p, t));
  }
  return res;
}

S getInTangent(C a, C b, double flg = 1.0) {
  P ap = a.first, bp = b.first;
  double ar = a.second, br = b.second;

  P base = bp - ap;
  double w = ar + br;
  double h = Sqrt(norm(base) - w * w);
  P k = base * P(w, h * flg) / norm(base);
  return S(ap + k * ar, bp - k * br);
}

S getOutTangent(C a, C b, double flg = 1.0) {
  P ap = a.first, bp = b.first;
  double ar = a.second, br = b.second;

  P base = bp - ap;
  double h = br - ar;

  double w = Sqrt(norm(base) - h * h);
  P k = base * P(w, h * flg) / norm(base) * P(0, flg);
  return S(ap + k * ar, bp + k * br);
}

vector<S> getTangentCC(C a, C b) {
  P ap = a.first, bp = b.first;
  double ar = a.second, br = b.second;

  vector<S> res;
  double dist = abs(ap - bp);

  if (dist > ar + br + eps)
    res.push_back(getInTangent(a, b, 1));

  if (dist > ar + br - eps)
    res.push_back(getInTangent(a, b, -1));

  if (dist > abs(ar - br) + eps)
    res.push_back(getOutTangent(a, b, 1));

  if (dist > abs(ar - br) - eps)
    res.push_back(getOutTangent(a, b, -1));

  return res;
}

vecP getCrossCL(C cir, L l) {
  P a = l.first, b = l.second;
  double cr = cir.second;
  P cp = cir.first;

  vecP res;
  P base = b - a, target = project(a, b, cp);

  double length = abs(base), h = abs(cp - target);
  base /= length;

  if (cr + eps < h)
    return res;
  double w = Sqrt(cr * cr - h * h);
  double L = getTime(normal(b - a), target - a) - w, R = L + w * 2.0;

  res.push_back(a + base * L);
  if (eq(L, R))
    return res;
  res.push_back(a + base * R);

  return res;
}

vecP getCrossCS(C cir, S s) {
  vecP tmp = getCrossCL(cir, s);
  vecP res;
  for (int i = 0; i < (int)tmp.size(); i++)
    if (ccw(s.first, s.second, tmp[i]) == 0)
      res.push_back(tmp[i]);
  return res;
}

double getArea(C c, P a, P b) {
  P cp = c.first;
  double cr = c.second;

  P va = cp - a, vb = cp - b;
  double A = abs(va), B = abs(vb);
  double f = cross(va, vb), d = distSP(S(a, b), cp), res = 0;

  if (eq(0, f))
    return 0;
  if (A < cr + eps && B < cr + eps)
    return f * 0.5;
  if (d > cr - eps)
    return cr * cr * PI * getArg(va, vb) / PI2;

  vecP u = getCrossCS(c, S(a, b));

  assert(!u.empty());
  u.insert(u.begin(), a), u.push_back(b);

  for (int i = 0; i + 1 < (int)u.size(); i++)
    res += getArea(c, u[i], u[i + 1]);
  return res;
}

double getCrossArea(vecP t, C c) {
  int n = t.size();
  if (n < 3)
    return 0;
  double res = 0;
  for (int i = 0; i < n; i++) {
    P a = t[i], b = t[(i + 1) % n];
    res += getArea(c, a, b);
  }
  return res;
}

double calcPolygonArea(const vecP &t) {
  double res = 0;
  int n = t.size();
  for (int i = 0; i < n; i++) {
    res += cross(t[(i + 1) % n], t[i]);
  }
  return abs(res) * 0.5;
}

P input() {
  double x, y;
  cin >> x >> y;
  return P(x, y);
}

void pr(P p, string str = "\\n") {
  printf("%.10f %.10f", p.real(), p.imag());
  cout << str;
}

int main() {
  int n;
  double r;
  cin >> n >> r;
  C a = C(P(0, 0), r);

  vecP t(n);
  for (int i = 0; i < n; i++)
    t[i] = input();

  printf("%.10f\\n", getCrossArea(t, a));
  return 0;
}