Problem p01705 (C++) — buggy submission, verdict: Time Limit Exceeded

#include <bits/stdc++.h>
using namespace std;
using Int = long long;

#define EPS (1e-10)
#define equals(a, b) (fabs((a) - (b)) < EPS)
#define PI 3.141592653589793238

// COUNTER CLOCKWISE
static const int CCW_COUNTER_CLOCKWISE = 1;
static const int CCW_CLOCKWISE = -1;
static const int CCW_ONLINE_BACK = 2;
static const int CCW_ONLINE_FRONT = -2;
static const int CCW_ON_SEGMENT = 0;

// Intercsect Circle & Circle
static const int ICC_SEPERATE = 4;
static const int ICC_CIRCUMSCRIBE = 3;
static const int ICC_INTERSECT = 2;
static const int ICC_INSCRIBE = 1;
static const int ICC_CONTAIN = 0;

struct Point {
  double x, y;
  Point() {}
  Point(double x, double y) : x(x), y(y) {}
  Point operator+(Point p) { return Point(x + p.x, y + p.y); }
  Point operator-(Point p) { return Point(x - p.x, y - p.y); }
  Point operator*(double k) { return Point(x * k, y * k); }
  Point operator/(double k) { return Point(x / k, y / k); }
  double norm() { return x * x + y * y; }
  double abs() { return sqrt(norm()); }

  bool operator<(const Point &p) const {
    return x != p.x ? x < p.x : y < p.y;
    // grid-point only
    // return !equals(x,p.x)?x<p.x:!equals(y,p.y)?y<p.y:0;
  }

  bool operator==(const Point &p) const {
    return fabs(x - p.x) < EPS && fabs(y - p.y) < EPS;
  }
};

istream &operator>>(istream &is, Point &p) {
  is >> p.x >> p.y;
  return is;
}

ostream &operator<<(ostream &os, Point p) {
  os << fixed << setprecision(12) << p.x << " " << p.y;
  return os;
}

bool sort_x(Point a, Point b) { return a.x != b.x ? a.x < b.x : a.y < b.y; }

bool sort_y(Point a, Point b) { return a.y != b.y ? a.y < b.y : a.x < b.x; }

typedef Point Vector;
typedef vector<Point> Polygon;

struct Segment {
  Point p1, p2;
  Segment() {}
  Segment(Point p1, Point p2) : p1(p1), p2(p2) {}
};
typedef Segment Line;

istream &operator>>(istream &is, Segment &s) {
  is >> s.p1 >> s.p2;
  return is;
}

struct Circle {
  Point c;
  double r;
  Circle() {}
  Circle(Point c, double r) : c(c), r(r) {}
};

istream &operator>>(istream &is, Circle &c) {
  is >> c.c >> c.r;
  return is;
}

double norm(Vector a) { return a.x * a.x + a.y * a.y; }
double abs(Vector a) { return sqrt(norm(a)); }
double dot(Vector a, Vector b) { return a.x * b.x + a.y * b.y; }
double cross(Vector a, Vector b) { return a.x * b.y - a.y * b.x; }

bool isOrthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }

bool isOrthogonal(Point a1, Point a2, Point b1, Point b2) {
  return isOrthogonal(a1 - a2, b1 - b2);
}

bool isOrthogonal(Segment s1, Segment s2) {
  return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

bool isParallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }

bool isParallel(Point a1, Point a2, Point b1, Point b2) {
  return isParallel(a1 - a2, b1 - b2);
}

bool isParallel(Segment s1, Segment s2) {
  return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

Point project(Segment s, Point p) {
  Vector base = s.p2 - s.p1;
  double r = dot(p - s.p1, base) / norm(base);
  return s.p1 + base * r;
}

Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }

double arg(Vector p) { return atan2(p.y, p.x); }

Vector polar(double a, double r) { return Point(cos(r) * a, sin(r) * a); }

int ccw(Point p0, Point p1, Point p2);
bool intersectSS(Point p1, Point p2, Point p3, Point p4);
bool intersectSS(Segment s1, Segment s2);
int intersectCC(Circle c1, Circle c2);
bool intersectSC(Segment s, Circle c);
double getDistanceLP(Line l, Point p);
double getDistanceSP(Segment s, Point p);
double getDistanceSS(Segment s1, Segment s2);
Point getCrossPointSS(Segment s1, Segment s2);
Point getCrossPointLL(Line l1, Line l2);
Polygon getCrossPointCL(Circle c, Line l);
Polygon getCrossPointCC(Circle c1, Circle c2);
int contains(Polygon g, Point p);
Polygon andrewScan(Polygon s);
Polygon convex_hull(Polygon ps);
double diameter(Polygon s);
bool isConvex(Polygon p);
double area(Polygon s);
Polygon convexCut(Polygon p, Line l);
Line bisector(Point p1, Point p2);
Vector translate(Vector v, double theta);
vector<Line> corner(Line l1, Line l2);

int ccw(Point p0, Point p1, Point p2) {
  Vector a = p1 - p0;
  Vector b = p2 - p0;
  if (cross(a, b) > EPS)
    return CCW_COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CCW_CLOCKWISE;
  if (dot(a, b) < -EPS)
    return CCW_ONLINE_BACK;
  if (a.norm() < b.norm())
    return CCW_ONLINE_FRONT;
  return CCW_ON_SEGMENT;
}

bool intersectSS(Point p1, Point p2, Point p3, Point p4) {
  return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
          ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}

bool intersectSS(Segment s1, Segment s2) {
  return intersectSS(s1.p1, s1.p2, s2.p1, s2.p2);
}

int intersectCC(Circle c1, Circle c2) {
  if (c1.r < c2.r)
    swap(c1, c2);
  double d = abs(c1.c - c2.c);
  double r = c1.r + c2.r;
  if (equals(d, r))
    return ICC_CIRCUMSCRIBE;
  if (d > r)
    return ICC_SEPERATE;
  if (equals(d + c2.r, c1.r))
    return ICC_INSCRIBE;
  if (d + c2.r < c1.r)
    return ICC_CONTAIN;
  return ICC_INTERSECT;
}

bool intersectSC(Segment s, Circle c) {
  double d = getDistanceSP(s, c.c);
  return d <= c.r;
}

double getDistanceLP(Line l, Point p) {
  return abs(cross(l.p2 - l.p1, p - l.p1) / abs(l.p2 - l.p1));
}

double getDistanceSP(Segment s, Point p) {
  if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
    return abs(p - s.p1);
  if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
    return abs(p - s.p2);
  return getDistanceLP(s, p);
}

double getDistanceSS(Segment s1, Segment s2) {
  if (intersectSS(s1, s2))
    return 0.0;
  return min(min(getDistanceSP(s1, s2.p1), getDistanceSP(s1, s2.p2)),
             min(getDistanceSP(s2, s1.p1), getDistanceSP(s2, s1.p2)));
}

Point getCrossPointSS(Segment s1, Segment s2) {
  Vector base = s2.p2 - s2.p1;
  double d1 = abs(cross(base, s1.p1 - s2.p1));
  double d2 = abs(cross(base, s1.p2 - s2.p1));
  double t = d1 / (d1 + d2);
  return s1.p1 + (s1.p2 - s1.p1) * t;
}

Point getCrossPointLL(Line l1, Line l2) {
  double a = cross(l1.p2 - l1.p1, l2.p2 - l2.p1);
  double b = cross(l1.p2 - l1.p1, l1.p2 - l2.p1);
  if (abs(a) < EPS && abs(b) < EPS)
    return l2.p1;
  return l2.p1 + (l2.p2 - l2.p1) * (b / a);
}

Polygon getCrossPointCL(Circle c, Line l) {
  Polygon p(2);
  Vector pr = project(l, c.c);
  Vector e = (l.p2 - l.p1) / abs(l.p2 - l.p1);
  double base = sqrt(c.r * c.r - norm(pr - c.c));
  p[0] = pr + e * base;
  p[1] = pr - e * base;
  return p;
}

Polygon getCrossPointCC(Circle c1, Circle c2) {
  Polygon p(2);
  double d = abs(c1.c - c2.c);
  double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
  double t = arg(c2.c - c1.c);
  p[0] = c1.c + polar(c1.r, t + a);
  p[1] = c1.c + polar(c1.r, t - a);
  return p;
}

// IN:2 ON:1 OUT:0
int contains(Polygon g, Point p) {
  int n = g.size();
  bool x = false;
  for (int i = 0; i < n; i++) {
    Point a = g[i] - p, b = g[(i + 1) % n] - p;
    if (fabs(cross(a, b)) < EPS && dot(a, b) < EPS)
      return 1;
    if (a.y > b.y)
      swap(a, b);
    if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
      x = !x;
  }
  return (x ? 2 : 0);
}

Polygon andrewScan(Polygon s) {
  Polygon u, l;
  if (s.size() < 3)
    return s;
  sort(s.begin(), s.end());
  u.push_back(s[0]);
  u.push_back(s[1]);
  l.push_back(s[s.size() - 1]);
  l.push_back(s[s.size() - 2]);
  for (int i = 2; i < (int)s.size(); i++) {
    for (int n = u.size();
         n >= 2 && ccw(u[n - 2], u[n - 1], s[i]) != CCW_CLOCKWISE; n--) {
      u.pop_back();
    }
    u.push_back(s[i]);
  }
  for (int i = s.size() - 3; i >= 0; i--) {
    for (int n = l.size();
         n >= 2 && ccw(l[n - 2], l[n - 1], s[i]) != CCW_CLOCKWISE; n--) {
      l.pop_back();
    }
    l.push_back(s[i]);
  }
  reverse(l.begin(), l.end());
  for (int i = u.size() - 2; i >= 1; i--)
    l.push_back(u[i]);
  return l;
}

Polygon convex_hull(Polygon ps) {
  int n = ps.size();
  sort(ps.begin(), ps.end(), sort_y);
  int k = 0;
  Polygon qs(n * 2);
  for (int i = 0; i < n; i++) {
    while (k > 1 && cross(qs[k - 1] - qs[k - 2], ps[i] - qs[k - 1]) < 0)
      k--;
    qs[k++] = ps[i];
  }
  for (int i = n - 2, t = k; i >= 0; i--) {
    while (k > t && cross(qs[k - 1] - qs[k - 2], ps[i] - qs[k - 1]) < 0)
      k--;
    qs[k++] = ps[i];
  }
  qs.resize(k - 1);
  return qs;
}

double diameter(Polygon s) {
  Polygon p = s;
  int n = p.size();
  if (n == 2)
    return abs(p[0] - p[1]);
  int i = 0, j = 0;
  for (int k = 0; k < n; k++) {
    if (p[i] < p[k])
      i = k;
    if (!(p[j] < p[k]))
      j = k;
  }
  double res = 0;
  int si = i, sj = j;
  while (i != sj || j != si) {
    res = max(res, abs(p[i] - p[j]));
    if (cross(p[(i + 1) % n] - p[i], p[(j + 1) % n] - p[j]) < 0.0) {
      i = (i + 1) % n;
    } else {
      j = (j + 1) % n;
    }
  }
  return res;
}

bool isConvex(Polygon p) {
  bool f = 1;
  int n = p.size();
  for (int i = 0; i < n; i++) {
    int t = ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]);
    f &= t != CCW_CLOCKWISE;
  }
  return f;
}

double area(Polygon s) {
  double res = 0;
  for (int i = 0; i < (int)s.size(); i++) {
    res += cross(s[i], s[(i + 1) % s.size()]) / 2.0;
  }
  return abs(res);
}

Polygon convexCut(Polygon p, Line l) {
  Polygon q;
  for (int i = 0; i < (int)p.size(); i++) {
    Point a = p[i], b = p[(i + 1) % p.size()];
    if (ccw(l.p1, l.p2, a) != -1)
      q.push_back(a);
    if (ccw(l.p1, l.p2, a) * ccw(l.p1, l.p2, b) < 0)
      q.push_back(getCrossPointLL(Line(a, b), l));
  }
  return q;
}

Line bisector(Point p1, Point p2) {
  Circle c1 = Circle(p1, abs(p1 - p2)), c2 = Circle(p2, abs(p1 - p2));
  Polygon p = getCrossPointCC(c1, c2);
  if (cross(p2 - p1, p[0] - p1) > 0)
    swap(p[0], p[1]);
  return Line(p[0], p[1]);
}

Vector translate(Vector v, double theta) {
  Vector res;
  res.x = cos(theta) * v.x - sin(theta) * v.y;
  res.y = sin(theta) * v.x + cos(theta) * v.y;
  return res;
}

vector<Line> corner(Line l1, Line l2) {
  vector<Line> res;
  if (isParallel(l1, l2)) {
    double d = getDistanceLP(l1, l2.p1) / 2.0;
    Vector v1 = l1.p2 - l1.p1;
    v1 = v1 / v1.abs() * d;
    Point p = l2.p1 + translate(v1, 90.0 * (PI / 180.0));
    double d1 = getDistanceLP(l1, p);
    double d2 = getDistanceLP(l2, p);
    if (abs(d1 - d2) > d) {
      p = l2.p1 + translate(v1, -90.0 * (PI / 180.0));
    }
    res.push_back(Line(p, p + v1));
  } else {
    Point p = getCrossPointLL(l1, l2);
    Vector v1 = l1.p2 - l1.p1, v2 = l2.p2 - l2.p1;
    v1 = v1 / v1.abs();
    v2 = v2 / v2.abs();
    res.push_back(Line(p, p + (v1 + v2)));
    res.push_back(Line(p, p + translate(v1 + v2, 90.0 * (PI / 180.0))));
  }
  return res;
}

Polygon tangent(Circle c1, Point p2) {
  Circle c2 = Circle(p2, sqrt(norm(c1.c - p2) - c1.r * c1.r));
  Polygon p = getCrossPointCC(c1, c2);
  sort(p.begin(), p.end());
  return p;
}
struct beet {
  int l, r;
  double x, h;
  beet(int l, int r, double x, double h) : l(l), r(r), x(x), h(h) {}
  bool operator<(const beet &a) const { return x < a.x; }
};
signed main() {
  int n;
  while (cin >> n, n) {
    vector<Circle> vc(n);
    for (int i = 0; i < n; i++) {
      double x, r;
      cin >> x >> r;
      vc[i] = Circle(Point(x, 0), r);
    }
    auto check = [&](double A) {
      double a = A / 2.0;
      // cout<<a<<endl;
      map<double, int> m;
      for (int i = 0; i < n; i++) {
        if (vc[i].r < a)
          continue;
        double d = sqrt(vc[i].r * vc[i].r - a * a);
        double l = vc[i].c.x - d, r = vc[i].c.x + d;
        m[l]++;
        m[r]--;
      }
      double res = 0, tmp = 0, pos = -1e9;
      int cur = 0;
      for (auto p : m) {
        double x = p.first;

        if (cur)
          tmp += x - pos;
        else
          tmp = 0;
        pos = x;
        res = max(res, tmp);

        cur += p.second;
      }
      return res >= A;
    };
    double l = 0, r = 1e9;
    for (int i = 0; i < 50; i++) {
      double m = (l + r) / 2;
      if (check(m))
        l = m;
      else
        r = m;
    }
    cout << fixed << setprecision(15) << l << endl;
  }
  return 0;
}

Error: TLE

Fixed submission:

#include <bits/stdc++.h>
#define EPS (1e-10)
#define equals(a, b) (fabs((a) - (b)) < EPS)
#define PI 3.141592653589793238
#define int long long

using namespace std;

// COUNTER CLOCKWISE
static const int CCW_COUNTER_CLOCKWISE = 1;
static const int CCW_CLOCKWISE = -1;
static const int CCW_ONLINE_BACK = 2;
static const int CCW_ONLINE_FRONT = -2;
static const int CCW_ON_SEGMENT = 0;

// Intercsect Circle & Circle
static const int ICC_SEPERATE = 4;
static const int ICC_CIRCUMSCRIBE = 3;
static const int ICC_INTERSECT = 2;
static const int ICC_INSCRIBE = 1;
static const int ICC_CONTAIN = 0;

struct Point {
  double x, y;
  Point() {}
  Point(double x, double y) : x(x), y(y) {}
  Point operator+(Point p) { return Point(x + p.x, y + p.y); }
  Point operator-(Point p) { return Point(x - p.x, y - p.y); }
  Point operator*(double k) { return Point(x * k, y * k); }
  Point operator/(double k) { return Point(x / k, y / k); }
  double norm() { return x * x + y * y; }
  double abs() { return sqrt(norm()); }

  bool operator<(const Point &p) const { return x != p.x ? x < p.x : y < p.y; }

  bool operator==(const Point &p) const {
    return fabs(x - p.x) < EPS && fabs(y - p.y) < EPS;
  }
};

istream &operator>>(istream &is, Point &p) {
  is >> p.x >> p.y;
  return is;
}

ostream &operator<<(ostream &os, Point p) {
  os << fixed << setprecision(12) << p.x << " " << p.y;
  return os;
}

bool sort_x(Point a, Point b) { return a.x != b.x ? a.x < b.x : a.y < b.y; }

bool sort_y(Point a, Point b) { return a.y != b.y ? a.y < b.y : a.x < b.x; }

typedef Point Vector;
typedef vector<Point> Polygon;

struct Segment {
  Point p1, p2;
  Segment() {}
  Segment(Point p1, Point p2) : p1(p1), p2(p2) {}
};
typedef Segment Line;

istream &operator>>(istream &is, Segment &s) {
  is >> s.p1 >> s.p2;
  return is;
}

struct Circle {
  Point c;
  double r;
  Circle() {}
  Circle(Point c, double r) : c(c), r(r) {}
};

istream &operator>>(istream &is, Circle &c) {
  is >> c.c >> c.r;
  return is;
}

double norm(Vector a) { return a.x * a.x + a.y * a.y; }
double abs(Vector a) { return sqrt(norm(a)); }
double dot(Vector a, Vector b) { return a.x * b.x + a.y * b.y; }
double cross(Vector a, Vector b) { return a.x * b.y - a.y * b.x; }

bool isOrthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }

bool isOrthogonal(Point a1, Point a2, Point b1, Point b2) {
  return isOrthogonal(a1 - a2, b1 - b2);
}

bool isOrthogonal(Segment s1, Segment s2) {
  return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

bool isParallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }

bool isParallel(Point a1, Point a2, Point b1, Point b2) {
  return isParallel(a1 - a2, b1 - b2);
}

bool isParallel(Segment s1, Segment s2) {
  return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

Point project(Segment s, Point p) {
  Vector base = s.p2 - s.p1;
  double r = dot(p - s.p1, base) / norm(base);
  return s.p1 + base * r;
}

Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }

double arg(Vector p) { return atan2(p.y, p.x); }

Vector polar(double a, double r) { return Point(cos(r) * a, sin(r) * a); }

int ccw(Point p0, Point p1, Point p2);
bool intersectSS(Point p1, Point p2, Point p3, Point p4);
bool intersectSS(Segment s1, Segment s2);
int intersectCC(Circle c1, Circle c2);
bool intersectSC(Segment s, Circle c);
double getDistanceLP(Line l, Point p);
double getDistanceSP(Segment s, Point p);
double getDistanceSS(Segment s1, Segment s2);
Point getCrossPointSS(Segment s1, Segment s2);
Point getCrossPointLL(Line l1, Line l2);
pair<Point, Point> getCrossPointCL(Circle c, Line l);
pair<Point, Point> getCrossPointCC(Circle c1, Circle c2);
int contains(Polygon g, Point p);
Polygon andrewScan(Polygon s);
Polygon convex_hull(Polygon ps);
double diameter(Polygon s);
bool isConvex(Polygon p);
double area(Polygon s);
Polygon convexCut(Polygon p, Line l);
Line bisector(Point p1, Point p2);
Vector translate(Vector v, double theta);
vector<Line> corner(Line l1, Line l2);

int ccw(Point p0, Point p1, Point p2) {
  Vector a = p1 - p0;
  Vector b = p2 - p0;
  if (cross(a, b) > EPS)
    return CCW_COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CCW_CLOCKWISE;
  if (dot(a, b) < -EPS)
    return CCW_ONLINE_BACK;
  if (a.norm() < b.norm())
    return CCW_ONLINE_FRONT;
  return CCW_ON_SEGMENT;
}

bool intersectSS(Point p1, Point p2, Point p3, Point p4) {
  return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
          ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}

bool intersectSS(Segment s1, Segment s2) {
  return intersectSS(s1.p1, s1.p2, s2.p1, s2.p2);
}

int intersectCC(Circle c1, Circle c2) {
  if (c1.r < c2.r)
    swap(c1, c2);
  double d = abs(c1.c - c2.c);
  double r = c1.r + c2.r;
  if (equals(d, r))
    return ICC_CIRCUMSCRIBE;
  if (d > r)
    return ICC_SEPERATE;
  if (equals(d + c2.r, c1.r))
    return ICC_INSCRIBE;
  if (d + c2.r < c1.r)
    return ICC_CONTAIN;
  return ICC_INTERSECT;
}

bool intersectSC(Segment s, Circle c) {
  double d = getDistanceSP(s, c.c);
  return d <= c.r;
}

double getDistanceLP(Line l, Point p) {
  return abs(cross(l.p2 - l.p1, p - l.p1) / abs(l.p2 - l.p1));
}

double getDistanceSP(Segment s, Point p) {
  if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
    return abs(p - s.p1);
  if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
    return abs(p - s.p2);
  return getDistanceLP(s, p);
}

double getDistanceSS(Segment s1, Segment s2) {
  if (intersectSS(s1, s2))
    return 0.0;
  return min(min(getDistanceSP(s1, s2.p1), getDistanceSP(s1, s2.p2)),
             min(getDistanceSP(s2, s1.p1), getDistanceSP(s2, s1.p2)));
}

Point getCrossPointSS(Segment s1, Segment s2) {
  Vector base = s2.p2 - s2.p1;
  double d1 = abs(cross(base, s1.p1 - s2.p1));
  double d2 = abs(cross(base, s1.p2 - s2.p1));
  double t = d1 / (d1 + d2);
  return s1.p1 + (s1.p2 - s1.p1) * t;
}

Point getCrossPointLL(Line l1, Line l2) {
  double a = cross(l1.p2 - l1.p1, l2.p2 - l2.p1);
  double b = cross(l1.p2 - l1.p1, l1.p2 - l2.p1);
  if (abs(a) < EPS && abs(b) < EPS)
    return l2.p1;
  return l2.p1 + (l2.p2 - l2.p1) * (b / a);
}

pair<Point, Point> getCrossPointCL(Circle c, Line l) {
  Vector pr = project(l, c.c);
  Vector e = (l.p2 - l.p1) / abs(l.p2 - l.p1);
  double base = sqrt(c.r * c.r - norm(pr - c.c));
  return make_pair(pr + e * base, pr - e * base);
}

pair<Point, Point> getCrossPointCC(Circle c1, Circle c2) {
  double d = abs(c1.c - c2.c);
  double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
  double t = arg(c2.c - c1.c);
  return make_pair(c1.c + polar(c1.r, t + a), c1.c + polar(c1.r, t - a));
}

int contains(Polygon g, Point p) {
  int n = g.size();
  bool x = false;
  for (int i = 0; i < n; i++) {
    Point a = g[i] - p, b = g[(i + 1) % n] - p;
    if (fabs(cross(a, b)) < EPS && dot(a, b) < EPS)
      return 1;
    if (a.y > b.y)
      swap(a, b);
    if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
      x = !x;
  }
  return (x ? 2 : 0);
}

Polygon andrewScan(Polygon s) {
  Polygon u, l;
  if (s.size() < 3)
    return s;
  sort(s.begin(), s.end());
  u.push_back(s[0]);
  u.push_back(s[1]);
  l.push_back(s[s.size() - 1]);
  l.push_back(s[s.size() - 2]);
  for (int i = 2; i < (int)s.size(); i++) {
    for (int n = u.size();
         n >= 2 && ccw(u[n - 2], u[n - 1], s[i]) != CCW_CLOCKWISE; n--) {
      u.pop_back();
    }
    u.push_back(s[i]);
  }
  for (int i = s.size() - 3; i >= 0; i--) {
    for (int n = l.size();
         n >= 2 && ccw(l[n - 2], l[n - 1], s[i]) != CCW_CLOCKWISE; n--) {
      l.pop_back();
    }
    l.push_back(s[i]);
  }
  reverse(l.begin(), l.end());
  for (int i = u.size() - 2; i >= 1; i--)
    l.push_back(u[i]);
  return l;
}

Polygon convex_hull(Polygon ps) {
  int n = ps.size();
  sort(ps.begin(), ps.end(), sort_y);
  int k = 0;
  Polygon qs(n * 2);
  for (int i = 0; i < n; i++) {
    while (k > 1 && cross(qs[k - 1] - qs[k - 2], ps[i] - qs[k - 1]) < 0)
      k--;
    qs[k++] = ps[i];
  }
  for (int i = n - 2, t = k; i >= 0; i--) {
    while (k > t && cross(qs[k - 1] - qs[k - 2], ps[i] - qs[k - 1]) < 0)
      k--;
    qs[k++] = ps[i];
  }
  qs.resize(k - 1);
  return qs;
}

double diameter(Polygon s) {
  Polygon p = s;
  int n = p.size();
  if (n == 2)
    return abs(p[0] - p[1]);
  int i = 0, j = 0;
  for (int k = 0; k < n; k++) {
    if (p[i] < p[k])
      i = k;
    if (!(p[j] < p[k]))
      j = k;
  }
  double res = 0;
  int si = i, sj = j;
  while (i != sj || j != si) {
    res = max(res, abs(p[i] - p[j]));
    if (cross(p[(i + 1) % n] - p[i], p[(j + 1) % n] - p[j]) < 0.0) {
      i = (i + 1) % n;
    } else {
      j = (j + 1) % n;
    }
  }
  return res;
}

bool isConvex(Polygon p) {
  bool f = 1;
  int n = p.size();
  for (int i = 0; i < n; i++) {
    int t = ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]);
    f &= t != CCW_CLOCKWISE;
  }
  return f;
}

double area(Polygon s) {
  double res = 0;
  for (int i = 0; i < (int)s.size(); i++) {
    res += cross(s[i], s[(i + 1) % s.size()]) / 2.0;
  }
  return abs(res);
}

Polygon convexCut(Polygon p, Line l) {
  Polygon q;
  for (int i = 0; i < (int)p.size(); i++) {
    Point a = p[i], b = p[(i + 1) % p.size()];
    if (ccw(l.p1, l.p2, a) != -1)
      q.push_back(a);
    if (ccw(l.p1, l.p2, a) * ccw(l.p1, l.p2, b) < 0)
      q.push_back(getCrossPointLL(Line(a, b), l));
  }
  return q;
}

Line bisector(Point p1, Point p2) {
  Circle c1 = Circle(p1, abs(p1 - p2)), c2 = Circle(p2, abs(p1 - p2));
  pair<Point, Point> p = getCrossPointCC(c1, c2);
  if (cross(p2 - p1, p.first - p1) > 0)
    swap(p.first, p.second);
  return Line(p.first, p.second);
}

Vector translate(Vector v, double theta) {
  Vector res;
  res.x = cos(theta) * v.x - sin(theta) * v.y;
  res.y = sin(theta) * v.x + cos(theta) * v.y;
  return res;
}

vector<Line> corner(Line l1, Line l2) {
  vector<Line> res;
  if (isParallel(l1, l2)) {
    double d = getDistanceLP(l1, l2.p1) / 2.0;
    Vector v1 = l1.p2 - l1.p1;
    v1 = v1 / v1.abs() * d;
    Point p = l2.p1 + translate(v1, 90.0 * (PI / 180.0));
    double d1 = getDistanceLP(l1, p);
    double d2 = getDistanceLP(l2, p);
    if (abs(d1 - d2) > d) {
      p = l2.p1 + translate(v1, -90.0 * (PI / 180.0));
    }
    res.push_back(Line(p, p + v1));
  } else {
    Point p = getCrossPointLL(l1, l2);
    Vector v1 = l1.p2 - l1.p1, v2 = l2.p2 - l2.p1;
    v1 = v1 / v1.abs();
    v2 = v2 / v2.abs();
    res.push_back(Line(p, p + (v1 + v2)));
    res.push_back(Line(p, p + translate(v1 + v2, 90.0 * (PI / 180.0))));
  }
  return res;
}

signed main() {
  int n;
  while (cin >> n, n) {
    Circle c[n];
    double ans = 0;
    for (int i = 0; i < n; i++) {
      cin >> c[i].c.x >> c[i].r;
      c[i].c.y = 0;
      ans = max(ans, c[i].r * sqrt(2.0));
    }
    for (int i = 0; i < n; i++) {
      for (int j = i + 1; j < n; j++) {
        if (abs(c[i].c - c[j].c) > c[i].r + c[j].r)
          break;
        auto ps = getCrossPointCC(c[i], c[j]);
        double tmp = abs(ps.first - ps.second);
        if (tmp < ans)
          break;
        Line l = Line(ps.first, ps.first + Point(1000, 0));
        double lb = 1e10, ub = -1e10;
        auto qs = getCrossPointCL(c[i], l);
        lb = min(lb, qs.first.x);
        ub = max(ub, qs.first.x);
        lb = min(lb, qs.second.x);
        ub = max(ub, qs.second.x);
        qs = getCrossPointCL(c[j], l);
        lb = min(lb, qs.first.x);
        ub = max(ub, qs.first.x);
        lb = min(lb, qs.second.x);
        ub = max(ub, qs.second.x);
        ans = max(ans, min(tmp, ub - lb));
        // cout<<i<<" "<<j<<tmp<<" "<<ub-lb<<endl;
      }
    }
    printf("%.12f\\n", ans);
  }
  return 0;
}