Problem p03168 (C++)

Buggy submission (status: Time Limit Exceeded):
// --------------------<optimizations>--------------------
#pragma GCC optimize("O3")
//(UNCOMMENT WHEN HAVING LOTS OF RECURSIONS)\
#pragma comment(linker, "/stack:200000000")
//(UNCOMMENT WHEN TRYING TO BRUTEFORCE WITH A LOT OF LOOPS)\
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>

using namespace std;

#define ll long long
#define mp make_pair
#define pb push_back
#define eb emplace_back
#define pii pair<ll, ll>
#define vpii vector<pair<ll, ll>>
#define F first
#define S second
#define ld long double
#define built __builtin_popcountll
#define mst(a, i) memset(a, i, sizeof(a))
#define all(x) x.begin(), x.end()
#define itit(it, a) for (auto it = (a).begin(); it != (a).end(); it++)
#define rep(i, a, b) for (ll i = a; i < b; i++)
#define repr(i, a, b) for (ll i = a; i > b; i--)
#define reprr(i, a, b) for (ll i = a; i >= b; i--)
#define pi 3.14159265358979323846264338327950288419716939937510582097494459230
ll max3(ll x, ll y, ll z) { return max(max(x, y), z); }
ll min3(ll x, ll y, ll z) { return min(min(x, y), z); }
const ll M = 2e5 + 10, M2 = 1e6 + 10, mod = 1e9 + 7, inf = 1e17 + 10;
void add(int &a, int b) {
  a += b;
  if (a >= mod) {
    a -= mod;
  }
}
#define trace1(x) cerr << #x << ": " << x << endl
#define trace2(x, y) cerr << #x << ": " << x << " | " << #y << ": " << y << endl
#define trace3(x, y, z)                                                        \
  cerr << #x << ":" << x << " | " << #y << ": " << y << " | " << #z << ": "    \
       << z << endl
#define trace4(a, b, c, d)                                                     \
  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": "   \
       << c << " | " << #d << ": " << d << endl
#define trace5(a, b, c, d, e)                                                  \
  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": "   \
       << c << " | " << #d << ": " << d << " | " << #e << ": " << e << endl
#define trace6(a, b, c, d, e, f)                                               \
  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": "   \
       << c << " | " << #d << ": " << d << " | " << #e << ": " << e << " | "   \
       << #f << ": " << f << endl
int X[] = {0, 1, 0, -1};
int Y[] = {-1, 0, 1, 0};
ll power(ll x, ll n) {
  ll result = 1;
  while (n > 0) {
    if (n % 2 == 1)
      result = (result * x) % mod;
    x = ((x % mod) * (x % mod)) % mod;
    n = n / 2;
  }
  return result;
}

ll n;
ld p[3000];
ld dp[3000][3000];

int main() {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  cin >> n;
  for (ll i = 1; i <= n; i++) {
    cin >> p[i];
  }
  dp[0][0] = 1;
  for (ll i = 0; i < n; i++) {
    for (ll j = 0; j <= n; j++) {
      dp[i + 1][j + 1] += dp[i][j] * p[i + 1];
      dp[i + 1][j] += dp[i][j] * (1.0 - p[i + 1]);
    }
  }
  ld ans = 0;
  for (ll i = 0; i < n; i++) {
    for (ll j = 0; j < n; j++)
      trace1(dp[i][j]);
  }
  for (ll i = n / 2 + 1; i <= n; i++) {
    ans += dp[n][i];
  }
  cout << fixed << setprecision(18) << ans;
  return 0;
}
/* The judge is never wrong! Your code is buggy
   Look for:
 * * Read the problem carefully.
 * * Don't forget to sort(), mod, ll!!!!
 * * Initial value = +/- infinity instead of zero!!!
 * * an easier and alternate approach
 * * read the problem statement carefully
 * * if concept is correct and still WA, try with a different implementation
 * * special cases (n=1?)
 * * if you have no idea just guess the appropriate well-known algorithm instead
 of doing nothing :/
 */

Fixed submission:
// --------------------<optimizations>--------------------
#pragma GCC optimize("O3")
//(UNCOMMENT WHEN HAVING LOTS OF RECURSIONS)\
#pragma comment(linker, "/stack:200000000")
//(UNCOMMENT WHEN TRYING TO BRUTEFORCE WITH A LOT OF LOOPS)\
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>

using namespace std;

#define ll long long
#define mp make_pair
#define pb push_back
#define eb emplace_back
#define pii pair<ll, ll>
#define vpii vector<pair<ll, ll>>
#define F first
#define S second
#define ld long double
#define built __builtin_popcountll
#define mst(a, i) memset(a, i, sizeof(a))
#define all(x) x.begin(), x.end()
#define itit(it, a) for (auto it = (a).begin(); it != (a).end(); it++)
#define rep(i, a, b) for (ll i = a; i < b; i++)
#define repr(i, a, b) for (ll i = a; i > b; i--)
#define reprr(i, a, b) for (ll i = a; i >= b; i--)
#define pi 3.14159265358979323846264338327950288419716939937510582097494459230
ll max3(ll x, ll y, ll z) { return max(max(x, y), z); }
ll min3(ll x, ll y, ll z) { return min(min(x, y), z); }
const ll M = 2e5 + 10, M2 = 1e6 + 10, mod = 1e9 + 7, inf = 1e17 + 10;
void add(int &a, int b) {
  a += b;
  if (a >= mod) {
    a -= mod;
  }
}
#define trace1(x) cerr << #x << ": " << x << endl
#define trace2(x, y) cerr << #x << ": " << x << " | " << #y << ": " << y << endl
#define trace3(x, y, z)                                                        \
  cerr << #x << ":" << x << " | " << #y << ": " << y << " | " << #z << ": "    \
       << z << endl
#define trace4(a, b, c, d)                                                     \
  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": "   \
       << c << " | " << #d << ": " << d << endl
#define trace5(a, b, c, d, e)                                                  \
  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": "   \
       << c << " | " << #d << ": " << d << " | " << #e << ": " << e << endl
#define trace6(a, b, c, d, e, f)                                               \
  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": "   \
       << c << " | " << #d << ": " << d << " | " << #e << ": " << e << " | "   \
       << #f << ": " << f << endl
int X[] = {0, 1, 0, -1};
int Y[] = {-1, 0, 1, 0};
ll power(ll x, ll n) {
  ll result = 1;
  while (n > 0) {
    if (n % 2 == 1)
      result = (result * x) % mod;
    x = ((x % mod) * (x % mod)) % mod;
    n = n / 2;
  }
  return result;
}

ll n;
ld p[3000];
ld dp[3000][3000];

int main() {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  cin >> n;
  for (ll i = 1; i <= n; i++) {
    cin >> p[i];
  }
  dp[0][0] = 1;
  for (ll i = 0; i < n; i++) {
    for (ll j = 0; j <= n; j++) {
      dp[i + 1][j + 1] += dp[i][j] * p[i + 1];
      dp[i + 1][j] += dp[i][j] * (1.0 - p[i + 1]);
    }
  }
  ld ans = 0;
  for (ll i = n / 2 + 1; i <= n; i++) {
    ans += dp[n][i];
  }
  cout << fixed << setprecision(18) << ans;
  return 0;
}
/* The judge is never wrong! Your code is buggy
   Look for:
 * * Read the problem carefully.
 * * Don't forget to sort(), mod, ll!!!!
 * * Initial value = +/- infinity instead of zero!!!
 * * an easier and alternate approach
 * * read the problem statement carefully
 * * if concept is correct and still WA, try with a different implementation
 * * special cases (n=1?)
 * * if you have no idea just guess the appropriate well-known algorithm instead
 of doing nothing :/
 */