Find the number of a given number in the Fibonacci sequence

The task itself: http://www.codeabbey.com/index/task_view/fibonacci-sequence--ru, my code:

#include <iostream>
#include <vector>
using namespace std;

int main()
{
    vector<long double> fib{0, 1};
    for(int i = 1; i < 1000; i++) // По условию все числа из первой тысячи
                                  // значений последовательности
    {
        fib.push_back(fib[i] + fib[i-1]);
    }

    int N; // Первая строчка должна содержать количество проводимых тестов
    cin >> N;

 long double value[N];
for(int i = 0; i < N; i++)
{
    cin >> value[i];
}

    int begin, end;
    for (int i = 0; i < N; i++)
    {
        if(value[i] > fib[fib.size() / 2]) // Если value находится во второй половине последовательности
        {
            begin = fib.size() / 2;
            while(value[i] > fib[begin]) 
            {
                begin /= 2;
            }
            begin *= 2; // Возвращаюсь на шаг назад
                        // ибо после выполнения цикла
                        // value выходит из диапозона
            end = fib.size();
            for( ; begin < end; begin++)
            {
                if(fib[begin] == value[i]) {cout << begin << " "; break;}
            }
        }

        else
        {
            end = fib.size() / 2;
            while (value[i] < fib[end])
            {
                end /= 2;
            }
            end *= 2; // То же самое что и в первой ветви
            begin = 0;
            for ( ; begin < end; begin++)
            {
                if(fib[begin] == value[i]) {cout << begin << " "; break;}
            }
        }
    }

    return 0;
}

It would be possible not to make such a game and just write

for(int i = 0; i < N; i++)
    for(int j = 0; j < fib.size(); j++)
        if(value[i] == fib[j]) cout << j + 1 << " ";

But while reading one smart book, I became interested in recreating a simplified version of the binary_search() algorithm, and in general, here :)

The site with the task also kindly offers data entry to check the program, I was given such:

For the sake of a small test, I tried to enter only a few of them, and more specifically - the very first, second and fourth. To my surprise, all I got on the way out was nothing.

Disappointed in my code, I redid the code to

vector<long double> fib{0, 1}; 
    long double value;
         cin >> value;

    int i = 1;
    int number_of_element = 1;
    while (value > fib.back()) {
        fib.push_back(fib[i] + fib[i-1]);
        i++;
        number_of_element++;
    }
    cout << fixed << "Element: " << fib[i-1] << " It's number: " << --number_of_element;

To find out what I'm wrong about and at the same time check if the numbers just come across more than the first thousand. I entered the same thing in turn: the first, second, and third numbers from Test data. In all three results the number obtained by the program was almost identical to the one entered, up to about 19 characters (I lost count many times on this it is not accurate), the element number also did not exceed a thousand. Based on this, I concluded that the machine simply had an error after a certain number of characters. Is there any way to fix this problem?