sorting algorithm

#include <iostream>
#include <vector>
#include <cmath>

using namespace std;

/**
 * Finds the vector index of the searched value OR the index of the greatest value less than the searched value.
 *
 * @param vector $haystack An ordered array of integers
 * @param int $needle The element to be found
 * @return int $index The index of the matching element.
*/
int vector_search(vector<int> haystack, int needle) {
    int start = 0, end = haystack.size() - 1, middle = ceil((start + end) / 2);

    while (end - start > 1) {
        if (haystack[middle] == needle) {
            return middle;
        }

        if (haystack[middle] > needle) {
            end = middle;
        } else {
            start = middle;
        }

        middle = ceil((start + end) / 2);
    }

    if (haystack[end] == needle) {
        return end;
    }

    return start;
}

int main() {
    int N;
    vector<int> numbers;
    numbers.reserve(N);

    for (int i = 0; i < N; i++) {
        int value, position;
        cin >> value;
        position = vector_search(numbers, value);
        numbers.insert(position + 1, value);
    }

    for (int i = 0; i < (N - 1); i++) {
        cout << numbers[i] << " ";
    }

    cout << numbers[N - 1] << endl;
}