为什么我的程序不近似 pi？

Question

我正在尝试编写一个通过使用两个随机数互质的概率来近似 pi 的程序，即 6/pi^2 所以我应该能够将 pi 近似为 sqrt(6/probability) 但出于某种原因它不倾向于 pi，而是倾向于 3.912。我试过 rand() 函数，也试过 mersenne twister 随机数生成器，但它们都给了我相似的结果。这是怎么回事？这是我在 C++ 中的代码

#include <iostream>
#include <ctime>
#include <cstdlib>
#include <cmath>
#include <algorithm>

using namespace std;

int main()
{
    srand(time(0));
    int times;

    cout << "enter number of times: ";
    cin >> times;

    for(int i = 0; i <= times; i ++)
    {
        double pi_approx, probability;
        int num1, num2, number_of_coprime;
        num1 = rand();
        num2 = rand();
        if(__gcd(num1, num2)>1)
        {
           number_of_coprime++;
        }

        probability = double(number_of_coprime)/double(i);
        pi_approx = sqrt(6/probability);

        cout <<100*i/times<< "% number of coprimes: "<< number_of_coprime << " pi approximation: " << pi_approx<<endl;

    }

    return 0;
}

Answer 1

您需要在适当的范围内声明变量，正确的公式是随着次数趋于无穷大，概率趋于 6/pi^2。试试下面的代码：

int main()
{
    srand(time(0));

    //for (int times = 100; times < 1000; ++times) 
    int times = 1000000;
    {
        double pi_approx, probability;
        int number_of_coprime = 0;
        for (int i = 0; i < times; i++)
        {
            int num1, num2;
            num1 = rand() % times;
            num2 = rand() % times;
            if (__gcd(num1, num2) == 1) // increment if coprime !!
            {
                number_of_coprime++;
            }
        }
        probability = double(number_of_coprime) / double(times);
        pi_approx = sqrt(6 / probability);
        cout << pi_approx << endl;
    }

    return 0;
}

输出是：

3.14179

Answer 2

你计算错了：你应该在它们互质时增加计数器 (gcd == 1)，但是当 gcd > 1 时你应该增加。