Different methods in C++/Rcpp code for calculating the constant Pi: Gregory Leibniz, Nilakantha, and Monte Carlo. Plotting is done in R.

pi.cpp

#include <Rcpp.h>
#include <iostream>
#include <random>
#include <chrono>
#include <math.h>
using namespace Rcpp;

// [[Rcpp::export]]
long double gregoryLeibnizCpp(int iterations) {
	long double pi = 1;
	bool op = true;
	for(int i = 3; iterations > i; i = i + 2) {
		if(op) {
			pi = pi - (1 / (long double) i);
			op = false;
		} else {
			pi = pi + (1 / (long double) i);
			op = true;
		}
	}
	return pi * 4;
}

// [[Rcpp::export]]
long double nilakanthaCpp(int iterations) {
	long double pi = 3;
	bool op = true;
	for(long double i = 2; iterations > i; i = i + 2) {
		if(op) {
			pi = pi + (4 / (i * (i + 1) * (i + 2)));
			op = false;
		} else {
			pi = pi - (4 / (i * (i + 1) * (i + 2)));
			op = true;
		}
	}
	return pi;
}


std::mt19937 rng(std::chrono::steady_clock::now().time_since_epoch().count());

// [[Rcpp::export]]
DataFrame monteCarloPi(double iterations) {
	NumericVector x(iterations);
	NumericVector y(iterations);
	NumericVector colour(iterations);
	
	double circle = 0;
	double width = 1;
	
	for(int i = 0; i < iterations; i++) {
		x[i] = std::uniform_real_distribution<double>(0, width)(rng);
		y[i] = std::uniform_real_distribution<double>(0, width)(rng);
		
		if(std::sqrt(x[i] * x[i] + y[i] * y[i]) <= width) {
			circle++;
			colour[i] = 1;
		} else {
			colour[i] = 0;
		}
	}
	
	double pi = (circle / iterations) * 4.0;
	std::cout << pi << std::endl;
	
	return DataFrame::create(_["x"] = x, _["y"] = y, _["colour"] = colour);
}

Plot Monte Carlo results:

res <- monteCarloPi(100000)

par(bg = "black", mar = rep(0, 4))
plot(res[res$colour != 1, "x"], res[res$colour != 1, "y"], cex = 0.1, col = "gray")
points(res[res$colour == 1, "x"], res[res$colour == 1, "y"], cex = 0.1, col = "red")