Simulating pi from R or C++ in about five lines

Dirk Eddelbuettel — written Dec 20, 2012 — source

This example takes the standard simulation for the value of pi and in implements it in both R and C++. We will see that the C++ version is very similar to the R version.

We start with an R version.

The basic idea is that for a point (x,y), we compute the distance to the origin using Pythargoras’ well-known expression, or in this context a standard distance norm. We do this repeatedly for a number of points, the ratio of those below one (“inside the unit circle”) to the number N of simulation will approach pi/4 – as we were filling the area of one quarter of the unit circle by limiting ourselves the first quadrant (as we forced x and y to be between 0 and 1).

The key here is that we do this in a vectorised way: by drawing all N x and y coordinates, then computing all distances and comparing to 1.0. After scaling up to the full circle, we have an estimate of pi.

piR <- function(N) {
    x <- runif(N)
    y <- runif(N)
    d <- sqrt(x^2 + y^2)
    return(4 * sum(d < 1.0) / N)

c(piR(1000), piR(10000), piR(100000), piR(1000000))
[1] 3.156 3.155 3.139 3.141

The neat thing about Rcpp sugar enables us to write C++ code that looks almost as compact.

#include <Rcpp.h>
using namespace Rcpp;

// [[Rcpp::export]]
double piSugar(const int N) {
    NumericVector x = runif(N);
    NumericVector y = runif(N);
    NumericVector d = sqrt(x*x + y*y);
    return 4.0 * sum(d < 1.0) / N;

Apart from using types (hey, this is C++) and assuring the RNG gets set and reset, the code is essentially identical.

And by using the same RNG, so are the results. Rcpp ensures that the RNG state is set and reset properly by instantiating an object of class RNGScope.

c(piSugar(1000), piSugar(10000), piSugar(100000), piSugar(1000000))
[1] 3.156 3.155 3.139 3.141

tags: sugar 

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