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Fast Monte Carlo Method with JavaScript

How many times should a random number from [0, 1] be drawn to have it sum over 1?

If you want to figure it out for yourself, stop reading now and come back when you're done.

The answer is e. When I came across this question I took the lazy programmer route and, rather than work out the math, I estimated the answer using the Monte Carlo method. I used the language I always use for these scratchpad computations: Emacs Lisp. All I need to do is switch to the *scratch* buffer and start hacking. No external program needed.

The downside is that Elisp is incredibly slow. Fortunately, Elisp is so similar to Common Lisp that porting to it is almost trivial. My preferred Common Lisp implementation, SBCL, is very, very fast so it's a huge speed upgrade with little cost, should I need it. As far as I know, SBCL is the fastest Common Lisp implementation.

Even though Elisp was fast enough to determine that the answer is probably e, I wanted to play around with it. This little test program doubles as a way to estimate the value of e, similar to estimating pi. The more trial runs I give it the more accurate my answer will get -- to a point.

Here's the Common Lisp version. (I love the loop macro, obviously.)

(defun trial ()
  (loop for count upfrom 1
     sum (random 1.0) into total
     until (> total 1)
     finally (return count)))

(defun monte-carlo (n)
  (loop repeat n
     sum (trial) into total
     finally (return (/ total 1.0 n))))

Using SBCL 1.0.57.0.debian on an Intel Core i7-2600 CPU, once everything's warmed up this takes about 9.4 seconds with 100 million trials.

(time (monte-carlo 100000000))
Evaluation took:
  9.423 seconds of real time
  9.388587 seconds of total run time (9.380586 user, 0.008001 system)
  99.64% CPU
  31,965,834,356 processor cycles
  99,008 bytes consed
2.7185063

Since this makes for an interesting benchmark I gave it a whirl in JavaScript,

function trial() {
    var count = 0, sum = 0;
    while (sum <= 1) {
        sum += Math.random();
        count++;
    }
    return count;
}

function monteCarlo(n) {
    var total = 0;
    for (var i = 0; i < n; i++) {
        total += trial();
    }
    return total / n;
}

I ran this on Chromium 24.0.1312.68 Debian 7.0 (180326) which uses V8, currently the fastest JavaScript engine. With 100 million trials, this only took about 2.7 seconds!

monteCarlo(100000000); // ~2.7 seconds, according to Skewer
// => 2.71850356

Whoa! It beat SBCL! I was shocked. Let's try using C as a baseline. Surely C will be the fastest.

#include <stdio.h>
#include <stdlib.h>

int trial() {
    int count = 0;
    double sum = 0;
    while (sum <= 1.0) {
        sum += rand() / (double) RAND_MAX;
        count++;
    }
    return count;
}

double monteCarlo(int n) {
    int i, total = 0;
    for (i = 0; i < n; i++) {
        total += trial();
    }
    return total / (double) n;
}

int main() {
    printf("%f\n", monteCarlo(100000000));
    return 0;
}

I used the highest optimization setting on the compiler.

$ gcc -ansi -W -Wall -Wextra -O3 temp.c
$ time ./a.out
2.718359

real    0m3.782s
user    0m3.760s
sys 0m0.000s

Incredible! JavaScript was faster than C! That was completely unexpected.

The Circumstances

Both the Common Lisp and C code could probably be carefully tweaked to improve performance. In Common Lisp's case I could attach type information and turn down safety. For C I could use more compiler flags to squeeze out a bit more performance. Then maybe they could beat JavaScript.

In contrast, as far as I can tell the JavaScript code is already as optimized as it can get. There just aren't many knobs to tweak. Note that minifying the code will make no difference, especially since I'm not measuring the parsing time. Except for the functions themselves, the variables are all local, so they are never "looked up" at run-time. Their name length doesn't matter. Remember, in JavaScript global variables are expensive, because they're (generally) hash table lookups on the global object at run-time. For any decent compiler, local variables are basically precomputed memory offsets -- very fast.

The function names themselves are global variables, but the V8 compiler appears to eliminate this cost (inlining?). Wrapping the entire thing in another function, turning the two original functions into local variables, makes no difference in performance.

While Common Lisp and C may be able to beat JavaScript if time is invested in optimizing them -- something to be done rarely -- in a casual implementation of this algorithm, JavaScript beats them both. I find this really exciting.

tags: [ emacs lisp c javascript ]
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