Notation as a Tool of Thought (1979) [pdf]

I have to confess that I don't find this sort of example very compelling; Box-Muller isn't much more than a one-liner in any language. I mean, here it is in C, not noted for its richly expressive terseness:

  double norm(void) {
    while (1) {
      double u=rand01(), v=rand01();
      double s = u*u+v*v;
      if (0 < s && s <= 1) return u * sqrt(-2*log(s)/s);
    }
  }

I bet the C implementation is distinctly easier to make sense of for most mathematicians and most programmers, and it's not even substantially longer. (APL: 94 non-blank characters. C: 111 non-blank characters. I remark that counting non-blank characters is about as favourable a metric for APL as one can reasonably ask for.) One can code-golf it further: combine the declaration of u,v with that of s (saves 5 non-blank characters --> 106); switch to your "u" instead of my "rand01" and rename u,v to (say) x,y (saves 10 characters --> 96); convert to do/while and use commas instead of {;;;} for body (saves 4 characters -> 92). That looks like this, and I think is still pretty easy to read:

  double norm(void) {
    double x,y,s;
    do x=u(), y=u(), s=x*x+y*y; while (s==0 || s>1);
    return x*sqrt(-2*log(s)/s);
  }

[EDITED to fix formatting screwage; aargh, I hate HN's treatment of asterisks]