April 3, 2018

constexpr is one of C++11’s features I’m starting to like very much. constexpr is a bit finicky, but it allows you to evaluate functions—including ctors—at compile time. This of course, allows computations to be replaced directly by results.

So in the best of cases, you could end-up with less code, or better yet, no code at all!

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Yes? No? Maybe? (Part II)

March 27, 2018

Last week, we had a look at how to implement a trool, or a tri-valued boolean what accepts true, false, and undefined. We remarked that the storage of an enum likely defaults to int, and that my poc wouldn’t play well with std::vector as that container has no specialization to deal with this new type.

A specialization would be interesting because we can do much better than using an integer to store three different values. We can do much, much better.

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Yes? No? Maybe? (Part I)

March 20, 2018

Initializing arrays, or any variable for that matter, is always kind of a problem. Most of the times, you can get away with a default value, typically zero in C#C++, but not always. For floats, for example, NaN makes much more sense. Indeed, it’s initialized to not a number: it clearly states that it is initialized, consciously, to not a value. That’s neat. What about integers? Clearly, there’s no way to encode a NaI (not an integer), maybe std::numeric_limits::min(), which is still better than zero. What about bools?

Bool is trickier. In C++, bool is either false or true, and weak typing makes everything not zero true. However, if you assign 3 to a bool, it will be “normalized” to true, that is, exactly 1. Therefore, and not that surprisingly, you can’t have true, false, and maybe. Well, let’s fix that.

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Paeth’s Method (Square Roots, Part VII)

March 13, 2018

In Graphics Gems [1], Paeth proposes a fast (but quite approximate) method for the rapid computation of hypotenuse,

\displaystyle h=\sqrt{x^2+y^2}.

The goal here is to get rid of the big bad \sqrt{} because it is deemed “too expensive”—I wonder if that’s still actually true. First, he transforms the above equation:

\displaystyle h=\sqrt{x^2+y^2}

\displaystyle =\sqrt{x^2\left(1+\frac{y^2}{x^2}\right)}

\displaystyle =x\sqrt{1+\frac{y^2}{x^2}}.

This leaves us with the special case $\sqrt{1+u^2}, for which we can find the Taylor series

\displaystyle \sqrt{1+u^2}=1+\frac{1}{2}u^2-\frac{1}{8}u^4+\cdots

The numerators are given by A002596 and the denominators by A046161. So we can rewrite the whole thing as

\displaystyle x\sqrt{1+\frac{y^2}{x^2}}=x\left(1+\frac{1}{2}\left(\frac{y}{x}\right)^2-\frac{1}{8}\left(\frac{y}{x}\right)^4+\cdots\right).

* *

We’ll now try to get away with only a few terms of the series, and hope the precision isn’t too damaged. say we keep only the first two terms, how does it compare to other methods?

Let’s break down the graph:

  • Blue: Circle of radius 10, the real hypotenuse;
  • Green: True radius, 10;
  • Dotted purple: Archytas, 3 iterations;
  • Purple: Bakhshali’s method;
  • Red: Paeth’s method.

Since Paeth’s method works well if y<x, we may tweak it a bit. Letting


then letting

P(x,y)=\begin{cases}  Q(x,y)&\textrm{if~}y<x,\\  Q(y,x)&\textrm{otherwise}.  \end{cases}

The imprecision is maximal when x\approx{y}, or when the hypotenuse is at an angle of about 45 degrees.

* *

Originally, this hack was for considered for fast collisions/proximity testing. If you absolutely need a circular (or spherical) region, you need a square root, but that’s not the only option:

  • L_1 metric, which is just the sum of absolute values |x|+|y|;
  • L_2 metric, the usual euclidean distance;
  • L_\infty metric, which is \max(x,y).

The L_2 metric requires the square root, but the others are (if branches are cheap), rather inexpensive to compute—but they have counter-intuitive behavior. They might be useful for a first faster check, followed by Paeth’s method, followed by a full test.

[1] lan W. Paeth — VIII.5 A Fast Approximation to the Hypotenuse — in : Andrew Glassnet, ed., Graphics Gems, Morgan Kaufmann (1993), p. 427– 431

Encoding seeds

March 6, 2018

I was discussing procedural generation with one of my students when he brought up The Binding of Isaac, that delightfully quirky and creepy rogue-like game. One of the interesting features of the game is that the dungeons are randomly generated and that you can get the seeds for the dungeons and share them. A typical seed looks something like this:


So what does it encode?

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Random Points on a Sphere (Generating Random Sequences III, Revisited)

February 27, 2018

While searching for old notes—that I haven’t found anyway—I returned to an old blog entry and I thought I was kind of unsatisfactory, with be best part being swept under the carpet with a bit a faery dust, and very handwavingly.

So let’s work-out how to uniformly distribute points on a sphere in a more satisfactory fashion.

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Square roots (Part VI)

February 20, 2018

I’ve discussed algorithms for computing square roots a couple of times already, and then some. While sorting notes, I’ve came across something interesting: Archytas’ method for computing square roots.

Archytas’ method is basically the old Babylonian method, where you first set



and iterate

\displaystyle a'=\frac{a+b}{2},

\displaystyle b'=\frac{n}{a'}=\frac{2n}{a+b},

until desired precision is achieved (or the scribe is exhausted).

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