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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Unary numbers.

February 13, 2018

A positional number system needs a base that is either greater than one, or smaller than minus one—yes, we can have a negative base for a number system. The system, however, seems to break down if the base we chose is base 1.

If the base is 1, then there are no permissible digits since the digits d, in a base b system, must be 0\leqslant{d}<b. But we can still represent numbers using just 1s. That's the unary numeral system, and numbers are just represented as repeated 1s. 15? Fifteen ones: 111111111111111. Operations? Not very complicated, just… laborious.

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Elementary Automata (Generating Random Sequences XI)

February 6, 2018

So 3-cells context elementary automata seem too “self-correcting” to be useful pseudo-random generators. What if we fixed that boundary problem and have the automaton run on a cylinder (with both end joined)? What if we augment the context from 3 to 5 cells?

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