#import "@local/handout:0.1.0": *

= Introduction

In 2005, ID Software published the source code of _Quake III Arena_, a popular game released in 1999. \
This caused quite a stir: ID Software was responsible for many games popular among old-school engineers (most notably _Doom_, which has a place in programmer humor even today).

#v(2mm)

Naturally, this community immediately began dissecting _Quake_'s source. \
One particularly interesting function is reproduced below, with original comments: \

#v(3mm)

```c
float Q_rsqrt( float number ) {
	long i;
	float x2, y;
	const float threehalfs = 1.5F;

	x2 = number * 0.5F;
	y  = number;
	i  = * ( long * ) &y;						// evil floating point bit level hacking
	i  = 0x5f3759df - ( i >> 1 );               // [redacted]
	y  = * ( float * ) &i;
	y  = y * ( threehalfs - ( x2 * y * y ) );   // 1st iteration
//	y  = y * ( threehalfs - ( x2 * y * y ) );   // 2nd iteration, this can be removed

	return y;
}
```

#v(3mm)

This code defines a function `Q_sqrt`, which was used as a fast approximation of the inverse square root in graphics routines. (in other words, `Q_sqrt` efficiently approximates $1 div sqrt(x)$)

#v(3mm)

The key word here is "fast": _Quake_ ran on very limited hardware, and traditional approximation techniques (like Taylor series)#footnote[Taylor series aren't used today, and for the same reason. There are better ways.] were too computationally expensive to be viable.

#v(3mm)

Our goal today is to understand how `Q_sqrt` works. \
To do that, we'll first need to understand how computers represent numbers. \
We'll start with simple binary integers---turn the page.