Added introduction

src/Advanced/Fast Inverse Root/main.typ

@@ -13,16 +13,19 @@
   by: "Mark",
 )
 
-#include "parts/00 int.typ"
+#include "parts/00 intro.typ"
 #pagebreak()
 
-#include "parts/01 float.typ"
+#include "parts/01 int.typ"
 #pagebreak()
 
-#include "parts/02 approx.typ"
+#include "parts/02 float.typ"
 #pagebreak()
 
-#include "parts/03 quake.typ"
+#include "parts/03 approx.typ"
 #pagebreak()
 
-#include "parts/04 bonus.typ"
+#include "parts/04 quake.typ"
+#pagebreak()
+
+#include "parts/05 bonus.typ"

src/Advanced/Fast Inverse Root/parts/00 intro.typ

@@ -0,0 +1,45 @@
+#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 techinques (like Taylor series)#footnote[In fact, 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---turn the page.

src/Advanced/Fast Inverse Root/parts/04 quake.typ

@@ -2,10 +2,9 @@
 
 = The Fast Inverse Square Root
 
+A simplified version of the _Quake_ routine we are studying is reproduced below.
 
-The following code is present in _Quake III Arena_ (1999):
-
-#v(5mm)
+#v(2mm)
 
 ```c
 float Q_rsqrt( float number ) {
@@ -15,7 +14,7 @@ float Q_rsqrt( float number ) {
 }
 ```
 
-#v(5mm)
+#v(2mm)
 
 This code defines a function `Q_rsqrt` that consumes a float `number` and approximates its inverse square root.
 If we rewrite this using notation we're familiar with, we get the following:
@@ -165,8 +164,7 @@ though it is fairly close to the ideal $epsilon$.
 
 #remark()
 And now, we're done! \
-We've shown that `Q_sqrt(x)` approximates $1/sqrt(x)$ fairly well, \
-thanks to the approximation $log(1+a) = a + epsilon$.
+We've shown that `Q_sqrt(x)` approximates $1/sqrt(x)$ fairly well. \
 
 #v(2mm)