Holy cow, I wrote a book!
Exercise: Why do we use the formula c = a + (b-a)/2 instead
of the simpler c = (a+b)/2?
c = a + (b-a)/2
c = (a+b)/2
Answer: To avoid integer overflow in the computation of a+b.
Here, a and b are window coordinates, and the window can
be anywhere. If the window were placed at extreme coordinates like (MAXLONG,MAXLONG),
then the arithmetic would overflow and the "midpoint" would be incorrectly computed.
Note that the alternate formula a+(b-a)/2 is also subject to overflow,
this time in the computation of the value b-a. However, in our case, b-a is
the width of our window, which is something that we can control.
Integer overflow was one of the Windows 95 application compatibility bugs that I had
to deal with. There was a DOS game that wanted to do a binary search, and instead
of using indices, they attempted to average the two pointers together:
BYTE *low = ...;
BYTE *high = ...;
BYTE *mid = ((UINT)low + (UINT)high)/2;
This worked as long as the game was being run under an operating system without virtual
memory, because the "low" and "high" pointers would both be comparatively small numbers
(nobody had machines with 2GB of RAM), so the sum low+high would not
Windows 95 ran these DOS games, but under a DPMI server that supported virtual memory.
The DPMI specification permits the server to put memory anywhere, and we put our memory
at the high end of the address space.
This program then overflowed in its attempt to average the two pointers and crashed.