Integer Overflow - Origin

Origin

The register width of a processor determines the range of values that can be represented. Typical binary register widths include:

8 bits: maximum representable value 28 − 1 = 255

16 bits: maximum representable value 216 − 1 = 65,535

32 bits: maximum representable value 232 − 1 = 4,294,967,295 (the most common width for personal computers as of 2005),

64 bits: maximum representable value 264 − 1 = 18,446,744,073,709,551,615 (the most common width for personal computers, but not necessarily their operating systems, as of 2012),

128 bits: maximum representable value 2128 − 1 = 340,282,366,920,938,463,463,374,607,431,768,211,455

Since an arithmetic operation may produce a result larger than the maximum representable value, a potential error condition may result. In the C programming language, signed integer overflow causes undefined behavior, while unsigned integer overflow causes the number to be reduced modulo a power of two, meaning that unsigned integers "wrap around" on overflow. This "wrap around" is the cause of the famous "Split Screen" in Pac-Man. A "wrap around" corresponds to the fact, that e.g. if the addition of two positive integers produces an overflow, it may result in a negative number. In counting, one just starts over again from the bottom. Example: 16 bit signed integer: 30000 + 30000 = −5536.

In computer graphics or signal processing, it is typical to work on data that ranges from 0 to 1 or from −1 to 1. An example of this is a grayscale image where 0 represents black, 1 represents white, and values in-between represent varying shades of gray. One operation that one may want to support is brightening the image by multiplying every pixel by a constant. Saturated arithmetic allows one to just blindly multiply every pixel by that constant without worrying about overflow by just sticking to a reasonable outcome that all these pixels larger than 1 (i.e. "brighter than white") just become white and all values "darker than black" just become black.