Saturation arithmetic
Saturation arithmetic is a version of arithmetic in which all operations such as addition and multiplication are limited to a fixed range between a minimum and maximum value.
If the result of an operation is greater than the maximum, it is set ("clamped") to the maximum; if it is below the minimum, it is clamped to the minimum. The name comes from how the value becomes "saturated" once it reaches the extreme values; further additions to a maximum or subtractions from a minimum will not change the result.
For example, if the valid range of values is from -100 to 100, the following operations produce the following values:
As can be seen from these examples, familiar properties like associativity and distributivity may fail in saturation arithmetic. This makes it unpleasant to deal with in abstract mathematics, but it has an important role to play in digital hardware and algorithms.
Typically, general-purpose microprocessors do not implement integer arithmetic operations using saturation arithmetic; instead, they use the easier-to-implement modular arithmetic, in which values exceeding the maximum value "wrap around" to the minimum value, like the hours on a clock passing from 12 to 1. In hardware, modular arithmetic with a minimum of zero and a maximum of rn-1, where r is the radix can be implemented by simply discarding all but the lowest n digits. For binary hardware, which the vast majority of modern hardware is, the radix is 2 and the digits are bits.
However, although more difficult to implement, saturation arithmetic has numerous practical advantages. The result is as numerically close to the true answer as possible; for 8-bit binary signed arithmetic, when the correct answer is 130, it is considerably less surprising to get an answer of 127 from saturating arithmetic than to get an answer of −126 from modular arithmetic. Likewise, for 8-bit binary unsigned arithmetic, when the correct answer is 258, it is less surprising to get an answer of 255 from saturating arithmetic than to get an answer of 2 from modular arithmetic.
Saturation arithmetic also enables overflow of additions and multiplications to be detected consistently without an overflow bit or excessive computation, by simple comparison with the maximum or minimum value (provided the datum is not permitted to take on these values).
...
Wikipedia