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 saturating arithmetic operations produce the following values:

• 60 + 30 → 90.
• 60 + 43 → 100. (not the expected 103.)
• (60 + 43) − (75 + 25) → 0. (not the expected 3.) (100 − 100 → 0.)
• 10 × 11 → 100. (not the expected 110.)
• 99 × 99 → 100. (not the expected 9801.)
• 30 × (5 − 1) → 100. (not the expected 120.) (30 × 4 → 100.)
• (30 × 5) − (30 × 1) → 70. (not the expected 120. not the previous 100.) (100 − 30 → 70.)

Here is another example for saturating subtraction when the valid range is from 0 to 100 instead:

• 30 - 60 → 0. (not the expected -30.)

As can be seen from these examples, familiar properties like associativity and distributivity may fail in saturation arithmetic.^[a] This makes it unpleasant to deal with in abstract mathematics, but it has an important role to play in digital hardware and algorithms where only values ranging from a minimum to a maximum value can be represented.
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