Computer Science – 13.3 Floating-point numbers, representation and manipulation | e-Consult

Underflow occurs in binary floating-point arithmetic when the result of a calculation is smaller than the smallest representable positive number. This happens when the exponent of the result is too small to be represented within the available exponent range of the floating-point format.

For example, consider a system with a minimum representable positive exponent of -3 and a maximum of +3. If a calculation results in a number with an exponent of -4, it cannot be represented. The system will typically represent this as zero. However, this can lead to inaccuracies.

Imagine calculating the square root of a very small positive number, say 0.0000000001. The result will be a number with a negative exponent. If the system underflows and represents this as zero, the subsequent calculations involving this value will be affected. For instance, if we add a small positive number to zero, the result will be approximately equal to the small positive number, but the precision of the small positive number might be lost due to the underflow. This can significantly impact the accuracy of the overall calculation.

Underflow can also occur when performing multiplications. If the product of two numbers is smaller than the smallest positive representable number, the result will underflow to zero. This can mask important information and lead to incorrect conclusions.