Why Does Double-Precision Arithmetic Sometimes Show More Than 15 Decimal Places of Precision?

Double Precision: Understanding the Extended Precision

When working with double-precision data types, it is commonly understood that the approximate precision is limited to 15 decimal places. However, certain types of numbers, such as 1.0/7.0, may display a precision of 17 decimal places when internally stored as a double.

This anomaly is attributable to the inherent design of the IEEE double data representation. Double-precision values have 53 significant bits, providing approximately 15.95 decimal digits of precision. However, the C library rounds this value down to 15, resulting in a theoretical precision of 15 decimal places. In reality, the value of DBL_DIG is set to 15 rather than 16 due to this rounding.

The significance of this extended precision is further elaborated using the nextafter() function. This function calculates the nearest representable number to a given value. When examining the value of 1.0/7.0 using this function, the output reveals three consecutive values with differing last digits: 0.14285714285714282, 0.14285714285714285, and 0.14285714285714288.

The precise value of the last displayed digit is determined by the rounding operation performed by the compiler. In this case, the digit changes by 3 with each step, indicating that the precision is approximately 16 decimal digits. Thus, while the theoretical precision of a double is 15 decimal places, the extended precision of nearly 16 digits provides greater accuracy for certain types of calculations.

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