Why Does Double-Precision Floating-Point Arithmetic Sometimes Offer 16 Decimal Digits of Precision Instead of 15?

Double Precision: Precision Mystery Unveiled

Despite the notion that double-precision data holds approximately 15 decimal places of precision, anomalies occur when dealing with repeating decimals like 1.0/7.0. Surprisingly, the variable assigned such a value stores a 17-digit representation internally. This difference warrants an exploration of the underlying mechanism.

An IEEE double precision value consists of 53 significant bits (DBL_MANT_DIG). This translates to nearly 15.95 decimal digits. However, the implementation rounds this down to 15 (DBL_DIG). Effectively, this grants an additional decimal digit of precision.

Utilizing the nextafter() function, which returns the nearest representable number to a given value, we can demonstrate the nuances of precision. For 1.0/7.0, nextafter() progressively prints the two adjacent representable values, indicating a precision of approximately 16 decimal digits, even though DBL_DIG suggests 15.

Conclusion:

The precision of double-precision values hinges on the trade-off between significant bits and representable digits. While the theoretical precision suggests 15 decimal places, it can occasionally extend to 16 due to rounding and the inherent nature of IEEE representation. Understanding these factors is crucial for handling decimal representations in numeric computation.

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