Why Do Double-Precision Numbers Seem More Precise Than 15 Decimal Places?

Double Precision Decimal Places

Question:

Despite documentation suggesting that double precision values have a precision of 15 decimal places, internal values often exhibit more precision. Why is this discrepancy observed?

Answer:

IEEE doubles have 53 significant bits, providing approximately 15.95 decimal digits of precision (based on the formula log10(2⁵³)). This theoretical precision is slightly greater than the 15 decimal digits indicated by the DBL_DIG constant. The additional precision is accounted for by the extra significant bit.

The nextafter() function demonstrates the true precision of a given double. By calculating the nearest representable numbers to a given value, it reveals that double precision values typically have 16 significant digits.

Here's a program that illustrates this:

#include <cstdio>
#include <cfloat>
#include <cmath>

int main() {
    double x = 1.0/7.0;
    printf("FLT_RADIX = %d\n", FLT_RADIX);
    printf("DBL_DIG = %d\n", DBL_DIG);
    printf("DBL_MANT_DIG = %d\n", DBL_MANT_DIG);
    printf("%.17g\n%.17g\n%.17g\n", nextafter(x, 0.0), x, nextafter(x, 1.0));
}

The output typically shows that:

• 1.0/7.0 is represented with approximately 16 significant digits.
• The last hexadecimal bit alternates between 0 and 1 for consecutive values.
• The mathematical value of 1.0/7.0 (0.142857142857142857) is only represented approximately in double precision.

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