Double precision - decimal places

Question

From what I have read, a value of data type double has an approximate precision of 15 decimal places. However, when I use a number whose decimal representation repeats, such as 1.0/7.0, I find that the variable holds the value of 0.14285714285714285 - which is 17 places (via the debugger).

I would like to know why it is represented as 17 places internally, and why a precision of 15 is always written at ~15?

Answer 1

An IEEE double has 53 significant bits (that's the value of DBL_MANT_DIG in <cfloat>). That's approximately 15.95 decimal digits (log10(2⁵³)); the implementation sets DBL_DIG to 15, not 16, because it has to round down. So you have nearly an extra decimal digit of precision (beyond what's implied by DBL_DIG==15) because of that.

The nextafter() function computes the nearest representable number to a given number; it can be used to show just how precise a given number is.

This program:

#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));
}

gives me this output on my system:

FLT_RADIX = 2
DBL_DIG = 15
DBL_MANT_DIG = 53
0.14285714285714282
0.14285714285714285
0.14285714285714288

(You can replace %.17g by, say, %.64g to see more digits, none of which are significant.)

As you can see, the last displayed decimal digit changes by 3 with each consecutive value. The fact that the last displayed digit of 1.0/7.0 (5) happens to match the mathematical value is largely coincidental; it was a lucky guess. And the correct rounded digit is 6, not 5. Replacing 1.0/7.0 by 1.0/3.0 gives this output:

FLT_RADIX = 2
DBL_DIG = 15
DBL_MANT_DIG = 53
0.33333333333333326
0.33333333333333331
0.33333333333333337

which shows about 16 decimal digits of precision, as you'd expect.