What is the maximum precision representable value for a given double value on a machine?

Question

Couple of questions from below code:

a) What does std::numeric_limits::digits10 denote? b) If (a) represents max 10 digits double number re-presentable on a machine then we can keep getting more and more precision by just adding +1, +2 and then whats the maximum re-presentable double value on a machine? How can be display that? i.e what std::numeric_limits::digits10 + MaxN?

#include <iostream>
#include <iomanip>
#include <cmath>
#include <limits>


int main(void)
{
  double d1 = 0.1;
  double d2 = 0.2;
  double d3 = d1 + d2;
  std::cout << std::setprecision(std::numeric_limits<long double>::digits10 + 1) << "Sum 0.1 + 0.2 = " << d3 << std::endl;
  std::cout << std::setprecision(std::numeric_limits<long double>::digits10 + 2) << "Sum 0.1 + 0.2 = " << d3 << std::endl;
  std::cout << std::setprecision(std::numeric_limits<long double>::digits10 + 3) << "Sum 0.1 + 0.2 = " << d3 << std::endl;
  std::cout << std::setprecision(std::numeric_limits<long double>::digits10 + 4) << "Sum 0.1 + 0.2 = " << d3 << std::endl;
  return 0;
}

Answer 1

What does std::numeric_limits::digits10 denote?

double is typically 64-bits and can exactly represent about 2⁶⁴ different values.

Text like "0.1", "3.1415926535897932384626433832795", etc. has infinite possibilities.

Convert text to double and then back to text, do we get the same text (round-tripping)? This will not work for all possible text.

Take a sub-set of text possibilities, say only those with up to N leading significant decimal digits and with a limited exponent range.

numeric_limits<double>::digits10 (e.g. 15) is that N that allow for all successful round-tripping.

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What is the maximum precision representable value for a given double value on a machine?

Typical double is encoded using base-2 and has a maximum precision of 53 significant binary digits.

can keep getting more and more precision by just adding ...

Select double values, written in decimal, take more than N significant digits to write exactly. I'd say up to about 754.

Writing a double to numeric_limits<double>::max_digits10 (e.g. 17) significant decimal digits is sufficient to distinguish it from all other double.

A quality double to text function will certainly and correctly write at least max_digits10 decimal digits. Beyond that it is up to the implementation. To meet IEEE 754, I am certain the minimum is max_digits10 + 3.

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numeric_limits<double>::digits10 differs from numeric_limits<double>::max_digits10 because the double encoding using base 2 differs from text as base 10. The double values are not nicely distributed along base 10 values.