Question

What is the biggest "no-floating" integer that can be stored in an IEEE 754 double type without losing precision?

In other words, at would the follow code fragment return:

UInt64 i = 0;
Double d = 0;

while (i == d)
{
        i += 1; 
        d += 1;
}
Console.WriteLine("Largest Integer: {0}", i-1);

Answer 1

The biggest/largest integer that can be stored in a double without losing precision is the same as the largest possible value of a double. That is, DBL_MAX or approximately 1.8 × 10³⁰⁸ (if your double is an IEEE 754 64-bit double). It’s an integer, and it’s represented exactly.

What you might want to know instead is what the largest integer is, such that it and all smaller non-negative integers can be stored in IEEE 64-bit doubles without losing precision. An IEEE 64-bit double has 52 bits of mantissa, so it’s 2⁵³ (and -2⁵³ on the negative side):

• 2⁵³ + 1 cannot be stored, because the 1 at the start and the 1 at the end have too many zeros in between.
• Anything less than 2⁵³ can be stored, with 52 bits explicitly stored in the mantissa, and then the exponent in effect giving you another one.
• 2⁵³ obviously can be stored, since it’s a small power of 2.

Or another way of looking at it: once the bias has been taken off the exponent, and ignoring the sign bit as irrelevant to the question, the value stored by a double is a power of 2, plus a 52-bit integer multiplied by 2^{exponent − 52}. So with exponent 52 you can store all values from 2⁵² through to 2⁵³ − 1. Then with exponent 53, the next number you can store after 2⁵³ is 2⁵³ + 1 × 2^{53 − 52}. So loss of precision first occurs with 2⁵³ + 1.