biggest integer that can be stored in a double

Question

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. It's represented exactly. What more do you want?

Go on, ask me what the largest integer is, such that it and all smaller integers can be stored in IEEE 64-bit doubles without losing precision. An IEEE 64-bit double has 52 bits of mantissa, so I think it's 2⁵³:

• 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.

9007199254740992 (that's 9,007,199,254,740,992 or 2^53) with no guarantees :)

Program

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#include <math.h>
#include <stdio.h>

int main(void) {
  double dbl = 0; /* I started with 9007199254000000, a little less than 2^53 */
  while (dbl + 1 != dbl) dbl++;
  printf("%.0f\n", dbl - 1);
  printf("%.0f\n", dbl);
  printf("%.0f\n", dbl + 1);
  return 0;
}

Result

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9007199254740991
9007199254740992
9007199254740992

The largest integer that can be represented in IEEE 754 double (64-bit) is the same as the largest value that the type can represent, since that value is itself an integer.

This is represented as 0x7FEFFFFFFFFFFFFF, which is made up of:

• The sign bit 0 (positive) rather than 1 (negative)
• The maximum exponent 0x7FE (2046 which represents 1023 after the bias is subtracted) rather than 0x7FF (2047 which indicates a NaN or infinity).
• The maximum mantissa 0xFFFFFFFFFFFFF which is 52 bits all 1.

In binary, the value is the implicit 1 followed by another 52 ones from the mantissa, then 971 zeros (1023 - 52 = 971) from the exponent.

The exact decimal value is:

179769313486231570814527423731704356798070567525844996598917476803157260780028538760589558632766878171540458953514382464234321326889464182768467546703537516986049910576551282076245490090389328944075868508455133942304583236903222948165808559332123348274797826204144723168738177180919299881250404026184124858368

This is approximately 1.8 x 10³⁰⁸.

Wikipedia has this to say in the same context with a link to IEEE 754:

On a typical computer system, a 'double precision' (64-bit) binary floating-point number has a coefficient of 53 bits (one of which is implied), an exponent of 11 bits, and one sign bit.

2^53 is just over 9 * 10^15.

You need to look at the size of the mantissa. An IEEE 754 64 bit floating point number (which has 52 bits, plus 1 implied) can exactly represent integers with an absolute value of less than or equal to 2^53.

1.7976931348623157 × 10^308

http://en.wikipedia.org/wiki/Double_precision_floating-point_format

It is true that, for 64-bit IEEE754 double, all integers up to 9007199254740992 == 2^53 can be exactly represented.

However, it is also worth mentioning that all representable numbers beyond 4503599627370496 == 2^52 are integers. Beyond 2^52 it becomes meaningless to test whether or not they are integers, because they are all implicitly rounded to a nearby representable value.

In the range 2^51 to 2^52, the only non-integer values are the midpoints ending with ".5", meaning any integer test after a calculation must be expected to yield at least 50% false answers.

Below 2^51 we also have ".25" and ".75", so comparing a number with its rounded counterpart in order to determine if it may be integer or not starts making some sense.

TLDR: If you want to test whether a calculated result may be integer, avoid numbers larger than 2251799813685248 == 2^51