What range of numbers can be represented in a 16-, 32- and 64-bit IEEE-754 systems?

For a given IEEE-754 floating point number X, if

2^E <= abs(X) < 2^(E+1)

then the distance from X to the next largest representable floating point number (epsilon) is:

epsilon = 2^(E-52)    % For a 64-bit float (double precision)
epsilon = 2^(E-23)    % For a 32-bit float (single precision)
epsilon = 2^(E-10)    % For a 16-bit float (half precision)

The above equations allow us to compute the following:

• For half precision...

  If you want an accuracy of +/-0.5 (or 2^-1), the maximum size that the number can be is 2^10. Any larger than this and the distance between floating point numbers is greater than 0.5.

  If you want an accuracy of +/-0.0005 (about 2^-11), the maximum size that the number can be is 1. Any larger than this and the distance between floating point numbers is greater than 0.0005.

• For single precision...

  If you want an accuracy of +/-0.5 (or 2^-1), the maximum size that the number can be is 2^23. Any larger than this and the distance between floating point numbers is greater than 0.5.

  If you want an accuracy of +/-0.0005 (about 2^-11), the maximum size that the number can be is 2^13. Any larger than this and the distance between floating point numbers is greater than 0.0005.

• For double precision...

  If you want an accuracy of +/-0.5 (or 2^-1), the maximum size that the number can be is 2^52. Any larger than this and the distance between floating point numbers is greater than 0.5.

  If you want an accuracy of +/-0.0005 (about 2^-11), the maximum size that the number can be is 2^42. Any larger than this and the distance between floating point numbers is greater than 0.0005.

For floating-point integers (I'll give my answer in terms of IEEE double-precision), every integer between 1 and 2^53 is exactly representable. Beyond 2^53, integers that are exactly representable are spaced apart by increasing powers of two. For example:

• Every 2nd integer between 2^53 + 2 and 2^54 can be represented exactly.
• Every 4th integer between 2^54 + 4 and 2^55 can be represented exactly.
• Every 8th integer between 2^55 + 8 and 2^56 can be represented exactly.
• Every 16th integer between 2^56 + 16 and 2^57 can be represented exactly.
• Every 32nd integer between 2^57 + 32 and 2^58 can be represented exactly.
• Every 64th integer between 2^58 + 64 and 2^59 can be represented exactly.
• Every 128th integer between 2^59 + 128 and 2^60 can be represented exactly.
• Every 256th integer between 2^60 + 256 and 2^61 can be represented exactly.
• Every 512th integer between 2^61 + 512 and 2^62 can be represented exactly. . . .

Integers that are not exactly representable are rounded to the nearest representable integer, so the worst case rounding is 1/2 the spacing between representable integers.

The precision quoted form Peter R's link to the MSDN ref is probably a good rule of thumb, but of course reality is more complicated.

The fact that the "point" in "floating point" is a binary point and not decimal point has a way of defeating our intuitions. The classic example is 0.1, which needs a precision of only one digit in decimal but isn't representable exactly in binary at all.

If you have a weekend to kill, have a look at What Every Computer Scientist Should Know About Floating-Point Arithmetic. You'll probably be particularly interested in the sections on Precision and Binary to Decimal Conversion.