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

Question

I know a little bit about how floating-point numbers are represented, but not enough, I'm afraid.

The general question is:

For a given precision (for my purposes, the number of accurate decimal places in base 10), what range of numbers can be represented for 16-, 32- and 64-bit IEEE-754 systems?

Specifically, I'm only interested in the range of 16-bit and 32-bit numbers accurate to +/-0.5 (the ones place) or +/- 0.0005 (the thousandths place).

Answer 1

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.