IEEE and Cray floating-point formats

This brief article introduces the formats and approximate ranges of floating-point numbers for computers with IEEE binary floating-point arithmetic and for Cray PVP (parallel vector processor) computers.

Formats

IEEE single (32 bits)

     1    8               23
    +-+--------+-----------------------+
    |s|   e    |          m            |
    +-+--------+-----------------------+

IEEE double (64 bits)

     1     11                            52
    +-+-----------+----------------------------------------------------+
    |s|     e     |                      m                             |
    +-+-----------+----------------------------------------------------+

Cray single (64 bits)

     1 1      14                              48
    +-+-+--------------+------------------------------------------------+
    |s|t|      e       |                       m                        |
    +-+-+--------------+------------------------------------------------+

Where:

    s = Sign of mantissa
    t = Sign of exponent
    e = Exponent
    m = Mantissa

Ranges

                   +------------+------------+-----------------+
                   |            |            | Significant     |
                   |   Min      |   Max      | Decimal places  |
    +--------------+------------+------------+-----------------+
    +--------------+------------+------------+-----------------+
    | IEEE single  | 1.17E-0038 | 3.40E+0038 |       7         |
    +--------------+------------+------------+-----------------+
    | IEEE double  | 2.22E-0308 | 1.79E+0308 |      16         |
    +--------------+------------+------------+-----------------+
    | Cray single  | 3.67E-2466 | 2.73E+2465 |      15         |
    +--------------+------------+------------+-----------------+

Discussion

From the first section above, we see that a 32-bit IEEE floating-point number has 1 bit for the sign, 8 bits for the exponent, and 23 bits for the mantissa.

From the second section, we can see that the smallest floating-point number on a Cray is approximately 3.67 * 10^-2466, and that the largest is approximately 2.73 * 10²⁵⁶⁵. The 48 bits of mantissa result in about 15 significant decimal places.

Notes

It is a bit involved to explain how you can convert a binary floating-point number to decimal (because of exponent bias, normalization, binary/hex arithmetic, etc.), so that discussion is deferred to the references, below.

Because of normalization, IEEE numbers actually provide 24-bit precision for single, and 53-bit precision for double.

Cray double-precision arithmetic is rarely used. See the second reference below if you want information about it.

References

"IEEE Standard for Binary Floating-Point Arithmetic," IEEE Std 754-1985.

"CF77 Fortran Language Reference Manual," SR-3772. See Appendix F, "Machine Representation of Data."