IEEE floating point format

[Andrew Koenig]

In article <3554 at buengc.BU.EDU>, bph at buengc.BU.EDU (Blair P. Houghton) writes:
> In article <9697 at alice.UUCP> ark at alice.UUCP (Andrew Koenig) writes:
> >	exponent all 1's	fraction == 0		infinity
> >	exponent all 1's	fraction != 0		NaN
> >	exponent all 0's				denormalized

> Fascinating; but, what does it mean to say "denormalized" in this context?

It means the number is being represented in `gradual underflow'
mode.  To be more specfic: the smallest positive number that can be
represented in IEEE 64-bit form without going into denormalized mode
is 2^-1022.  That number is represented this way:

	0  00000000001  0000000000000000000000000000000000000000000000000000

If you count the way I did in my last note, this means an exponent
of -1021 and a fraction of .(1)0000000....

The 1 is in parentheses because it's the `hidden bit' -- it's not
actually stored.

The next smaller number is represented this way:

	0  00000000000  1111111111111111111111111111111111111111111111111111

This is the largest denormalized number: its value is 2^-1021 times
.(0)11111...    That is, the hidden bit becomes 0 when all the exponent
bits are 0.  Thus it is possible to represent numbers that are too
small for the normal exponent range, albeit with reduced precision.

As a result of this notatation, the smallest positive number that
can be represented in IEEE 64-bit floating-point is 2^-1074.
-- 
				--Andrew Koenig
				  ark at europa.att.com