Extreme Issues

As mentioned previously, and repeated for your convenience, an IEEE single precision floating point number, z, is stored in 32 bits, with parts s, the sign bit; e, the 8-bit excess-127 exponent; and, f, the 23 bit fractional part as shown below.

$$z = (-1)^s \;\;\; 1.f\;\;\; \times 2^{e-127}$$

Since all of the $2^{32}$ bit patterns are accounted for above, we must set aside several exceptional or signal values which are interpreted by the rules below, instead of the simple-minded formula above.

s	e	f	Meaning
0/1	$00_H$	$00_H$	Zero
0/1	$00_H$	non-zero	DeNormalized
0	$FF_H$	$00_H$	$+\infty$
1	$FF_H$	$00_H$	$-\infty$
0/1	$FF_H$	non-zero	Not a Number (NaN)

1. Why must zero be a ``signal'' value?

2. What is the largest exponent that can be represented for a normalized floating point number? What value of e is stored corresponding to this exponent?

3. What is the smallest (most negative) exponent that can be represented, and what value of e is stored for this exponent?

4. Give an example of a denormalized floating point number and indicate how it would be stored.