Confusion regarding floating point representation

Question

I've been reading an article by Dr William T. Verts titled An 8-Bit Floating Point Representation where the author defines, as the title suggests, an 8-bit floating point representation with one bit for sign, three bits for exponent and the remaining four bits for the mantissa.

Question 1

Now, in the article, he writes:

The range of legal exponents (those representing neither denormalized nor infinite numbers) is then between 2^{-2} and 2^{+3}.

I'm confused as to what he means by legal in this context. And, why can't 2^{-3} can't be a 'legal' value? As when 2^{-3} offset by 3, gives 0 which is 000 in the exponent field.

Question 2

Next, he lists all possible values of numbers representable in this system. I'm attaching the screenshot of the first few denormalized numbers below.

I understand that we cannot represent small numbers in the normalized form, forcing us to detour from the norm and use 0 before the decimal point rather than the usual 1, which gives us what we call a denormalized form.

But, why are we using -2 in the exponent of 2(rather than -3)?

According to the rules of excess-3, the exponent field should have -2+3=1 which is 001 in the exponent field, but it has 000 in the exponent field instead(it does have 001 in the normalized numbers). But, if we use 2^{-3}, the exponent field indeed should have 000 as -3+3=0.

A nice summary of my second question would be: Why are we not using 2^{-3} as the exponent from N=1 to N=15? The exponent field does have 000 which corresponds to -3 as the exponent.

Note: I understand that the nature of the topic itself is pretty confusing. Thus, if there is any vagueness in my question, please point it out in the comments.

Answer 1

But, why are we using -2 in the exponent of 2(rather than -3)?

Suppose you did that. With exponent field 000₂ for subnormal numbers (leading bit 0) and primary significand¹ field values from 0000₂ to 1111₂, you could represent the numbers 0.0000₂•2⁻³ to 0.1111₂•2⁻³. And with exponent field 001₂ for normal numbers (leading bit 1) and primary significand field values from 0000₂ to 1111₂, you could represent the numbers 1.0000₂•2⁻² to 1.1111₂•2⁻².

Where is 1.0000₂•2⁻³? The subnormals stop at 0.1111₂•2⁻³, and the normals start at 1.0000₂•2⁻². All the numbers from 1.0000₂•2⁻³ to 1.1111₂•2⁻³ were skipped.

When you are going down from the lowest subnormal, 1.0000₂•2⁻², you cannot both decrease the exponent and change the leading digit from 1 to 0.

That is why the 000₂ exponent code means the same exponent code as 001₂. It is a code for changing the leading bit to 0 but using the same exponent as the lowest normals. Which answers your first question too.

The exponent field actually encodes both the exponent and the leading bit of the significand:

Exponent field bits	Meaning
000	Leading bit 0, exponent −2
001	Leading bit 1, exponent −2
010	Leading bit 1, exponent −1
011	Leading bit 1, exponent 0
100	Leading bit 1, exponent +1
101	Leading bit 1, exponent +2
110	Leading bit 1, exponent +3
111	Infinity or NaN

Footnote

¹ “Significand” is the preferred term for the fraction part of a floating-point number. “Mantissa” is an old term for the fraction part of a logarithm. Mantissas are logarithmic; adding to the mantissa multiplies the number. Significands are linear; adding to the significand adds to the number (as scaled by the exponent).