Why does the exponent IEEE754 (single) limit between 2^{-126}<= e <=2^{127}?

I am referring to this question but I'm not satisfied with the answer: IEEE-754 32 Bit (single precision) exponent -126 instead of -127

Does the answer imply that actually the exponent can be but because of the representation in denormalized form the smallest possitive number is $1\cdot2^{-127}\cdot1.\cdots$ equal to $1\cdot2^{-126}\cdot0.1\cdots$ .

Does this mean than that the smallest representable positive number is:

or $1\cdot2^{-126}0.1(21\cdot0)1$

Which of the last two options?

Thank you very much in advance.

Thank you again to everyone, you all gave great explanations!

Solution

Why does the exponent IEEE754 (single) limit between 2^{-126}<= e <=2^{127}?
Reworded:
Why does the exponent IEEE754 (single) limit between -126 <= exponent <= 127?

binary32 has an 8-bit biased exponent field allowing for 256 values. A wider/smaller exponent field could have been selected, yet it is 8 here. 2 values, 0 and 255, have special meaning leaving 254.

In 1970s, it was concluded to symmetrically distribute these biased exponent values +/- about zero leading to the 254 values -126 to +127 by employing a 127 offset.

Normal values have a form of: sign * (1.xxx... total 23 x's...xxx) * 2^{exponent - offset} providing a 24-bit binary precision.

For various numerical computational reasons (after much debate), it was concluded that |values| less than the smallest normal positive number 1.0 * 2^{1 - 127} should have a gradual loss of precision. These are sub-normal or denormal numbers. They are encoded with a biased exponent of 0 and with the same resultant exponent of -126 as the smallest normal number.

         v--------------------------------------- Implied valued              
         | v--------------------------v---------- Significant explicitly encoded
         | |                          |     v---- Biased exponent
         | |                          |     | v-v Implied offset  
2^-126 = 1.000 0000 0000 0000 0000 0000 * 2^1-127 // smallest normal
         0.111 1111 1111 1111 1111 1111 * 2^0-126 // largest sub-normal
2^-127 = 0.100 0000 0000 0000 0000 0000 * 2^0-126
2^-128 = 0.010 0000 0000 0000 0000 0000 * 2^0-126
2^-129 = 0.001 0000 0000 0000 0000 0000 * 2^0-126
...
2^-149 = 0.000 0000 0000 0000 0000 0001 * 2^0-126 // smallest sub-normal
0.0f   = 0.000 0000 0000 0000 0000 0000 * 2^0-126 // zero

So values down to 2^-127 have 24-bit precision and values down to 2^-149 have ever decreasing precision.