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What is theoretically possible the smallest floating point format?

Assuming you are considering the IEEE-754 format for floating point numbers for things like single and double precision, what is the minimum floating point format you could have?

I know there are half-floats and mini-fleets, but how small still makes sense? I know that applications may not be there to make the format great for any practical use.

I'm trying to determine what is the minimum mantissa bit width you could have and the smallest exponent width?

For example, does it make sense to have a mantissa that is in XX format (it is assumed that single precision will be represented as X.XXXXXXXXXXXXXXXXXXXXXXX)? Also, does it make sense to have a measure with a width of 1?

As an example of what I think:

If you had XX format and no exponent, then your only possible numbers are +/- {1.0,1.1}, but is there anything fundamental with respect to numbers or the floating point format that makes it impossible to consider them?

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I sometimes used the four-bit FP format: 2 exponent bits and 1 significant bit. This gives you the following set of values:

encoding    value
  x000     +/-0.0
  x001     +/-0.5
  x010     +/-1.0
  x011     +/-1.5
  x100     +/-2.0
  x101     +/-3.0
  x110     +/-Inf
  x111        NaN

Obviously, you cannot do a lot of useful calculations with this format, but it is useful for training, because it is the smallest format that gives you all the interesting edge cases (without NaN signaling, although if you like it, you want to make an alarm " -NaN ").

, " " , , , , 4- , . , , ( 8b- ).

; +/- 0, +/- 1, +/- 2 +/- Inf, NaN, IEEE-754. b010 Inf b011 NaN, ( 1 + 1 ) , .

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, , .

, , - 8- . :

[ 1-bit sign ] [ 4-bit exponent ] [ 3-bit mantissa/fraction ]

1/64 128 ( NaN/). , FP sign x (1 + mantissa) x 2^(exponent - bias).

IEEE-754 6- :

[ 1-bit sign ] [ 3-bit exponent ] [ 2-bit mantissa/fraction ]

, , , (.. , , , , ).

, , (, ), - " ".

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- (, j-law, , ). , . 4 + 1- +/- 31 +/- 1,984 (31 * 64); 4 + 1- 507,904 (31 * 16,384). , , , , . , . , 2 + 1 32 80 (32, 40, 48, 56, 64, 80); 5: 4 (1.250) 8: 7 (1.143).

" " , (32, 38, 45, 54, 64, 76), (1.188, 1.184, 1.200, 1.185, 1.188). , , 64 , 64 ; , , .

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