Not a program Imagine a system where the Sign, Exponent and Fraction bits are 1,

Question

Not a program

Imagine a system where the Sign, Exponent and Fraction bits are 1, 5, and 10 bits in size respectively (in total 16 bits). Write down the bits representation for the number 6.5 used in IEEE Floating-point Representation (Normalized value). What would the value of the representation in short int data type. What is the floating point equivalent for the following bits representation in Normalized form 1100000001001010 Hand simulation is needed to verify your result. (Bonus Points) According to you, what is the maximum and minimum floating point value can be represented using this 16 bits representation. Please prove your claim.

Explanation / Answer

Bits representation of value 6.5 : 0 0110.1000 0000 00

Value representation in short int: 00000000 00000110

Value representation of 1100000001001010 : 49226
1*215 +1*214 +0 +0 +0 +0 +0 +0 +0 +1*26 +0 +0 +1*23 +0 +1*21 +0
=> 32768 + 16384 + 64+ 8+2
=> 49226

Half-precision binary floating-point format for binary16
Sign bit:1 ; Exponent:5 ; Fraction:10 ;

The format is assumed to have an implicit lead bit with value 1 unless the exponent field is stored with all zeros. Thus only 10 bits of the significand appear in the memory format but the total precision is 11 bits. In IEEE 754 parlance, there are 10 bits of significand, but there are 11 bits of significand precision (log10(211) 3.311 decimal digits, or 4 digits ± slightly less than 5 units in the last place.

So Emax= 111102-011112= 15

   Emin= 000012-011112=-14

Here 011112 is the exponent bias.

Thus, as per the offset binary representation, in order to get the true exponent the offset of 15 has to be subtracted from the stored exponent.

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