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I have read that for IEEE-754 Single Precision

For E = 0, N = (-1)^S × 0.F × 2^(-126). These numbers are in the so-called denormalized form. The exponent of 2^-126 evaluates to a very small number. Denormalized form is needed to represent zero (with F=0 and E=0). It can also represents very small positive and negative number close to zero.

My Question is

Do numbers with Exponent E is not 0 can also have De-normalized Representation Possible ? If so how please explain with binary number ?

In IEEE 754 Representation I could write N = (-1)^S × 0.F × 2^(-126) to denote the denormal numbers . What about other Representation say Given the following Representation

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My Question is

Do numbers with Exponent E is not 0 can also have De-normalized 
Representation Possible ?

If so how please explain with binary number ?

the answer is Yes, but not within IEEE754

Don't forget that a bit pattern is not a number, a bit pattern represents a number, according to conversion rules.

In 754, E==0 is the definition of a denormalised number.

Therefore E!=0 means the number represented is normalised, the bit pattern is interpretted as a number according to the rules for normalised numbers.

The way the mapping is defined, all possible bit patterns have a unique representation as a number (except for the signed zeroes and the NaN codes). If E==0, then the mantissa is calculated as 0.F. If E!=0, then the mantissa is 0.1F, an extra implied '1' bit is prepended to F before binary conversion. It's therefore not logically possible under this mapping for a number with E!=0 to be non-normed. There is no code space, staying within IEEE754, for an alternative mapping. Anything else is outside of 754!

If you want to invent an alternative mapping between numbers and bit patterns, then that's fine, you can do it. However, any bit level data you interchange with programs that use IEEE754 will map to numbers that may be different to the ones you intended. That's why IEEE754 was created, so that bit level data could be interchanged between programs, and mean the same thing, represent the same number, for everyone. If you use a 16 bit representation such as the one you've suggested for IEEE754 single reals, then you will have to approximate the numbers, as you do not have the precision to represent them fully.

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  • \$\begingroup\$ Without normalization and De normalized numbers are they same ? or different concept \$\endgroup\$ Commented Jan 24, 2017 at 7:55
  • \$\begingroup\$ That 16 bit representation is not IEEE 754 . It is another representation for Floating point number. I want to know how denormal numbers are reprenented in the given reprentation \$\endgroup\$ Commented Jan 24, 2017 at 7:56
  • \$\begingroup\$ Old x87 FPUs used to support for 80bits floating point numbers, which is not really IEEE, the weirdo "pseudo-denormal" and "unnormal" formats. \$\endgroup\$
    – Grabul
    Commented Nov 11, 2017 at 21:56

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