Question

IEEE 754-2008 contains a half precision that is only 16 bits wide. Th e left most bit is still the sign bit, the exponent is 5 bits wide and has a bias of 15, and the mantissa is 10 bits long. A hidden 1 is assumed. Write down the bit pattern to represent 1.5625 101 assuming a version of this format, which uses an excess-16 format to store the exponent. Comment on how the range and accuracy of this 16-bit fl oating point format compares to the single precision IEEE 754 standard.

Answer 1

Answer:

The representation of -1.5625 * 10^-1 is 1 01100 0100000000

Explanation:

A bit to represent -1.5625 * 10^-1

Convert -1.5626 base 10 to base 2.

It gives -0.00101

First, we normalise the above number representation.

If 1 number is assumed to be hidden

This is given as;

-0.00101 base 2

= -1.01 * 2^-3 -- base 2

The sign, exponent and fraction are as follows;

The left most bit is still the sign bit, so we have;

Sign: 1 ---- Negative Number

The exponent is 5 bits wide and has a bias of 15, so we have

Exponent: 12 ----. (-3 = 12 - 15; 15 being bias)

The mantissa is 10 bits long.

Fraction; 1.01

So, the representation of -1.5625 * 10^-1 is as follows

1 01100 0100000000

Comment:

Since, exponent is much shorter; the range is also shorter than the single precision IEEE754 standard.

And also, the fraction is also shorter; so, the half precision standard is not accurate.