Assume you have The IEEE Half Precision (16 bits) floating point Format. This fo

Question

Assume you have The IEEE Half Precision (16 bits) floating point Format. This format has 1 bit for the sign, 5 bits for the exponent and 10 bits for the fraction. Assume we use the IEEE standard to represent Zero, Infinity and NaN. (a) What is the bias of this format. (b) Represent the number - 1101.01110010000 in this format. (c) What is the smallest normalized number that this Half precision format can represent? (d) What is the biggest normalized number that this Half precision format can represent? (e) What does the hex number 4545 representation mean if it is given in this IEEE format.

Explanation / Answer

A) The bias of the half precision floating point format also called exponent bias

and it is = 011112 = 15

B)

C)

The smallest normalized number is

= (?1)signbit × 2exponent?15 × 1.significantbits2

Which gives 2-14 = 6.1 x 10-5

D)

The biggest normalized number

from the formula (?1)signbit × 2exponent?15 × 1.significantbits2

is calculated to be ( 2 - 2-14) x 215 = +65504

E)

The hexadecimal representation of 4545 is 0100 0101 0100 0101

so it is divided in sign bit, exponent and mantissa as follows

sign bit : 0

mantissa : 1.0101000101

exponent = 10001 = 17 - ( 24 -1 ) = 2

so mantissa denormalized = 101.01000101

which represents 5.328

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