Question

Inspired of the IEEE 754 standard, a floating point format that is only 10 bits wide is defined for a special computer. The leftmost bit is still the sign bit, the exponent is 5 bits wide and has a bias of 15, and the fractions is 4 bits long. A hidden 1 is assumed for the normal number, but not for the denormalized number.

c) Construct a case to show that floating point addition is not associative

Answer 1

(c) Example to show that floating point addition is not associative

Let suppose we want to add three numbers

let a=16, b =17, c=-4

we want to find Y = a + b + c

According to associative law

(a+b)+c = (16 + 17) +(-4)

=33 + (-4) [Since 33 is out of range it is truncated to 32]

=32 + (-4)

=28 (Ans)

a+(b+c) = 16 + (17 +(-4) )

= 16 + 13

= 29 (Ans)

It's clear that (a+b)+c is not equal to a+(b+c)

Hence floating point addition is not associative.

Now let us do the calculation using  IEEE 754 standard, a floating point format that is only 10 bits wide is defined for a special computer.

(a+b)+c = (16 + 17) +(-4)

=33 + (-4)

Let us represent 33 in IEEE 754 standard, a floating point format that is only 10 bits wide is defined for a special computer.

33 = 100001₂ x 2⁰ = 1.00001 x 2⁵ = 1.M x 2⁵ (Where 1 in 1.M is the hidden bit )

Sing bit, S = 0

Actual exponent, e = 5

biased exponent, E = e + 15 = 5 + 15 =20 = 10100₂

Mantissa, M= 00001 = 0000 (Mantissa is 4 -bit)

Therefore, 33 in IEEE 754 standard, a floating point format that is only 10 bits wide is defined for a special computer.

33 = 0 10100 0000

Let us convert A=0 10100 0000 into decimal

A = (-1)^S x 1.M x 2^E-15

S = 0, M=0000 i.e. 1.M=1.0000

e=E-15 = 20 - 15 =5

A = (-1)⁰ x 1.0000 x 2^20-15

=1 x 1.0000 x 2⁵

=32

Since 33 is out of range it's truncated to 32

i.e.

(a+b)+c = (16 + 17) +(-4)

=33 + (-4) (33 is out of range and is truncated to 32 as shown above)

=32 + (-4)

=28 (Wrong result)

On the other hand , a+(b+c )= 16 + (17 +(-4))

=16 + 13 (All the numbers are within the range i.e. <= 32 , so there is no loss of precision)

= 29 (Correct result)

Hence floating point addition is not associative.