Find the binary representation of x 103 Find the floatingpo

Find the binary representation of x = 10/3. Find the floating-point representation of x = 10/3. Find the exact difference fl(x) - x. What is the relative rounding error?

Solution

Solution: Given decimal number is x= 10/3= 3.333

Now       

3/2    quotient =1 remainder = 1

1/2   quotient =0 remainder = 1

Here integer part 3 is equivalent to 11₂

So 3₁₀ = 11₂

Also

0.333x2   product = 0.666   integer part = 0

0.666x2   product = 1.332 integer part = 1

0.332x2   product = 0.664 integer part = 0

0.664x2   product = 0.664 integer part = 0

0.664x2   product = 1.328 integer part = 1

0.328x2   product = 0.656 integer part = 0

0.656x2   product = 1.312 integer part = 1

So 0.333₁₀ =0.0100101₂

So 3.333₁₀ = 11.0100101₂

Floating point representation:

Here decimal number is x= 10/3= 3.333=2x1.666

Now 1₁₀ = 1₂ and 0.666₁₀ = 0.100101₂ (from above)

So 1.666₁₀= (1.100101₂ )x2¹

Hence floating representation is

The presence of the factor 2^E is inconvenient, and so the relative

rounding error associated with x is defined to be
\\delta = (round(x)/x - 1)

=\\frac{round(x)}{x} - 1 = \\frac{round(x) -x}{x} =

0

E = 1

1.10010100