EXERCISE THREE: Addition in Base $2$

$1.$ Write a function z $=$ AddIntegral $($x$,$y$)$ which takes as input two vectors x and y representing two positive integers in base $2$ and adds them. It returns the sum z$,$ which is also a vector in base $2.$ Example: For x $=[100111]$ and y $=[1111],$ your function should return z $=[110110]$

$2.$ Write a function $[$IF$]=$ AddFractional $($x$,$y$)$ which takes as input two vectors x and y representing two fractional parts in base $2$ and adds them. The result might have an integral part I which is not zero. The function therefore returns two vectors: the integral part I of the sum and its fractional part F$,$ both in base $2.$

Example: For x $=[1111]$ and y $=[111],$ your function should return: I $=1$ and F $=[1101]$

$3.$ Write a function $[$IF$]=$ AddBinary $($Ix$,$ Fx$,$ Iy$,$ Fy$)$ which will operate on two

fractional binary numbers x and y whose integral parts are the binary vectors Ix and Iy

respectively, and their fractional parts are the binary vectors Fx and Fy respectively.

It returns $[$I F$],$ which are two vectors containing respectively the integral and the

fractional part of the sum of x and y$.$ Example: For Ix $=[100111],$ Iy $=[1111],$ Fx $=[1111],$ and Fy $=[111],$ your function should return: I $=[110111]$ and F $=[1101]$

$4.$ Write a function $[$IFE$]=$ main$($Fx$,$ Fy$),$ which will operate on two positive numbers x and y in IEEE single precision assuming integral parts equal to $1$ and an exponent equal to zero. It takes as input Fx and Fy$,$ which are respectively the fractional parts of x and y$,$ in the IEEE single representation. It returns $[$IFE$],$ which are $3$ vectors containing respectively the integral, the fractional part, and the exponent of the sum of x and y$,$ in a normalized format and rounded to the closest in the IEEE single precision representation. Example: For Fy $=[11111111111111111111111]$ and

Fx $=[00000000000000000000001]$ your function should return: I $=1,$ F $=[100000000000000]$ and E $=1$