Exercise Points$]$:

a$)$ Consider:

$1-\sqrt[\phantom{2}]{1-lon}.$

Identify the problematic subtraction in this expression and give an algebraically equivalent formulation of $1-\sqrt[\phantom{2}]{1-lon}$ that does not suffer from this problematic subtraction for $|lon|1.$ Briefly explain your choice. Hint: Multiply by a fraction $($equal to one$)$ which removes the problematic subtraction

b$)$ Use $lon=0\mathrm{.}0002$ and write it in floating point representation with a mantissa of $k=4$ digits precision and base $b=10.$ Show that your equivalent formulation is much more accurate for $lon=0\mathrm{.}0002$ compared to the original formulation in a$).$

c$)$ Use the theorem that was shown in class to predict the number of lost significant bits $($i$.$e$.$ in base $2)$ when executing the problematic subtraction for $lon=0\mathrm{.}0002.$