Assume that we are considering enhancing a machine by adding a vector mode to

it$.$ When a computation is performed in vector mode, it is $40$ times faster than the

normal mode of execution. We call the percentage of time that the machine spends

using vector mode the percentage of vectorization.

a$.$ What percentage of vectorization is needed to achieve one$-$half of the

maximum speedup attainable from using vector mode? $(7$ points$)$

b$.$ Suppose the percentage of vectorization for a program is $60\%.$ The hardware

design group could double the speed of vector mode with a significant

additional engineering investment. The compiler crew could increase the use

of vector mode as another approach to increasing performance. How much

of an increase in the percentage of vectorization $($relative to current usage$)$

would the compiler team need to obtain the same performance gain? $(8$

points$)$