In this exercise, assume that we are considering enhancing a machine by adding vector hardware to it. When a computation is run in vector mode on the vector hardware, it is 10 times faster than the normal mode of execution. We call the percentage of time that could be spent using vector mode the percentage of vectorization. Vectors are discussed in Chapter 4, but you don’t need to know anything about how they work to answer this question!

a. What percentage of vectorization is needed to achieve a speedup of 2?

b. What percentage of the computation run time is spent in vector mode if a speedup of 2 is achieved?

c. What percentage of vectorization is needed to achieve one-half the maximum speedup attainable from using vector mode?

d. Suppose you have measured the percentage of vectorization of the program to be 70%. The hardware design group estimates it can speed up the vector hardware even more with significant additional investment. You wonder whether the compiler crew could increase the percentage of vectorization, instead. What percentage of vectorization would the compiler team need to achieve in order to equal an addition 2× speedup in the vector unit (beyond the initial 10×)?