Question: When making changes to optimize part of a processor, it is often the case that speeding up one type of instruction comes at the cost of slowing down something else. For example, if we put in a complicated fast floating$-$point unit, that takes space, and something might have to be moved farther away from the middle to accommodate it$,$ adding an extra cycle in

When making changes to optimize part of a processor, it is often the case that speeding up one type of instruction comes at the cost of slowing down something else. For example, if we put in a complicated fast floating$-$point unit, that takes space, and something might have to be moved farther away from the middle to accommodate it$,$ adding an extra cycle in delay to reach that unit. The basic Amdahl$$s law equation does not take into account this trade$-$ off.

a$.$ If the new fast floating$-$ point unit speeds up floating$-$ point operations by$,$ on average, $2,$ and floating$-$ point operations take $20\%$ of the original program$$s execution time, what is the overall speedup $($ ignoring the penalty to any other instructions$)?$

b$.$ Now assume that speeding up the floating$-$ point unit slowed down data cache accesses, resulting in a $1\mathrm{.}5$ slowdown $($or $2/3$ speedup$).$ Data cache accesses consume $10\%$ of the execution time. What is the overall speedup now?

c$.$ After implementing the new floating$-$ point operations, what percentage of execution time is spent on floating$-$ point operations? What percentage is spent on data cache accesses?