Problem $4.$

$1\mathrm{.}14$ 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, $2x,$ 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 $1\mathrm{.}5x$ slowdown $($or $\frac{2}{3}$ 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?