Gene Amdahl, one of the early pioneers in computing, made a simple but insightful observation about the effectiveness of improving the performance of one part of a system. This observation has come to be known as Amdahl’s law. The main idea is that when we speed up one part of a system, the effect on the overall system performance depends on both how significant this part was and how much it sped up.

That law says that a fraction (α) of the code can be speed up by a factor of $k$, the total system speedup.

S= 1/(1+α)+ α/k

The marketing department at your company has promised your customers that the next software release will show a 2-fold performance improvement. You have been assigned the task of delivering on that promise. You have determined that only 80% of the system can be improved. How much (i. e., what value of k to a precision of 0.02) would you need to improve this part to meet the overall performance target?