Example 3
Calculate f(x) = m65536
f(x)= m×m = m2.
m2×m2 = m4
m4×m4 = m8
m8 ×m8 = m16
m16×m16 = m32
m32×m32 = m64
m64×m64 = m128
m128×m128 = m256
m256×m256 = m512
m512×m512 = m1024
m2048×m2048 = m4096
m4096×m4096 = m8192
m8192×m8192 = m16384
m16384×m16384 = m32768
m32768×m32768 = m65536
Thus we have used only 15 multiplication operations instead of the 65,535 multiplication operations that we would have used in the normal method. Let us calculate how efficient the calculation of this third example was. We used only 15 multiplication operations instead of 65,535 multiplication operations and the difference is 65,535-15 = 65,520. Therefore, we have reduced the number of multiplication operations by 65,520. Therefore, in percentage terms we have reduced inefficient multiplication operations by 65,520÷65,535×100= 99.98%. I do not think that there is any method of exponential calculation out there that is more efficient than this method.
Notice that there is a relationship between the exponents in the 3 examples. The three exponents are 16, 256 and 65,536. The peculiar thing about these three numbers is that they all belong to the same sequence. They belong to the 2n sequence. This is an important fact.
Despite how efficient this method is, this method tends to be slightly more inefficient if the value of the exponent does not belong to the 2n sequence. I will give several examples of exponential functions whose exponents do not belong to the 2n sequence. However, this increase in inefficiency is only slight and will not greatly affect the overall efficiency of this new method. Let us calculate the following function using the ngazi method, f(x) = m310.