Metaprogramming in Julia is the practice of writing code that generates code. There are many uses for metaprogramming, but it generally falls into two categories:
For new language features, you can do things like create inheritance from abstract types if you want. But we'll focus on 2. Syntactic sugar metaprogramming is the generation of new syntax rules which makes it easy to do common tasks.
Evluation of a polynomial is a common task in many disciplines. Julia's Base provides @evalpoly x a0 a1 a2 ... that implements a0 + x*a1 + x^2 * a2 + ... using Horner's rule, which is writing it out as: (((an*x) + a(n-1))*x + ...) * x + a0.
Implement your own version of the @evalpoly macro called @myevalpoly.
Wilkinson's polynomial has the form $$ w(x)=\underbrace{\prod_{i=1}^{20}(x-i)}_\text{root form}=\underbrace{a_1+a_2x+a_3x^2+\cdots+a_{21}x^{20}}_\text{coefficient form}. $$ It is a famous example of ill-conditioning in numerical analysis. One can show this visually by plotting the roots of polynomials with perturbed coefficients $\hat{a}_k=a_k(1+10^{-10}r_k)$, where $r_k$ is a normally distributed random number.
This problem has three parts, which are