I am trying to prove the fundamental axiom of floating point arithmetic also applies to complex number multiplication.
First, let $fl$ be a function that maps a number to its closest floating point approximation. If we restricted the domain of $fl$ to be only $\mathbb{R}$, then $fl$ satisfies the following property called fundamental axiom of floating point arithmetic.
$$fl(x+y) = (x+y)(1+ \epsilon_1)$$ $$fl(x-y) = (x-y)(1 + \epsilon_2)$$ $$fl(x\cdot y) = (x\cdot y)(1 + \epsilon_3)$$ $$fl(x \div y) = (x \div y)(1+ \epsilon_4)$$
where $|\epsilon_1| \le \epsilon_{machine}$, and so does $\epsilon_2, \epsilon_3, \epsilon_4$.
Given that computer stores a complex number in a length 2 vector ($a+bi$ will be stored as $[a,b]$), I need to show that
$$\frac{|fl(z_1 \cdot z_2)|}{|z_1 \cdot z_2|} \le 4\epsilon_{machine}$$
where $z_1, z_2 \in \mathbb{C}$.
In other words, there exists a $\hat{\epsilon}$ such that $|\hat{\epsilon} |\le 4\epsilon_{machine}$, and
$$fl(z_1 \cdot z_2) = (z_1 \cdot z_2)(1 + \hat{\epsilon})$$
