Understanding a Set Theory Equation on Decoding a Linear value Into a Logarithmic value I'm currently writing a series of equations into python code (I should note, for fun!) and am having trouble understanding this. From my ignorant perspective it looks like a set with a series of variables. The following equation encodes a 16-bit integer value Vp (unsigned, only positive values up to 65,536 ) as an 12-bit (4096) integer value Vi, theoretically taking a value from a linear to logarithmic space.

If anyone could explain would be incredibly grateful!
 A: I have attempted to unveil a part of the mysterious definition. Here is what I have "deciphered".
Let us start from the definition of $q$:
$$q=\log_2(v_p)-9+\varepsilon \ \ \ \ \text{where}\ \ 0\le \varepsilon < 1$$
As a consequence:
$$2^q=2^{\log_2(v_p)}2^{-9}2^{\varepsilon}$$
which is equivalent to :
$$2^q=\frac{v_p}{512} K  \ \ \ \ \text{where}\ \ 1\le K < 2\tag{1}$$
The curly brackets clearly mean the two different expressions $v_i$ can take according to the position of $v_p$ with respect to $1024$.
Let us consider the interesting case $v_p \ge 1024$, where  we can convert the RHS expression into:
$$v_i=512 q + \frac{v_p}{2^q}\tag{2}$$
(explanations below)
Plugging (1) into (2):
$$v_i=512 q + \frac{512}{K}$$
$$v_i=512(q+K')  \ \ \ \ \text{where}\ \ 0.5 < K':=\frac{1}{K} \le 1\tag{3}$$
$$v_i=512(\log_2(v_p)-9+\varepsilon+K')  \ \ \ \ \text{where}\ \ \begin{cases}0.5 < K' \le 1 \\ 0\le \varepsilon < 1 \end{cases}\tag{4}$$
where, indeed, $v_i$ is "asymptotically proportional" to $\log(v_p)$.
Explanation for expression (2): Notation N >> q means "right shift by $q$ places of the binary expression of $N$", amounting to a division by $2^q$.
