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$\DeclareMathOperator{\sign}{sign}$ The principal branch of Lambert-W function is defined like so: $$W(x) = f^{-1}(x)\\ \text{where} f(x)=xe^x$$ So $W(e) = 1$, $W(2e^2)$ is 2, $W(3e^3)$ is 3 etc.

We define Lambert-W number system to represent a real number $X$: $$X \rightarrow \{s=\sign(x),x=W(|x|)\}$$ For instance, $-2e^{2}$ is represented $\{-1,2\}$, and $1$ is represented as $\{1,W(1)\}$.
Additionally, we define double Lambert-W number system as this: $$X \rightarrow \{s=\sign(x),x=W(W(|x|))\}$$ For instance, $(2e^{2})\cdot{e^{(2e^{2})}}$ is represented as $\{1,2\}$.

My question: How to add, subtract, multiply and divide numbers represented in these two systems?

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$\DeclareMathOperator{\sign}{sign}$ Let us consider $x \mapsto (\sign(x),W(|x|))$ and $y \mapsto (\sign(y),W(|y|))$. This means that $$ |x| = f(f^{-1}(|x|)) = f(W(|x|)) , $$ so $$ x = \sign(x) |x| = \sign(x) f(W(|x|)) = \sign(x) W(|x|) e^{W(|x|)} . $$

At the same way, $$ y = \sign(y) W(|y|) e^{W(|y|)} . $$

So, knowing the representation of $x$ and $y$ in the Lambert-W number system, we can reconstruct the value of $x$ and $y$.

Now assume that $\sign(x)=\sign(y)$.

$$ x+y = \sign(x) \left( W(|x|) e^{W(|x|)} + W(|y|) e^{W(|y|)} \right) , $$ which means that $$ |x+y| = W(|x|) e^{W(|x|)} + W(|y|) e^{W(|y|)} . $$

The representation of $x+y$ in the Lambert-W number system will be then $$ x+y \mapsto \left(\sign(x+y), W(|x+y|)\right) = \left(\sign(x+y), W\left( W(|x|) e^{W(|x|)} + W(|y|) e^{W(|y|)} \right) \right) . $$

We just proved that, if $(s_1,x_1)$ and $(s_2,x_2)$ are two numbers in the Lambert-W number system, if $s_1 = s_2$, we can write $$ (s_1,x_1)+(s_2,x_2) = \left(s_1,W\left(x_1 e^{x_1} + x_2 e^{x_2} \right) \right) . $$

When $s_1 \neq s_2$, that is when $\sign(x) \neq \sign(y)$, you can do a similar calculation using the fact that the function $f$ is monotone, so if $x > y$, then $x e^x > y e^y$.

You can derive the other operations with similar arguments.

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    $\begingroup$ Nice answer, but there are two small problems: 1. I think you meant $(s_1,x_1)+(s_2,x_2) = \left(s_1,W\left(x_1 e^{x_1} + x_2 e^{x_2} \right) \right) .$ 2. If you numerically compute this, there may be an overflow for sufficiently large $x_1$ or $x_2$. Does anyone know how to mitigate this issue? $\endgroup$
    – Nirvana
    Mar 17, 2021 at 6:50

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