A fast formula for the inverse of \(x[s]y\)!
I thought this would be infeasible, but I think I've found a better way of approximating the inverse of the modified Bennet equations. This only works for the case between multiplication and exponentiation, and I think it's pretty cool that this formula works. I thought it'd be a little dumb at first. But, here we are:
Let's write:
\[
y=x [s+1] \omega(y) = \exp^{\circ s}_{\omega(y)^{1/\omega(y)}}\left(\log^{\circ s}_{\omega(y)^{1/\omega(y)}}(x)\cdot \omega(y)\right)\\
\]
Rearranging, we are given the fixed point equation:
\[
\omega = \frac{\log^{\circ s}_{\omega^{1/\omega}}(y)}{\log^{\circ s}_{\omega^{1/\omega}}(x)}\\
\]
Therefore, if you can guess a value \(\omega_0\) sufficiently close to \(\omega\). And we define:
\[
f(\omega) = \frac{\log^{\circ s}_{\omega^{1/\omega}}(y)}{\log^{\circ s}_{\omega^{1/\omega}}(x)}\\
\]
Then \(f^{\circ n}(\omega_0) \to \omega\)!
So far, I've found that \(\omega_0 = y\) works perfectly!!!!!!!!!
YES!
This greatly simplifies looking for inverses. OMFG YES! This has been bugging me for a while. Thought I'd post!
Now onto better studying the object:
\[
x [s+1] (x[s+1]^{-1}y) + 1 \approx x[s] y\\
\]
I thought this would be infeasible, but I think I've found a better way of approximating the inverse of the modified Bennet equations. This only works for the case between multiplication and exponentiation, and I think it's pretty cool that this formula works. I thought it'd be a little dumb at first. But, here we are:
Let's write:
\[
y=x [s+1] \omega(y) = \exp^{\circ s}_{\omega(y)^{1/\omega(y)}}\left(\log^{\circ s}_{\omega(y)^{1/\omega(y)}}(x)\cdot \omega(y)\right)\\
\]
Rearranging, we are given the fixed point equation:
\[
\omega = \frac{\log^{\circ s}_{\omega^{1/\omega}}(y)}{\log^{\circ s}_{\omega^{1/\omega}}(x)}\\
\]
Therefore, if you can guess a value \(\omega_0\) sufficiently close to \(\omega\). And we define:
\[
f(\omega) = \frac{\log^{\circ s}_{\omega^{1/\omega}}(y)}{\log^{\circ s}_{\omega^{1/\omega}}(x)}\\
\]
Then \(f^{\circ n}(\omega_0) \to \omega\)!
So far, I've found that \(\omega_0 = y\) works perfectly!!!!!!!!!
YES!
This greatly simplifies looking for inverses. OMFG YES! This has been bugging me for a while. Thought I'd post!
Now onto better studying the object:
\[
x [s+1] (x[s+1]^{-1}y) + 1 \approx x[s] y\\
\]