\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.715908683995168 \cdot 10^{+83}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{-t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le 7.5252300969984295 \cdot 10^{-208} \lor \neg \left(t \le 5.822595306361626 \cdot 10^{-159}\right) \land t \le 1.6869758261951005 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -4.715908683995168e+83
1. Initial program 47.7
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification47.7
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(t \cdot t\right) \cdot 2 + \left(\ell \cdot \ell\right))_*\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(-\ell \cdot \ell\right))_*}}\]
3. Taylor expanded around -inf 3.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{-\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}}\]
if -4.715908683995168e+83 < t < 7.5252300969984295e-208 or 5.822595306361626e-159 < t < 1.6869758261951005e+135
1. Initial program 34.9
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification34.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(t \cdot t\right) \cdot 2 + \left(\ell \cdot \ell\right))_*\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(-\ell \cdot \ell\right))_*}}\]
3. Taylor expanded around inf 15.7
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \frac{{\ell}^{2}}{x}\right)}}}\]
4. Simplified10.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}}\]
if 7.5252300969984295e-208 < t < 5.822595306361626e-159 or 1.6869758261951005e+135 < t
1. Initial program 57.6
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification57.6
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(t \cdot t\right) \cdot 2 + \left(\ell \cdot \ell\right))_*\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(-\ell \cdot \ell\right))_*}}\]
3. Taylor expanded around inf 5.6
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]
4. Simplified5.6
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(t \cdot \sqrt{2}\right))_*}}\]
Recombined 3 regimes into one program.
Final simplification8.3
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.715908683995168 \cdot 10^{+83}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{-t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le 7.5252300969984295 \cdot 10^{-208} \lor \neg \left(t \le 5.822595306361626 \cdot 10^{-159}\right) \land t \le 1.6869758261951005 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Error

Derivation

`if t < -4.715908683995168e+83`

`if -4.715908683995168e+83 < t < 7.5252300969984295e-208 or 5.822595306361626e-159 < t < 1.6869758261951005e+135`

`if 7.5252300969984295e-208 < t < 5.822595306361626e-159 or 1.6869758261951005e+135 < t`

Runtime