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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.445246428600248 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le 1.4916293325529389 \cdot 10^{-307} \lor \neg \left(t \le 2.178127081867776 \cdot 10^{-161}\right) \land t \le 6.415149196452262 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -6.445246428600248e+94
1. Initial program 50.2
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification50.2
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(2 \cdot t\right) \cdot t + \left(\ell \cdot \ell\right))_*\right) \cdot \left(\frac{1 + x}{-1 + x}\right) + \left(\ell \cdot \left(-\ell\right)\right))_*}}\]
3. Taylor expanded around -inf 3.0
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{-\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}}\]
if -6.445246428600248e+94 < t < 1.4916293325529389e-307 or 2.178127081867776e-161 < t < 6.415149196452262e+136
1. Initial program 31.1
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification31.1
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(2 \cdot t\right) \cdot t + \left(\ell \cdot \ell\right))_*\right) \cdot \left(\frac{1 + x}{-1 + x}\right) + \left(\ell \cdot \left(-\ell\right)\right))_*}}\]
3. Taylor expanded around inf 13.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \frac{{\ell}^{2}}{x}\right)}}}\]
4. Simplified9.3
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}}\]
if 1.4916293325529389e-307 < t < 2.178127081867776e-161 or 6.415149196452262e+136 < t
1. Initial program 58.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification58.4
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(2 \cdot t\right) \cdot t + \left(\ell \cdot \ell\right))_*\right) \cdot \left(\frac{1 + x}{-1 + x}\right) + \left(\ell \cdot \left(-\ell\right)\right))_*}}\]
3. Taylor expanded around inf 14.2
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]
4. Simplified14.2
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(t \cdot \sqrt{2}\right))_*}}\]
Recombined 3 regimes into one program.
Final simplification9.2
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.445246428600248 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le 1.4916293325529389 \cdot 10^{-307} \lor \neg \left(t \le 2.178127081867776 \cdot 10^{-161}\right) \land t \le 6.415149196452262 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Error

Derivation

`if t < -6.445246428600248e+94`

`if -6.445246428600248e+94 < t < 1.4916293325529389e-307 or 2.178127081867776e-161 < t < 6.415149196452262e+136`

`if 1.4916293325529389e-307 < t < 2.178127081867776e-161 or 6.415149196452262e+136 < t`

Runtime