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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.985468283655096 \cdot 10^{+110}:\\ \;\;\;\;\frac{t}{\frac{\frac{t}{x}}{x + x} - \left(\frac{t}{1} + \frac{t}{\frac{x}{1}}\right)}\\ \mathbf{if}\;t \le -1.5333499902358367 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{4 \cdot t}{\frac{x}{t}} + \left(\left(t \cdot t + t \cdot t\right) + \left(\ell + \ell\right) \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \le -1.1597735907264873 \cdot 10^{-224}:\\ \;\;\;\;\frac{t}{\frac{\frac{t}{x}}{x + x} - \left(\frac{t}{1} + \frac{t}{\frac{x}{1}}\right)}\\ \mathbf{if}\;t \le 4.06565598290016 \cdot 10^{+141}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{4 \cdot t}{\frac{x}{t}} + \left(\left(t \cdot t + t \cdot t\right) + \left(\ell + \ell\right) \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -2.985468283655096e+110 or -1.5333499902358367e-133 < t < -1.1597735907264873e-224
1. Initial program 53.3
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 46.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
3. Taylor expanded around -inf 63.5
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \color{blue}{\frac{e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{t}\right)\right)}}{x}}\right)}}\]
4. Applied simplify44.3
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{4 \cdot t}{\frac{x}{t}} + \left(\left(t \cdot t + t \cdot t\right) + \left(\ell + \ell\right) \cdot \frac{\ell}{x}\right)}}}\]
5. Taylor expanded around -inf 9.5
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}}\]
6. Applied simplify9.5
  \[\leadsto \color{blue}{\frac{t}{\frac{\frac{t}{x}}{x + x} - \left(\frac{t}{1} + \frac{t}{\frac{x}{1}}\right)}}\]
if -2.985468283655096e+110 < t < -1.5333499902358367e-133 or -1.1597735907264873e-224 < t < 4.06565598290016e+141
1. Initial program 33.9
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 15.9
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
3. Taylor expanded around -inf 62.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \color{blue}{\frac{e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{t}\right)\right)}}{x}}\right)}}\]
4. Applied simplify11.9
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{4 \cdot t}{\frac{x}{t}} + \left(\left(t \cdot t + t \cdot t\right) + \left(\ell + \ell\right) \cdot \frac{\ell}{x}\right)}}}\]
if 4.06565598290016e+141 < t
1. Initial program 58.7
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 2.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]
3. Applied simplify2.7
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if t < -2.985468283655096e+110 or -1.5333499902358367e-133 < t < -1.1597735907264873e-224`

`if -2.985468283655096e+110 < t < -1.5333499902358367e-133 or -1.1597735907264873e-224 < t < 4.06565598290016e+141`

`if 4.06565598290016e+141 < t`

Runtime