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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.3861410773366907 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{elif}\;t \le 1.2176509874903433 \cdot 10^{+118}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\sqrt[3]{\frac{\ell}{x}} \cdot \sqrt[3]{\frac{\ell}{x}}\right) \cdot \left(\left(\ell \cdot 2\right) \cdot \sqrt[3]{\frac{\ell}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 - 1\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -1.3861410773366907e+109
1. Initial program 51.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around -inf 2.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right)}}\]
3. Simplified2.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(1 - 2\right) - t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right)}}\]
if -1.3861410773366907e+109 < t < 1.2176509874903433e+118
1. Initial program 36.4
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 17.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
3. Simplified13.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
4. Using strategy rm
5. Applied add-cube-cbrt13.2
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{x}} \cdot \sqrt[3]{\frac{\ell}{x}}\right) \cdot \sqrt[3]{\frac{\ell}{x}}\right)} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
6. Applied associate-*l*13.2
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\sqrt[3]{\frac{\ell}{x}} \cdot \sqrt[3]{\frac{\ell}{x}}\right) \cdot \left(\sqrt[3]{\frac{\ell}{x}} \cdot \left(\ell \cdot 2\right)\right)} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
if 1.2176509874903433e+118 < t
1. Initial program 53.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 2.6
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]
3. Simplified2.6
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(2 - 1\right) + t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right)}}\]
Recombined 3 regimes into one program.
Final simplification9.3
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3861410773366907 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{elif}\;t \le 1.2176509874903433 \cdot 10^{+118}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\sqrt[3]{\frac{\ell}{x}} \cdot \sqrt[3]{\frac{\ell}{x}}\right) \cdot \left(\left(\ell \cdot 2\right) \cdot \sqrt[3]{\frac{\ell}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 - 1\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \end{array}\]

Error

Try it out

Derivation

`if t < -1.3861410773366907e+109`

`if -1.3861410773366907e+109 < t < 1.2176509874903433e+118`

`if 1.2176509874903433e+118 < t`

Runtime

In
Out