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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.2713468600972515 \cdot 10^{+111}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 7.31184934192413 \cdot 10^{-234} \lor \neg \left(t \le 1.4603304138205092 \cdot 10^{-154} \lor \neg \left(t \le 1.73537962790826 \cdot 10^{+86}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{1}{\frac{\frac{x}{\ell}}{\ell \cdot 2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -1.2713468600972515e+111
1. Initial program 51.7
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around -inf 2.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right)}}\]
3. Applied simplify2.7
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(1 - 2\right) - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}}\]
if -1.2713468600972515e+111 < t < 7.31184934192413e-234 or 1.4603304138205092e-154 < t < 1.73537962790826e+86
1. Initial program 35.1
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 16.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
3. Applied simplify11.8
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\]
4. Using strategy rm
5. Applied clear-num11.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \color{blue}{\frac{1}{\frac{\frac{x}{\ell}}{2 \cdot \ell}}}}}\]
if 7.31184934192413e-234 < t < 1.4603304138205092e-154 or 1.73537962790826e+86 < t
1. Initial program 50.4
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 9.2
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]
3. Applied simplify9.3
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}}\]
Recombined 3 regimes into one program.
Applied simplify9.5
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -1.2713468600972515 \cdot 10^{+111}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 7.31184934192413 \cdot 10^{-234} \lor \neg \left(t \le 1.4603304138205092 \cdot 10^{-154} \lor \neg \left(t \le 1.73537962790826 \cdot 10^{+86}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{1}{\frac{\frac{x}{\ell}}{\ell \cdot 2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \end{array}}\]

Error

Derivation

`if t < -1.2713468600972515e+111`

`if -1.2713468600972515e+111 < t < 7.31184934192413e-234 or 1.4603304138205092e-154 < t < 1.73537962790826e+86`

`if 7.31184934192413e-234 < t < 1.4603304138205092e-154 or 1.73537962790826e+86 < t`

Runtime