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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.0455787307621947 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le 5.5216541725427274 \cdot 10^{-288} \lor \neg \left(t \le 3.2837604703102175 \cdot 10^{-231} \lor \neg \left(t \le 4.088619728431962 \cdot 10^{+114}\right)\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \frac{2}{x}\right) \cdot \ell + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \left(2 - 1\right) \cdot \frac{\frac{t}{x \cdot x}}{\sqrt{2}}}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -3.0455787307621947e+145
1. Initial program 60.6
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around -inf 1.9
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right)}}\]
3. Applied simplify1.9
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(1 - 2\right) - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}}\]
if -3.0455787307621947e+145 < t < 5.5216541725427274e-288 or 3.2837604703102175e-231 < t < 4.088619728431962e+114
1. Initial program 33.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 15.8
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
3. Applied simplify15.8
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\sqrt{\left(\ell \cdot \ell\right) \cdot \frac{2}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}}\]
4. Using strategy rm
5. Applied associate-*l*11.3
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\]
if 5.5216541725427274e-288 < t < 3.2837604703102175e-231 or 4.088619728431962e+114 < t
1. Initial program 54.6
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 8.3
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right) - 2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}}}}\]
3. Applied simplify8.3
  \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x \cdot x}}{\sqrt{2}} \cdot \left(2 - 1\right) + t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right)}}\]
Recombined 3 regimes into one program.
Applied simplify9.1
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -3.0455787307621947 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le 5.5216541725427274 \cdot 10^{-288} \lor \neg \left(t \le 3.2837604703102175 \cdot 10^{-231} \lor \neg \left(t \le 4.088619728431962 \cdot 10^{+114}\right)\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \frac{2}{x}\right) \cdot \ell + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \left(2 - 1\right) \cdot \frac{\frac{t}{x \cdot x}}{\sqrt{2}}}\\ \end{array}}\]

Error

Try it out

Derivation

`if t < -3.0455787307621947e+145`

`if -3.0455787307621947e+145 < t < 5.5216541725427274e-288 or 3.2837604703102175e-231 < t < 4.088619728431962e+114`

`if 5.5216541725427274e-288 < t < 3.2837604703102175e-231 or 4.088619728431962e+114 < t`

Runtime

In
Out