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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.7710362359613171 \cdot 10^{+78}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \left(\left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + 2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}}\right) + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 1.2921528621201326 \cdot 10^{-276} \lor \neg \left(t \le 1.0963571751894683 \cdot 10^{-162} \lor \neg \left(t \le 5.35556584533854 \cdot 10^{+37}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\sqrt{\frac{\ell}{\frac{x}{\ell}} + t \cdot t} \cdot \sqrt{\frac{\ell}{\frac{x}{\ell}} + t \cdot t}\right) \cdot 2 + 4 \cdot \frac{t}{\frac{x}{t}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + 2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}}\right) + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}}}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -1.7710362359613171e+78
1. Initial program 46.8
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around -inf 4.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right)}}\]
if -1.7710362359613171e+78 < t < 1.2921528621201326e-276 or 1.0963571751894683e-162 < t < 5.35556584533854e+37
1. Initial program 36.7
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 15.3
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
3. Simplified11.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}}\]
4. Using strategy rm
5. Applied add-sqr-sqrt11.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left(\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}} \cdot \sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}\right)} + \frac{t}{\frac{x}{t}} \cdot 4}}\]
if 1.2921528621201326e-276 < t < 1.0963571751894683e-162 or 5.35556584533854e+37 < t
1. Initial program 47.3
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Taylor expanded around inf 10.9
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]
Recombined 3 regimes into one program.
Final simplification9.7
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.7710362359613171 \cdot 10^{+78}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \left(\left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + 2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}}\right) + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 1.2921528621201326 \cdot 10^{-276} \lor \neg \left(t \le 1.0963571751894683 \cdot 10^{-162} \lor \neg \left(t \le 5.35556584533854 \cdot 10^{+37}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\sqrt{\frac{\ell}{\frac{x}{\ell}} + t \cdot t} \cdot \sqrt{\frac{\ell}{\frac{x}{\ell}} + t \cdot t}\right) \cdot 2 + 4 \cdot \frac{t}{\frac{x}{t}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + 2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}}\right) + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}}}\\ \end{array}\]

Error

Try it out

Derivation

`if t < -1.7710362359613171e+78`

`if -1.7710362359613171e+78 < t < 1.2921528621201326e-276 or 1.0963571751894683e-162 < t < 5.35556584533854e+37`

`if 1.2921528621201326e-276 < t < 1.0963571751894683e-162 or 5.35556584533854e+37 < t`

Runtime

In
Out