Derivation

Split input into 2 regimes
if t < -8.002343278121557e-150 or 6.45713653533628e-266 < t
1. Initial program 39.9
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification39.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\left(\ell \cdot \ell + \left(2 \cdot t\right) \cdot t\right) \cdot \frac{1 + x}{x + -1} - \ell \cdot \ell}}\]
3. Taylor expanded around inf 30.5
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
4. Simplified26.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
5. Using strategy rm
6. Applied div-inv26.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\left(\ell \cdot \frac{1}{x}\right)} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
7. Applied associate-*l*26.9
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right)} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
if -8.002343278121557e-150 < t < 6.45713653533628e-266
1. Initial program 59.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
2. Initial simplification59.5
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\left(\ell \cdot \ell + \left(2 \cdot t\right) \cdot t\right) \cdot \frac{1 + x}{x + -1} - \ell \cdot \ell}}\]
3. Taylor expanded around inf 30.5
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]
4. Simplified28.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
5. Using strategy rm
6. Applied div-inv28.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\left(\ell \cdot \frac{1}{x}\right)} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
7. Applied associate-*l*28.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right)} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
8. Using strategy rm
9. Applied flip-+28.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right) + \color{blue}{\frac{2 \cdot 2 - \frac{4}{x} \cdot \frac{4}{x}}{2 - \frac{4}{x}}} \cdot \left(t \cdot t\right)}}\]
10. Applied associate-*l/28.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right) + \color{blue}{\frac{\left(2 \cdot 2 - \frac{4}{x} \cdot \frac{4}{x}\right) \cdot \left(t \cdot t\right)}{2 - \frac{4}{x}}}}}\]
11. Applied associate-*l/28.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \color{blue}{\frac{1 \cdot \left(\ell \cdot 2\right)}{x}} + \frac{\left(2 \cdot 2 - \frac{4}{x} \cdot \frac{4}{x}\right) \cdot \left(t \cdot t\right)}{2 - \frac{4}{x}}}}\]
12. Applied associate-*r/30.5
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \left(1 \cdot \left(\ell \cdot 2\right)\right)}{x}} + \frac{\left(2 \cdot 2 - \frac{4}{x} \cdot \frac{4}{x}\right) \cdot \left(t \cdot t\right)}{2 - \frac{4}{x}}}}\]
13. Applied frac-add30.8
  \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\frac{\left(\ell \cdot \left(1 \cdot \left(\ell \cdot 2\right)\right)\right) \cdot \left(2 - \frac{4}{x}\right) + x \cdot \left(\left(2 \cdot 2 - \frac{4}{x} \cdot \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right)}{x \cdot \left(2 - \frac{4}{x}\right)}}}}\]
14. Applied sqrt-div27.5
  \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{\frac{\sqrt{\left(\ell \cdot \left(1 \cdot \left(\ell \cdot 2\right)\right)\right) \cdot \left(2 - \frac{4}{x}\right) + x \cdot \left(\left(2 \cdot 2 - \frac{4}{x} \cdot \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right)}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}}\]
15. Simplified21.6
  \[\leadsto \frac{t \cdot \sqrt{2}}{\frac{\color{blue}{\sqrt{\left(2 - \frac{4}{x}\right) \cdot \left(\left(\ell \cdot 2\right) \cdot \ell + \left(t \cdot \left(t \cdot x\right)\right) \cdot \left(\frac{4}{x} + 2\right)\right)}}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}\]
Recombined 2 regimes into one program.
Final simplification26.2
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.002343278121557 \cdot 10^{-150} \lor \neg \left(t \le 6.45713653533628 \cdot 10^{-266}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{1}{x} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(2 \cdot \ell\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot \left(t \cdot x\right)\right)\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}\\ \end{array}\]

Error

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Derivation

`if t < -8.002343278121557e-150 or 6.45713653533628e-266 < t`

`if -8.002343278121557e-150 < t < 6.45713653533628e-266`

Runtime

In
Out