Toniolo and Linder, Equation (13)

if l < -3.35e184

1. Initial program 6.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow222.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified42.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 27.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*27.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow227.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified27.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 83.0%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]
8. Step-by-step derivation

   1. div-inv83.0%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)\right) \cdot \frac{1}{Om}}}\right) \]

   2. *-commutative83.0%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}\right) \cdot \frac{1}{Om}}\right) \]

   3. *-commutative83.0%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} - 2\right)\right)\right) \cdot \frac{1}{Om}}\right) \]

   4. associate-*r/83.0%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} - 2\right)\right)\right) \cdot \frac{1}{Om}}\right) \]

   5. fma-neg83.0%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(U* - U, \frac{n}{Om}, -2\right)}\right)\right) \cdot \frac{1}{Om}}\right) \]

   6. metadata-eval83.0%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(n \cdot \left(U \cdot \mathsf{fma}\left(U* - U, \frac{n}{Om}, \color{blue}{-2}\right)\right)\right) \cdot \frac{1}{Om}}\right) \]

9. Applied egg-rr83.0%

   \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(U* - U, \frac{n}{Om}, -2\right)\right)\right) \cdot \frac{1}{Om}}}\right) \]

if -3.35e184 < l < -6.7999999999999997e-87 or 2.1e-72 < l < 2.5000000000000001e188

1. Initial program 49.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*55.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*58.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow258.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*63.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified64.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in64.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr64.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 62.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*72.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified72.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -6.7999999999999997e-87 < l < 2.1e-72

1. Initial program 67.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

if 2.5000000000000001e188 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 27.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out27.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*27.1%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*29.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified29.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
10. Step-by-step derivation

    1. associate-/l*80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]

    2. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]

    3. sub-neg80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]

    4. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} + \left(-2\right)\right)}}} \]

    5. associate-*r/84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}} \]

    6. metadata-eval84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}} \]

    7. +-commutative84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

11. Simplified84.0%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

Final simplification72.7%

if l < -5.99999999999999988e185

1. Initial program 6.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow222.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified42.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 27.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*27.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow227.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified27.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 83.0%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

if -5.99999999999999988e185 < l < -2.4e-87 or 7.2000000000000005e-74 < l < 2.5000000000000001e188

1. Initial program 49.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*55.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*58.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow258.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*63.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified64.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in64.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr64.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 62.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*72.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified72.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -2.4e-87 < l < 7.2000000000000005e-74

1. Initial program 67.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-/l*67.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. associate-/r/67.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Applied egg-rr67.5%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

if 2.5000000000000001e188 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 27.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out27.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*27.1%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*29.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified29.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
10. Step-by-step derivation

    1. associate-/l*80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]

    2. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]

    3. sub-neg80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]

    4. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} + \left(-2\right)\right)}}} \]

    5. associate-*r/84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}} \]

    6. metadata-eval84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}} \]

    7. +-commutative84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

11. Simplified84.0%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

Final simplification72.7%

if l < -8.09999999999999965e186

1. Initial program 6.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow222.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified42.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 27.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*27.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow227.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified27.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 83.0%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

if -8.09999999999999965e186 < l < -1.64999999999999992e-84 or 2.0999999999999999e-73 < l < 3.60000000000000021e188

1. Initial program 49.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*55.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in55.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative61.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*58.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow258.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*63.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified64.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in64.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr64.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 62.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg62.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*72.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified72.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -1.64999999999999992e-84 < l < 2.0999999999999999e-73

1. Initial program 67.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

if 3.60000000000000021e188 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 27.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out27.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*27.1%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*29.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified29.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
10. Step-by-step derivation

    1. associate-/l*80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]

    2. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]

    3. sub-neg80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]

    4. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} + \left(-2\right)\right)}}} \]

    5. associate-*r/84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}} \]

    6. metadata-eval84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}} \]

    7. +-commutative84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

11. Simplified84.0%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

Final simplification72.7%

if l < -3.8e187

1. Initial program 6.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow219.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified40.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 28.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*28.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow228.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified28.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 82.3%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

8. Taylor expanded in U around 0 81.6%

   \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}}}\right) \]
9. Step-by-step derivation

   1. associate-/l*81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U}}}}\right) \]

   2. *-commutative81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}}}}\right) \]

   3. sub-neg81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot U*}{Om} + \left(-2\right)\right)}}}}\right) \]

   4. associate-/l*81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{\frac{Om}{U*}}} + \left(-2\right)\right)}}}\right) \]

   5. metadata-eval81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + \color{blue}{-2}\right)}}}\right) \]

10. Simplified81.8%

    \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + -2\right)}}}}\right) \]

if -3.8e187 < l < -2.6999999999999999e-28 or 4.99999999999999998e-74 < l < 1.26e190

1. Initial program 47.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*54.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*59.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow259.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*64.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified65.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in65.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr65.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 61.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*73.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified73.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -2.6999999999999999e-28 < l < 4.99999999999999998e-74

1. Initial program 66.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*63.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow256.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 64.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

if 1.26e190 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
7. Step-by-step derivation

   1. associate-*l*68.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

   2. associate-/l*80.2%

      \[\leadsto \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}}\right) \]

   3. *-commutative80.2%

      \[\leadsto \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}}\right) \]

   4. sub-neg80.2%

      \[\leadsto \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}}\right) \]

   5. associate-*l/83.9%

      \[\leadsto \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)} + \left(-2\right)\right)}}}\right) \]

   6. *-commutative83.9%

      \[\leadsto \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}}\right) \]

   7. metadata-eval83.9%

      \[\leadsto \sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}}\right) \]

8. Simplified83.9%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + -2\right)}}}\right)} \]

Final simplification71.2%

if l < -2.69999999999999996e191

1. Initial program 6.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow219.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified40.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 28.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*28.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow228.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified28.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 82.3%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

8. Taylor expanded in U around 0 81.6%

   \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}}}\right) \]
9. Step-by-step derivation

   1. associate-/l*81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U}}}}\right) \]

   2. *-commutative81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}}}}\right) \]

   3. sub-neg81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot U*}{Om} + \left(-2\right)\right)}}}}\right) \]

   4. associate-/l*81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{\frac{Om}{U*}}} + \left(-2\right)\right)}}}\right) \]

   5. metadata-eval81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + \color{blue}{-2}\right)}}}\right) \]

10. Simplified81.8%

    \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + -2\right)}}}}\right) \]

if -2.69999999999999996e191 < l < -5.0000000000000002e-28 or 6.9999999999999995e-73 < l < 3.60000000000000021e188

1. Initial program 47.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*54.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*59.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow259.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*64.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified65.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in65.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr65.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 61.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*73.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified73.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -5.0000000000000002e-28 < l < 6.9999999999999995e-73

1. Initial program 66.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*63.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow256.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 64.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

if 3.60000000000000021e188 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 27.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out27.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*27.1%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*29.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified29.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
10. Step-by-step derivation

    1. associate-/l*80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]

    2. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]

    3. sub-neg80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]

    4. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} + \left(-2\right)\right)}}} \]

    5. associate-*r/84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}} \]

    6. metadata-eval84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}} \]

    7. +-commutative84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

11. Simplified84.0%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

Final simplification71.2%

if l < -2.69999999999999996e191

1. Initial program 6.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow219.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified40.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 28.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*28.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow228.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified28.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 82.3%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]
8. Step-by-step derivation

   1. mul-1-neg82.3%

      \[\leadsto \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]

   2. associate-*l*82.4%

      \[\leadsto -\color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

   3. associate-/l*82.4%

      \[\leadsto -\sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}}\right) \]

   4. *-commutative82.4%

      \[\leadsto -\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}}\right) \]

   5. sub-neg82.4%

      \[\leadsto -\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}}\right) \]

   6. associate-*l/82.4%

      \[\leadsto -\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)} + \left(-2\right)\right)}}}\right) \]

   7. *-commutative82.4%

      \[\leadsto -\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}}\right) \]

   8. metadata-eval82.4%

      \[\leadsto -\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}}\right) \]

9. Simplified82.4%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + -2\right)}}}\right)} \]

if -2.69999999999999996e191 < l < -5.0000000000000002e-28 or 4.99999999999999998e-74 < l < 2.3499999999999999e188

1. Initial program 47.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*54.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative60.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*59.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow259.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*64.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified65.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in65.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr65.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 61.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg61.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*73.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified73.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -5.0000000000000002e-28 < l < 4.99999999999999998e-74

1. Initial program 66.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*63.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow256.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 64.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

if 2.3499999999999999e188 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 27.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out27.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*27.1%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*29.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified29.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
10. Step-by-step derivation

    1. associate-/l*80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]

    2. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]

    3. sub-neg80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]

    4. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} + \left(-2\right)\right)}}} \]

    5. associate-*r/84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}} \]

    6. metadata-eval84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}} \]

    7. +-commutative84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

11. Simplified84.0%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

Final simplification71.2%

if l < -6.4999999999999998e182

1. Initial program 6.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow222.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*22.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified42.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 27.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative27.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*27.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow227.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified27.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 83.0%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

if -6.4999999999999998e182 < l < -2.6999999999999999e-28 or 4.99999999999999998e-74 < l < 2.25000000000000005e188

1. Initial program 48.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*55.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg55.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+55.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative55.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in55.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/61.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*61.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative61.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative61.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*59.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow259.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*63.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified64.7%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in64.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr64.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 62.6%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg62.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*62.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg62.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/65.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg65.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/67.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow267.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac67.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub067.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/65.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-65.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval65.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/67.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative67.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*74.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified74.4%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -2.6999999999999999e-28 < l < 4.99999999999999998e-74

1. Initial program 66.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*63.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow256.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 64.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

if 2.25000000000000005e188 < l

1. Initial program 6.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*7.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in7.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative22.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow218.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*18.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified27.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in27.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr27.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 27.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out27.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*27.1%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*29.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative29.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified29.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in l around inf 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
10. Step-by-step derivation

    1. associate-/l*80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U}}}} \]

    2. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}}}} \]

    3. sub-neg80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(-2\right)\right)}}}} \]

    4. *-commutative80.3%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{\color{blue}{\left(U* - U\right) \cdot n}}{Om} + \left(-2\right)\right)}}} \]

    5. associate-*r/84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}} + \left(-2\right)\right)}}} \]

    6. metadata-eval84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + \color{blue}{-2}\right)}}} \]

    7. +-commutative84.0%

       \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

11. Simplified84.0%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \left(U* - U\right) \cdot \frac{n}{Om}\right)}}}} \]

Final simplification71.6%

if l < -3.8e187

1. Initial program 6.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*6.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in6.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative18.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow219.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*19.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified40.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 28.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative28.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*28.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow228.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified28.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

7. Taylor expanded in l around -inf 82.3%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}\right)} \]

8. Taylor expanded in U around 0 81.6%

   \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}}}\right) \]
9. Step-by-step derivation

   1. associate-/l*81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U}}}}\right) \]

   2. *-commutative81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{\color{blue}{U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}}}}\right) \]

   3. sub-neg81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\frac{n \cdot U*}{Om} + \left(-2\right)\right)}}}}\right) \]

   4. associate-/l*81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\color{blue}{\frac{n}{\frac{Om}{U*}}} + \left(-2\right)\right)}}}\right) \]

   5. metadata-eval81.8%

      \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + \color{blue}{-2}\right)}}}\right) \]

10. Simplified81.8%

    \[\leadsto -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U*}} + -2\right)}}}}\right) \]

if -3.8e187 < l < -2.5000000000000001e-28 or 6.00000000000000014e-74 < l

1. Initial program 38.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*44.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg44.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+44.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative44.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in44.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/52.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*52.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative52.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative52.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*51.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow251.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*54.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in57.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr57.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 52.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg52.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*52.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg52.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/54.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg54.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/56.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow256.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac56.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub056.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/54.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-54.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval54.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*64.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified64.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -2.5000000000000001e-28 < l < 6.00000000000000014e-74

1. Initial program 66.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*63.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow256.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 64.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

Final simplification66.1%

if Om < -5e52 or 1.8e109 < Om

1. Initial program 47.1%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*46.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg46.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+46.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative46.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in46.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/57.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*57.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative57.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative57.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*49.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow249.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*49.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified49.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in n around 0 45.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r*49.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]

   2. +-commutative49.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right) \cdot U\right)} \]

   3. unpow249.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right)\right) \cdot U\right)} \]

   4. fma-def49.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\right) \cdot U\right)} \]

   5. associate-*r/66.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\ell \cdot \frac{\ell}{Om}}, t\right)\right) \cdot U\right)} \]

6. Simplified66.0%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right) \cdot U\right)}} \]

if -5e52 < Om < 5.00000000000000035e-230

1. Initial program 48.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*47.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative47.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*46.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow246.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*50.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified56.3%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 57.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

if 5.00000000000000035e-230 < Om < 1.8e109

1. Initial program 49.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*55.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg55.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+55.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative55.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in55.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/57.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*57.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative57.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative57.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*57.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow257.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*58.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified62.8%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in62.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr62.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 61.5%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg61.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*62.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg62.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/62.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg62.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/64.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow264.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac64.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub064.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/64.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-64.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval64.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/64.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative64.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*69.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified69.6%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

Final simplification64.4%

if l < -3.99999999999999988e-28 or 1.9999999999999999e-72 < l

1. Initial program 33.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*37.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg37.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+37.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative37.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in37.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*45.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow245.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*48.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified54.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in54.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr54.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 48.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg48.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*48.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg48.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/50.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg50.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow251.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub051.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/50.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-50.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval50.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/51.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative51.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*59.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified59.4%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

if -3.99999999999999988e-28 < l < 1.9999999999999999e-72

1. Initial program 66.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*63.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative63.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*56.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow256.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 64.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

Final simplification61.5%

if Om < -1.8e53

1. Initial program 47.8%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*46.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg46.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+46.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative46.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in46.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/51.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*51.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative51.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative51.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*44.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow244.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*46.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified46.2%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U* around 0 45.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(\left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right) \cdot U\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r*52.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right)\right) \cdot U\right)}} \]

   2. +-commutative52.4%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(t + \frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om}\right)}\right) \cdot U\right)} \]

6. Simplified58.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{\ell \cdot U}}}}\right)\right) \cdot U\right)}} \]

if -1.8e53 < Om < 1.6999999999999999e86

1. Initial program 48.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*50.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative50.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*49.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow249.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*52.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified58.1%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in U around 0 59.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

if 1.6999999999999999e86 < Om

1. Initial program 48.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*49.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/65.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*65.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative65.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative65.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*57.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow257.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*57.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified57.4%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in57.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr57.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 36.0%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out36.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*39.2%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*40.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*44.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative44.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified44.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in n around 0 46.4%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
10. Step-by-step derivation

    1. associate-/l*47.9%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

    2. associate-/l/50.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)} \]

    3. unpow250.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)} \]

    4. associate-/r*63.9%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

    5. associate-/r*68.6%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\color{blue}{\frac{Om}{U \cdot \ell}}}{\ell}}\right)} \]

    6. *-commutative68.6%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot U}}}{\ell}}\right)} \]

11. Simplified68.6%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\ell}}}\right)} \]

Final simplification61.2%

if l < -1.29999999999999996e81

1. Initial program 21.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*24.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg24.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+24.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative24.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in24.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/34.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*34.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative34.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative34.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*34.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow234.1%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*34.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified49.4%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 37.6%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative37.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*37.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow237.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified37.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity37.3%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

   2. associate-*l*37.3%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}} \]

   3. associate-/r/37.3%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)}\right)} \]

   4. associate-/r/37.3%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

8. Applied egg-rr37.3%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity37.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

   2. distribute-frac-neg37.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   3. unpow237.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

   4. distribute-lft-neg-in37.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}} \cdot {\ell}^{2}\right)}\right)} \]

   5. associate-/r/37.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\color{blue}{\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   6. sub-neg37.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{\color{blue}{2 + \left(-\frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   7. associate-*l/37.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \left(-\color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   8. mul-1-neg37.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \color{blue}{-1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   9. unpow237.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   10. distribute-frac-neg37.3%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

10. Simplified42.4%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)}} \]

11. Taylor expanded in Om around 0 49.8%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\color{blue}{\frac{Om}{\ell \cdot U}}}{\ell}}\right)} \]

if -1.29999999999999996e81 < l < 8.39999999999999955e-13

1. Initial program 63.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*61.6%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative61.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*55.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow255.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*58.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified58.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in57.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr57.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in l around -inf 51.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg51.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

   2. associate-/l*51.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)\right)} \]

   3. mul-1-neg51.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   4. associate-*l/55.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 + \left(-\color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   5. sub-neg55.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{\color{blue}{2 - \frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)\right)} \]

   6. associate-/l/54.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   7. unpow254.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   8. distribute-neg-frac54.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

   9. neg-sub054.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{0 - \left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   10. associate-*l/51.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{0 - \left(2 - \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   11. associate--r-51.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{\left(0 - 2\right) + \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   12. metadata-eval51.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{\color{blue}{-2} + \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   13. associate-*l/54.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   14. *-commutative54.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \color{blue}{\left(U* - U\right) \cdot \frac{n}{Om}}}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)} \]

   15. associate-/r*55.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

8. Simplified55.5%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

9. Taylor expanded in U* around inf 45.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{n \cdot \left({\ell}^{2} \cdot \left(U* \cdot U\right)\right)}{{Om}^{2}}}\right)} \]
10. Step-by-step derivation

    1. unpow211.5%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(U* \cdot U\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]

    2. times-frac13.9%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{n}{Om} \cdot \frac{{\ell}^{2} \cdot \left(U* \cdot U\right)}{Om}\right)}\right)} \]

    3. associate-*r*14.6%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot U}}{Om}\right)\right)} \]

    4. unpow214.6%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U*\right) \cdot U}{Om}\right)\right)} \]

11. Simplified58.3%

    \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\frac{n}{Om} \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot U}{Om}}\right)} \]

if 8.39999999999999955e-13 < l

1. Initial program 30.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*37.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg37.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+37.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative37.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in37.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*48.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow248.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*50.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified53.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in53.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr53.5%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 45.3%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out45.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*45.2%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*46.9%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*47.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative47.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified47.0%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in n around 0 34.3%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
10. Step-by-step derivation

    1. associate-/l*36.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

    2. associate-/l/39.1%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)} \]

    3. unpow239.1%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)} \]

    4. associate-/r*54.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

    5. associate-/r*57.0%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\color{blue}{\frac{Om}{U \cdot \ell}}}{\ell}}\right)} \]

    6. *-commutative57.0%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot U}}}{\ell}}\right)} \]

11. Simplified57.0%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\ell}}}\right)} \]

Final simplification56.6%

if Om < -1.80000000000000003e-190 or 5.90000000000000002e-71 < Om

1. Initial program 50.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*51.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/58.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*58.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative58.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative58.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*52.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow252.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*53.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified54.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in n around 0 48.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)}} \]
5. Step-by-step derivation

   1. *-commutative48.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   2. unpow248.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   3. associate-*r/54.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

6. Simplified54.5%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

if -1.80000000000000003e-190 < Om < 5.90000000000000002e-71

1. Initial program 42.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*43.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative43.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*44.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow244.5%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*49.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified59.8%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 40.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg40.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*40.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac40.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg40.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg40.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*35.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative35.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*37.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow237.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified37.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity37.3%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

   2. associate-*l*37.3%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}} \]

   3. associate-/r/38.6%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)}\right)} \]

   4. associate-/r/41.9%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

8. Applied egg-rr41.9%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity41.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

   2. distribute-frac-neg41.9%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   3. unpow241.9%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

   4. distribute-lft-neg-in41.9%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}} \cdot {\ell}^{2}\right)}\right)} \]

   5. associate-/r/42.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\color{blue}{\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   6. sub-neg42.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{\color{blue}{2 + \left(-\frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   7. associate-*l/42.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \left(-\color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   8. mul-1-neg42.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \color{blue}{-1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   9. unpow242.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   10. distribute-frac-neg42.1%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

10. Simplified45.1%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)}} \]

11. Taylor expanded in U* around inf 45.2%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{-2 + \color{blue}{\frac{n \cdot U*}{Om}}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)} \]

Final simplification52.1%

if U* < -2.29999999999999994e162 or 1.4e271 < U*

1. Initial program 40.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*46.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in46.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/50.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*50.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative50.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative50.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*44.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow244.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*44.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified53.8%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 38.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg38.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*40.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac40.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg40.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg40.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*34.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative34.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*34.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow234.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified34.6%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity34.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

   2. associate-*l*34.6%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}} \]

   3. associate-/r/34.6%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)}\right)} \]

   4. associate-/r/38.4%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

8. Applied egg-rr38.4%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity38.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

   2. distribute-frac-neg38.4%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   3. unpow238.4%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

   4. distribute-lft-neg-in38.4%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}} \cdot {\ell}^{2}\right)}\right)} \]

   5. associate-/r/40.2%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\color{blue}{\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   6. sub-neg40.2%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{\color{blue}{2 + \left(-\frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   7. associate-*l/40.0%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \left(-\color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   8. mul-1-neg40.0%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \color{blue}{-1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   9. unpow240.0%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   10. distribute-frac-neg40.0%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

10. Simplified43.6%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)}} \]

11. Taylor expanded in U* around inf 43.6%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{-2 + \color{blue}{\frac{n \cdot U*}{Om}}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)} \]

if -2.29999999999999994e162 < U* < 1.4e271

1. Initial program 50.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*50.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*51.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow251.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*54.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified56.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in55.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr55.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 54.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out54.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*54.4%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*54.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*55.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative55.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified55.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in n around 0 46.6%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
10. Step-by-step derivation

    1. associate-/l*48.7%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

    2. associate-/l/48.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)} \]

    3. unpow248.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)} \]

    4. associate-/r*54.2%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

    5. associate-/r*56.9%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\color{blue}{\frac{Om}{U \cdot \ell}}}{\ell}}\right)} \]

    6. *-commutative56.9%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot U}}}{\ell}}\right)} \]

11. Simplified56.9%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\ell}}}\right)} \]

Final simplification54.0%

if U* < -2.29999999999999994e162

1. Initial program 47.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*47.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg47.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+47.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative47.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in47.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*44.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow244.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*44.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified54.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 35.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg35.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*33.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative33.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*33.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow233.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified33.5%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity33.5%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

   2. associate-*l*33.5%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}} \]

   3. associate-/r/33.5%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)}\right)} \]

   4. associate-/r/36.1%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

8. Applied egg-rr36.1%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity36.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

   2. distribute-frac-neg36.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   3. unpow236.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

   4. distribute-lft-neg-in36.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}} \cdot {\ell}^{2}\right)}\right)} \]

   5. associate-/r/38.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\color{blue}{\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   6. sub-neg38.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{\color{blue}{2 + \left(-\frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   7. associate-*l/38.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \left(-\color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   8. mul-1-neg38.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \color{blue}{-1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   9. unpow238.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   10. distribute-frac-neg38.1%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

10. Simplified42.8%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)}} \]

11. Taylor expanded in Om around 0 45.0%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\color{blue}{\frac{Om}{\ell \cdot U}}}{\ell}}\right)} \]

if -2.29999999999999994e162 < U* < 1.4e271

1. Initial program 50.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*50.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in50.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*51.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow251.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*54.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified56.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-in55.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

5. Applied egg-rr55.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot U\right)}} \]

6. Taylor expanded in t around 0 54.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}}} \]
7. Step-by-step derivation

   1. distribute-lft-out54.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}} \]

   2. associate-*r*54.4%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{n \cdot \left(\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}{Om}\right)} \]

   3. associate-/l*54.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{n}{\frac{Om}{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}}\right)} \]

   4. associate-/l*55.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

   5. *-commutative55.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}\right) \cdot \left(\ell \cdot U\right)}}\right)} \]

8. Simplified55.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U + \frac{n}{\frac{Om}{\left(\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2\right) \cdot \left(\ell \cdot U\right)}}\right)}} \]

9. Taylor expanded in n around 0 46.6%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\right)} \]
10. Step-by-step derivation

    1. associate-/l*48.7%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

    2. associate-/l/48.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)} \]

    3. unpow248.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)} \]

    4. associate-/r*54.2%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\color{blue}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}}\right)} \]

    5. associate-/r*56.9%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\color{blue}{\frac{Om}{U \cdot \ell}}}{\ell}}\right)} \]

    6. *-commutative56.9%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + -2 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot U}}}{\ell}}\right)} \]

11. Simplified56.9%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{-2 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot U}}{\ell}}}\right)} \]

if 1.4e271 < U*

1. Initial program 13.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*44.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative44.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*44.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow244.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*44.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified52.7%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 46.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg46.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*46.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac46.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg46.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg46.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*38.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative38.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*38.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow238.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified38.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity38.8%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

   2. associate-*l*38.8%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}} \]

   3. associate-/r/38.7%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)}\right)} \]

   4. associate-/r/46.7%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

8. Applied egg-rr46.7%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity46.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

   2. distribute-frac-neg46.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   3. unpow246.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

   4. distribute-lft-neg-in46.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}} \cdot {\ell}^{2}\right)}\right)} \]

   5. associate-/r/46.9%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\color{blue}{\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   6. sub-neg46.9%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{\color{blue}{2 + \left(-\frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   7. associate-*l/46.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \left(-\color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   8. mul-1-neg46.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \color{blue}{-1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   9. unpow246.7%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   10. distribute-frac-neg46.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

10. Simplified46.9%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)}} \]

11. Taylor expanded in U* around inf 47.3%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{-2 + \color{blue}{\frac{n \cdot U*}{Om}}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)} \]

Final simplification54.4%

if Om < -2.90000000000000016e-192 or 1.2e-94 < Om

1. Initial program 49.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*51.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/57.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*57.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative57.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative57.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*52.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow252.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*53.8%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified54.4%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in n around 0 47.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)}} \]
5. Step-by-step derivation

   1. *-commutative47.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   2. unpow247.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   3. associate-*r/53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

6. Simplified53.9%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

if -2.90000000000000016e-192 < Om < 1.2e-94

1. Initial program 45.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l*42.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l*42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l*44.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow244.0%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l*47.2%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified59.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in l around -inf 39.6%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg39.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

   2. associate-/l*39.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

   3. distribute-neg-frac39.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

   4. mul-1-neg39.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   5. unsub-neg39.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   6. associate-/l*34.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{\frac{Om}{{\ell}^{2} \cdot U}}} \]

   7. *-commutative34.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

   8. associate-/r*36.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\color{blue}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}} \]

   9. unpow236.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}} \]

6. Simplified36.1%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity36.1%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}} \]

   2. associate-*l*36.1%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}} \]

   3. associate-/r/36.0%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{-\left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)}\right)} \]

   4. associate-/r/39.3%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \color{blue}{\frac{n}{Om} \cdot \left(U* - U\right)}\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

8. Applied egg-rr39.3%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity39.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(\frac{-\left(2 - \frac{n}{Om} \cdot \left(U* - U\right)\right)}{\frac{Om}{U}} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

   2. distribute-frac-neg39.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   3. unpow239.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

   4. distribute-lft-neg-in39.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{Om}{U}} \cdot {\ell}^{2}\right)}\right)} \]

   5. associate-/r/41.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\color{blue}{\frac{2 - \frac{n}{Om} \cdot \left(U* - U\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}}\right)\right)} \]

   6. sub-neg41.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{\color{blue}{2 + \left(-\frac{n}{Om} \cdot \left(U* - U\right)\right)}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   7. associate-*l/41.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \left(-\color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}\right)}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   8. mul-1-neg41.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + \color{blue}{-1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{\frac{Om}{U}}{{\ell}^{2}}}\right)\right)} \]

   9. unpow241.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{\frac{Om}{U}}{\color{blue}{\ell \cdot \ell}}}\right)\right)} \]

   10. distribute-frac-neg41.1%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}}\right)} \]

10. Simplified44.3%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \frac{-2 + \left(U* - U\right) \cdot \frac{n}{Om}}{\frac{\frac{\frac{Om}{U}}{\ell}}{\ell}}\right)}} \]

11. Taylor expanded in U* around inf 30.1%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\frac{n \cdot \left({\ell}^{2} \cdot \left(U* \cdot U\right)\right)}{{Om}^{2}}}\right)} \]
12. Step-by-step derivation

    1. unpow230.1%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(U* \cdot U\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]

    2. times-frac32.8%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{n}{Om} \cdot \frac{{\ell}^{2} \cdot \left(U* \cdot U\right)}{Om}\right)}\right)} \]

    3. associate-*r*34.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot U}}{Om}\right)\right)} \]

    4. unpow234.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U*\right) \cdot U}{Om}\right)\right)} \]

13. Simplified34.7%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\frac{n}{Om} \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot U}{Om}\right)}\right)} \]