if n < -6.237596765273558e-170

1. Started with
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   31.5
2. Using strategy rm
   31.5
3. Applied square-mult to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   31.5
4. Applied associate-/l* to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   29.1
5. Using strategy rm
   29.1
6. Applied associate-*l* to get
   \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
   29.6
7. Using strategy rm
   29.6
8. Applied square-mult to get
   \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{red}{{\left(\frac{\ell}{Om}\right)}^2}\right) \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\]
   29.6
9. Applied associate-*r* to get
   \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{red}{\left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)\right)}\]
   28.7
10. Applied taylor to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}\]
    29.6
11. Taylor expanded around 0 to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\color{red}{\frac{n \cdot \ell}{Om}} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\color{blue}{\frac{n \cdot \ell}{Om}} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}\]
    29.6
12. Applied simplify to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{n \cdot \ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(t - (\left(n \cdot \frac{\ell}{Om}\right) * \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}}\right))_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\]
    27.4

---

13. Applied final simplification

if -6.237596765273558e-170 < n < 1.4561613451310695e-242

1. Started with
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   38.5
2. Using strategy rm
   38.5
3. Applied square-mult to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   38.5
4. Applied associate-/l* to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   36.0
5. Using strategy rm
   36.0
6. Applied associate-*l* to get
   \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
   36.1
7. Using strategy rm
   36.1
8. Applied sub-neg to get
   \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{red}{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}\]
   36.1
9. Applied associate--l+ to get
   \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{red}{\left(\left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}\right)}\]
   36.1
10. Applied distribute-lft-in to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \color{red}{\left(U \cdot \left(t + \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)\right)}} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t + U \cdot \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
    36.1
11. Applied simplify to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \color{red}{U \cdot \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \color{blue}{\left(U \cdot \frac{\ell}{Om}\right) \cdot \left(\left(-2 \cdot \ell\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)}\right)}\]
    30.9
12. Applied taylor to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \left(U \cdot \frac{\ell}{Om}\right) \cdot \left(\left(-2 \cdot \ell\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot U}{Om} \cdot \left(\left(-2 \cdot \ell\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)}\]
    31.5
13. Taylor expanded around 0 to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \color{red}{\frac{\ell \cdot U}{Om}} \cdot \left(\left(-2 \cdot \ell\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \color{blue}{\frac{\ell \cdot U}{Om}} \cdot \left(\left(-2 \cdot \ell\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)}\]
    31.5
14. Applied simplify to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\ell \cdot U}{Om} \cdot \left(\left(-2 \cdot \ell\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)} \leadsto \sqrt{(\left(\frac{\left(n \cdot 2\right) \cdot \ell}{\frac{Om}{U}}\right) * \left(\ell \cdot \left(\left(-2\right) - \frac{U - U*}{\frac{Om}{n}}\right)\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\]
    28.7

---

15. Applied final simplification

if 1.4561613451310695e-242 < n < 4.3250250802722524e+66

1. Started with
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   31.5
2. Using strategy rm
   31.5
3. Applied square-mult to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   31.5
4. Applied associate-/l* to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   28.2
5. Using strategy rm
   28.2
6. Applied associate-*l* to get
   \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
   28.4
7. Using strategy rm
   28.4
8. Applied sub-neg to get
   \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{red}{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}\]
   28.4
9. Applied associate--l+ to get
   \[\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{red}{\left(\left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}\right)}\]
   28.4
10. Applied distribute-rgt-in to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \color{red}{\left(U \cdot \left(t + \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)\right)}} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + \left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \cdot U\right)}}\]
    28.4
11. Applied simplify to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{red}{\left(\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \cdot U}\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(U \cdot \frac{\ell}{Om}\right) \cdot \left(\left(-\ell \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{n}{Om} \cdot \ell\right)\right)}\right)}\]
    24.0
12. Applied simplify to get
    \[\sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(U \cdot \frac{\ell}{Om}\right) \cdot \color{red}{\left(\left(-\ell \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{n}{Om} \cdot \ell\right)\right)}\right)} \leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \left(U \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\left(\ell \cdot \left(\left(-2\right) - \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}\right)}\]
    25.3

if 4.3250250802722524e+66 < n

1. Started with
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   34.0
2. Using strategy rm
   34.0
3. Applied square-mult to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   34.0
4. Applied associate-/l* to get
   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
   31.9
5. Using strategy rm
   31.9
6. Applied associate-*l* to get
   \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
   32.2
7. Using strategy rm
   32.2
8. Applied sqrt-prod to get
   \[\color{red}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}} \leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}}\]
   21.2