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

↓

\[\begin{array}{l} \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right) \le -1.2957983155408528 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{\left(t + t\right) \cdot \left(U \cdot n\right) - \left(\left(\frac{4}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right) \cdot U}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right) \le -1.0026389104891047 \cdot 10^{-186}:\\ \;\;\;\;{\left(n \cdot \left(\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right) \le 8.639091583402157 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}\right)}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right) \le 1.618184842449678 \cdot 10^{+253}:\\ \;\;\;\;{\left(n \cdot \left(\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + t\right) \cdot \left(U \cdot n\right) - \left(\left(\frac{4}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right) \cdot U}\\ \end{array}\]

Derivation

Split input into 3 regimes
if (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < -1.2957983155408528e+268 or 1.618184842449678e+253 < (* (+ U U) (- t (/ (+ l l) (/ Om l))))
1. Initial program 48.3
  \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
2. Taylor expanded around 0 48.4
  \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]
3. Applied simplify45.4
  \[\leadsto \color{blue}{\sqrt{\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)}}\]
4. Using strategy rm
5. Applied pow1/245.4
  \[\leadsto \color{blue}{{\left(\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)\right)}^{\frac{1}{2}}}\]
6. Taylor expanded around inf 54.5
  \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}}^{\frac{1}{2}}\]
7. Applied simplify37.2
  \[\leadsto \color{blue}{\sqrt{\left(t + t\right) \cdot \left(U \cdot n\right) - \left(\frac{4}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(\ell \cdot U\right)}}\]
8. Using strategy rm
9. Applied associate-*r*37.7
  \[\leadsto \sqrt{\left(t + t\right) \cdot \left(U \cdot n\right) - \color{blue}{\left(\left(\frac{4}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right) \cdot U}}\]
if -1.2957983155408528e+268 < (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < -1.0026389104891047e-186 or 8.639091583402157e-81 < (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < 1.618184842449678e+253
1. Initial program 26.0
  \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
2. Taylor expanded around 0 28.6
  \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]
3. Applied simplify25.5
  \[\leadsto \color{blue}{\sqrt{\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)}}\]
4. Using strategy rm
5. Applied pow1/225.5
  \[\leadsto \color{blue}{{\left(\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)\right)}^{\frac{1}{2}}}\]
6. Using strategy rm
7. Applied associate-*l*18.8
  \[\leadsto {\color{blue}{\left(n \cdot \left(\left(U + U\right) \cdot \left(t - \frac{\ell + \ell}{\frac{Om}{\ell}}\right)\right)\right)}}^{\frac{1}{2}}\]
if -1.0026389104891047e-186 < (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < 8.639091583402157e-81
1. Initial program 30.1
  \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
2. Using strategy rm
3. Applied add-cube-cbrt30.2
  \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \sqrt[3]{U - U*}\right)}\right)}\]
4. Applied associate-*r*30.2
  \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}}\right)}\]
Recombined 3 regimes into one program.

Error

Derivation

`if (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < -1.2957983155408528e+268 or 1.618184842449678e+253 < (* (+ U U) (- t (/ (+ l l) (/ Om l))))`

`if -1.2957983155408528e+268 < (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < -1.0026389104891047e-186 or 8.639091583402157e-81 < (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < 1.618184842449678e+253`

`if -1.0026389104891047e-186 < (* (+ U U) (- t (/ (+ l l) (/ Om l)))) < 8.639091583402157e-81`

Runtime