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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.3783442159694274 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right) - \left(\frac{n}{Om} \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot 4\right)}\\ \mathbf{if}\;t \le 2.6726825940096086 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Derivation

Split input into 3 regimes
if t < -1.3783442159694274e-151
1. Initial program 31.2
  \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
2. Taylor expanded around 0 33.8
  \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]
3. Applied simplify31.7
  \[\leadsto \color{blue}{\sqrt{\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)}}\]
4. Using strategy rm
5. Applied pow1/231.7
  \[\leadsto \color{blue}{{\left(\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)\right)}^{\frac{1}{2}}}\]
6. Taylor expanded around inf 34.3
  \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}}^{\frac{1}{2}}\]
7. Applied simplify30.0
  \[\leadsto \color{blue}{\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right) - \left(\frac{n}{Om} \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot 4\right)}}\]
if -1.3783442159694274e-151 < t < 2.6726825940096086e-71
1. Initial program 35.6
  \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
2. Using strategy rm
3. Applied associate-*l*36.6
  \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]
if 2.6726825940096086e-71 < t
1. Initial program 31.7
  \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
2. Using strategy rm
3. Applied associate-/l*29.2
  \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
4. Using strategy rm
5. Applied sqrt-prod25.7
  \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}}\]
Recombined 3 regimes into one program.
Removed slow pow expressions.

Error

Derivation

`if t < -1.3783442159694274e-151`

`if -1.3783442159694274e-151 < t < 2.6726825940096086e-71`

`if 2.6726825940096086e-71 < t`

Runtime