Error

Derivation

1. Split input into 2 regimes

2. if U* < 1.069014172200058e+80

   1. Initial program 32.3

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

   3. Applied associate-/l*29.8

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

   5. Applied unpow229.8

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

   6. Applied associate-*r*29.0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)}\]
   7. Using strategy rm

   8. Applied pow129.0

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

   9. Applied pow129.0

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

   10. Applied pow-prod-down29.0

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

   11. Simplified28.8

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

   13. Applied sub-neg28.8

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

   14. Applied distribute-rgt-in28.8

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

   15. Simplified27.1

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

   if 1.069014172200058e+80 < U*

   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*33.4

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

   5. Applied unpow233.4

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

   6. Applied associate-*r*32.3

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)}\]
   7. Using strategy rm

   8. Applied associate-*l*31.7

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification28.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \le 1.069014172200058 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(-2 \cdot n\right)\right)\right) \cdot (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n + \left(2 \cdot \ell\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))

Details

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