Error

Derivation

1. Split input into 3 regimes

2. if l < -5.60009757992504e-10

   1. Initial program 44.1

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

   3. Applied *-un-lft-identity44.1

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

   4. Applied associate-*r*44.1

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

   5. Simplified36.6

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

   7. Applied sub-neg36.6

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

   8. Applied distribute-rgt-in36.6

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

   9. Simplified28.6

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

   11. Applied add-cube-cbrt28.8

       \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt28.8

       \[\leadsto \sqrt{\color{blue}{\sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \left(\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \left(\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}}}\]

   14. Applied rem-sqrt-square28.8

       \[\leadsto \color{blue}{\left|\sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \left(\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\right|}\]

   15. Simplified27.8

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

   if -5.60009757992504e-10 < l < -1.002460517490867e-116

   1. Initial program 29.8

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

   3. Applied *-un-lft-identity29.8

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

   4. Applied associate-*r*29.8

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

   5. Simplified28.2

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

   7. Applied sub-neg28.2

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

   8. Applied distribute-rgt-in28.2

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

   9. Simplified25.5

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

   11. Applied add-sqr-sqrt25.5

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

   12. Applied rem-sqrt-square25.5

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

   13. Simplified28.1

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

   if -1.002460517490867e-116 < l

   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 *-un-lft-identity30.1

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

   4. Applied associate-*r*30.1

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

   5. Simplified27.3

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

   7. Applied sub-neg27.3

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

   8. Applied distribute-rgt-in27.3

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

   9. Simplified24.2

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

   11. Applied add-cube-cbrt24.3

       \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification25.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -5.60009757992504 \cdot 10^{-10}:\\ \;\;\;\;\left|\sqrt{\left(\left(U \cdot 2\right) \cdot \left(t \cdot n\right) + -2 \cdot \left(\left(\left(U \cdot 2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \ell\right)\right) + \left(\left(\left(-2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)}\right|\\ \mathbf{elif}\;\ell \le -1.002460517490867 \cdot 10^{-116}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(\left(t \cdot U\right) \cdot n + \left(-2 \cdot \ell - n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot n\right)\right)\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\left(U \cdot 2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right) + t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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*))))))