Error

Derivation

1. Split input into 3 regimes

2. if n < -2.834938554161553e-44

   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. Using strategy rm

   3. Applied associate-/l*28.6

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

   5. Applied unpow228.6

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

   6. Applied associate-*r*27.6

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

   8. Applied pow127.6

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

   9. Applied pow127.6

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

   10. Applied pow-prod-down27.6

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

   11. Simplified27.7

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

   13. Applied pow1/227.7

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

   if -2.834938554161553e-44 < n < 2.8590652022574e-310

   1. Initial program 34.2

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

   3. Applied associate-/l*31.2

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

   5. Applied unpow231.2

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

   6. Applied associate-*r*30.3

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

   8. Applied pow130.3

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

   9. Applied pow130.3

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

   10. Applied pow-prod-down30.3

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

   11. Simplified30.2

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

   13. Applied associate-*l*27.0

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

   if 2.8590652022574e-310 < n

   1. Initial program 33.3

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

   3. Applied associate-/l*30.7

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

   5. Applied unpow230.7

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

   6. Applied associate-*r*29.7

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

   8. Applied pow129.7

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

   9. Applied pow129.7

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

   10. Applied pow-prod-down29.7

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

   11. Simplified28.8

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

   13. Applied unpow-prod-down28.8

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

   14. Applied sqrt-prod21.5

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

   15. Simplified21.4

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

4. Final simplification24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.834938554161553 \cdot 10^{-44}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;n \le 2.8590652022574 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right) + \left(\ell \cdot 2\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)} \cdot \sqrt{n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 +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*))))))