Error

Derivation

1. Split input into 4 regimes

2. if n < -1.7752693844118032e+206

   1. Initial program 36.8

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

   2. Simplified36.6

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

   4. Applied pow136.6

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

   5. Applied pow136.6

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

   6. Applied pow-prod-down36.6

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

   7. Simplified33.3

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

   9. Applied associate-*l*31.2

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

   11. Applied associate-*l*32.5

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

   13. Applied associate-*l/32.8

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

   14. Applied associate-*l/33.5

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

   if -1.7752693844118032e+206 < n < -1.515254705194873e+97

   1. Initial program 33.0

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

   2. Simplified33.0

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

   4. Applied pow133.0

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

   5. Applied pow133.0

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

   6. Applied pow-prod-down33.0

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

   7. Simplified29.4

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

   9. Applied unpow-prod-down29.4

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

   10. Applied sqrt-prod39.5

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

   11. Simplified39.5

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

   if -1.515254705194873e+97 < n < 1.259049375927267e-309

   1. Initial program 33.0

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

   2. Simplified33.0

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

   4. Applied pow133.0

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

   5. Applied pow133.0

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

   6. Applied pow-prod-down33.0

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

   7. Simplified30.7

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

   9. Applied associate-*l*30.1

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

   11. Applied associate-*l*29.6

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

   13. Applied associate-*l*28.6

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

   if 1.259049375927267e-309 < n

   1. Initial program 33.7

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

   2. Simplified33.7

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

   4. Applied pow133.7

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

   5. Applied pow133.7

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

   6. Applied pow-prod-down33.7

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

   7. Simplified31.7

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

   9. Applied associate-*l*31.0

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

   11. Applied associate-*l*31.1

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

   13. Applied unpow-prod-down31.1

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

   14. Applied sqrt-prod24.0

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

   15. Simplified24.0

       \[\leadsto \color{blue}{\sqrt{n}} \cdot \sqrt{{\left(U \cdot \left(2 \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell\right) + \left(t - \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot n\right)\right)\right)\right)}^{1}}\]
3. Recombined 4 regimes into one program.

4. Final simplification27.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.7752693844118032 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot \left(\left(t - \frac{\left(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \ell\right) \cdot n}{Om}\right) + \left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U\right) \cdot n}\\ \mathbf{elif}\;n \le -1.515254705194873 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{2 \cdot \left(\left(t - \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot n\right) + \left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;n \le 1.259049375927267 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{U \cdot \left(\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + \left(t - n \cdot \left(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 +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*))))))