Error

Derivation

1. Split input into 3 regimes

2. if n < -5.684438850513913e-58

   1. Initial program 30.7

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

   3. Applied *-un-lft-identity30.7

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

   4. Applied associate-*r*30.7

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

   5. Simplified28.4

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

   7. Applied pow128.4

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

   8. Applied pow128.4

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

   9. Applied pow-prod-down28.4

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

   10. Simplified27.9

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

   12. Applied add-cube-cbrt27.9

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

   if -5.684438850513913e-58 < n < 1.0479084281615305e-308 or 3.203356352341686e-193 < n < 4.144151915742766e+87

   1. Initial program 33.4

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

   3. Applied *-un-lft-identity33.4

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

   4. Applied associate-*r*33.4

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

   5. Simplified30.8

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

   7. Applied pow130.8

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

   8. Applied pow130.8

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

   9. Applied pow-prod-down30.8

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

   10. Simplified28.8

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

   12. Applied pow128.8

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

   13. Applied pow128.8

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

   14. Applied pow-prod-down28.8

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

   15. Simplified29.8

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

   17. Applied add-sqr-sqrt29.8

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

   18. Applied rem-sqrt-square29.8

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

   19. Simplified26.4

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

   if 1.0479084281615305e-308 < n < 3.203356352341686e-193 or 4.144151915742766e+87 < n

   1. Initial program 37.0

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

   3. Applied *-un-lft-identity37.0

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

   4. Applied associate-*r*37.0

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

   5. Simplified34.9

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

   7. Applied pow134.9

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

   8. Applied pow134.9

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

   9. Applied pow-prod-down34.9

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

   10. Simplified33.3

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

   12. Applied unpow-prod-down33.3

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

   13. Applied sqrt-prod22.0

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

   14. Simplified21.6

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

4. Final simplification25.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.684438850513913 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \left(2 \cdot \ell + \left(\left(\sqrt[3]{\left(U - U*\right) \cdot \frac{\ell}{Om}} \cdot \sqrt[3]{\left(U - U*\right) \cdot \frac{\ell}{Om}}\right) \cdot \sqrt[3]{\left(U - U*\right) \cdot \frac{\ell}{Om}}\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \le 1.0479084281615305 \cdot 10^{-308}:\\ \;\;\;\;\left|\sqrt{\left(2 \cdot U\right) \cdot \left(\left(t - (\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right) + \left(2 \cdot \ell\right))_* \cdot \frac{\ell}{Om}\right) \cdot n\right)}\right|\\ \mathbf{elif}\;n \le 3.203356352341686 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \frac{\ell}{Om} \cdot (2 \cdot \ell + \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}\\ \mathbf{elif}\;n \le 4.144151915742766 \cdot 10^{+87}:\\ \;\;\;\;\left|\sqrt{\left(2 \cdot U\right) \cdot \left(\left(t - (\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right) + \left(2 \cdot \ell\right))_* \cdot \frac{\ell}{Om}\right) \cdot n\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \frac{\ell}{Om} \cdot (2 \cdot \ell + \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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*))))))