Error

Derivation

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. Simplified34.4

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

4. Applied *-un-lft-identity34.4

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

5. Applied times-frac32.0

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

6. Simplified32.0

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

8. Applied associate-*l*31.2

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

10. Applied add-cube-cbrt31.3

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

11. Applied *-un-lft-identity31.3

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

12. Applied prod-diff31.3

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

13. Applied distribute-rgt-in31.3

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

14. Simplified31.3

    \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* - n \cdot \left(\frac{\ell}{Om} \cdot \left((1 \cdot U + \left(-\sqrt[3]{U*} \cdot \left(\sqrt[3]{U*} \cdot \sqrt[3]{U*}\right)\right))_* \cdot \frac{\ell}{Om} + \color{blue}{0}\right)\right)\right)\right)\right)}\]
15. Using strategy rm

16. Applied sub-neg31.3

    \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* + \left(-n \cdot \left(\frac{\ell}{Om} \cdot \left((1 \cdot U + \left(-\sqrt[3]{U*} \cdot \left(\sqrt[3]{U*} \cdot \sqrt[3]{U*}\right)\right))_* \cdot \frac{\ell}{Om} + 0\right)\right)\right)\right)}\right)\right)}\]

17. Applied distribute-rgt-in31.3

    \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* \cdot n + \left(-n \cdot \left(\frac{\ell}{Om} \cdot \left((1 \cdot U + \left(-\sqrt[3]{U*} \cdot \left(\sqrt[3]{U*} \cdot \sqrt[3]{U*}\right)\right))_* \cdot \frac{\ell}{Om} + 0\right)\right)\right) \cdot n\right)}\right)}\]

18. Applied distribute-lft-in31.3

    \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* \cdot n\right) + U \cdot \left(\left(-n \cdot \left(\frac{\ell}{Om} \cdot \left((1 \cdot U + \left(-\sqrt[3]{U*} \cdot \left(\sqrt[3]{U*} \cdot \sqrt[3]{U*}\right)\right))_* \cdot \frac{\ell}{Om} + 0\right)\right)\right) \cdot n\right)\right)}}\]

19. Simplified28.2

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

20. Final simplification28.2

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

Reproduce

herbie shell --seed 2019090 +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*))))))