Error

Derivation

1. Initial program 32.8

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

2. Initial simplification31.5

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

4. Applied sub-neg31.5

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

5. Applied distribute-rgt-in31.5

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

6. Simplified28.7

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

8. Applied add-sqr-sqrt28.7

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

9. Applied rem-sqrt-square28.7

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

10. Simplified26.9

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

12. Applied pow1/226.9

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

13. Final simplification26.9

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018220 +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*))))))