Error

Derivation

1. Split input into 3 regimes

2. if n < -3.97587947150416e-124 or 1.0007928738937381e-73 < n

   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. Simplified33.7

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

   4. Applied *-un-lft-identity33.7

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

   5. Applied times-frac31.9

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

   6. Simplified31.9

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

   8. Applied associate-*l*31.0

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

   10. Applied add-cube-cbrt31.0

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

   11. Applied *-un-lft-identity31.0

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

   12. Applied prod-diff31.0

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

   13. Applied distribute-lft-in31.0

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

   14. Simplified31.0

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

   16. Applied pow131.0

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

   17. Applied pow131.0

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

   18. Applied pow-prod-down31.0

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

   19. Simplified27.2

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

   if -3.97587947150416e-124 < n < -1.0360432425541344e-215

   1. Initial program 36.5

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

   2. Simplified33.4

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

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

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

   5. Applied times-frac29.2

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

   6. Simplified29.2

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

   8. Applied associate-*l*28.6

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

   9. Taylor expanded around -inf 41.2

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

   10. Simplified33.8

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

   if -1.0360432425541344e-215 < n < 1.0007928738937381e-73

   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. Simplified36.0

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

   4. Applied *-un-lft-identity36.0

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

   5. Applied times-frac32.8

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

   6. Simplified32.8

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

   8. Applied associate-*l*32.3

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

   10. Applied add-cube-cbrt32.3

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

   11. Applied *-un-lft-identity32.3

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

   12. Applied prod-diff32.3

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

   13. Applied distribute-lft-in32.3

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

   14. Simplified32.3

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

   16. Applied sub-neg32.3

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

   17. Applied distribute-rgt-in32.3

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

   18. Applied distribute-lft-in32.3

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

   19. Simplified29.5

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

4. Final simplification28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.97587947150416 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell\right) + \left(t - n \cdot \left(\left(\left(U + \left(-U*\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right))_*\right)}\\ \mathbf{elif}\;n \le -1.0360432425541344 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\left(\left((t \cdot n + \left(\frac{U*}{\frac{Om}{n \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}\right))_* - \frac{U}{\frac{Om}{n \cdot \ell} \cdot \frac{Om}{n \cdot \ell}}\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;n \le 1.0007928738937381 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(-U\right) + U*\right)\right) \cdot \left(U \cdot n\right) + U \cdot \left(n \cdot (\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell\right) + \left(t - n \cdot \left(\left(\left(U + \left(-U*\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right))_*\right)}\\ \end{array}\]

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*))))))