Error

Derivation

1. Split input into 2 regimes

2. if n < -7.462202010042364e-257 or 1.6373050537097972e-156 < n

   1. Initial program 32.1

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

   3. Applied *-un-lft-identity32.1

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

   4. Applied associate-*r*32.1

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

   5. Simplified28.4

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

   7. Applied sub-neg28.4

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

   8. Applied distribute-rgt-in28.4

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

   9. Simplified25.0

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

   11. Applied pow125.0

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

   12. Applied pow125.0

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

   13. Applied pow-prod-down25.0

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

   14. Simplified24.9

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

   16. Applied add-cube-cbrt25.0

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

   17. Applied associate-*r*25.0

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

   if -7.462202010042364e-257 < n < 1.6373050537097972e-156

   1. Initial program 37.7

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

   3. Applied *-un-lft-identity37.7

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

   4. Applied associate-*r*37.7

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

   5. Simplified34.8

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

   7. Applied sub-neg34.8

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

   8. Applied distribute-rgt-in34.8

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

   9. Simplified29.4

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

   11. Applied pow129.4

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

   12. Applied pow129.4

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

   13. Applied pow-prod-down29.4

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

   14. Simplified30.2

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

   16. Applied pow130.2

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

   17. Applied sqrt-pow130.2

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

   18. Simplified24.4

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

4. Final simplification24.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -7.462202010042364 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\sqrt[3]{U} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right)\right) + \left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;n \le 1.6373050537097972 \cdot 10^{-156}:\\ \;\;\;\;{\left(\left(U \cdot \left(\frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)}{Om} + t \cdot n\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\sqrt[3]{U} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right)\right) + \left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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*))))))