Error

Derivation

1. Split input into 3 regimes

2. if n < -3.5354787725804784e-50

   1. Initial program 30.8

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

   3. Applied *-un-lft-identity30.8

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

   4. Applied associate-*r*30.8

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

   5. Simplified26.7

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

   7. Applied sub-neg26.7

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

   8. Applied distribute-rgt-in26.7

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

   9. Simplified25.7

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

   11. Applied add-sqr-sqrt25.7

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

   12. Applied rem-sqrt-square25.7

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

   13. Simplified27.4

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

   if -3.5354787725804784e-50 < n < 2.477461544080291e-184

   1. Initial program 35.6

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

   3. Applied *-un-lft-identity35.6

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

   4. Applied associate-*r*35.6

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

   5. Simplified32.7

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

   7. Applied sub-neg32.7

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

   8. Applied distribute-rgt-in32.7

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

   9. Simplified26.2

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

   10. Taylor expanded around -inf 26.3

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

   11. Simplified24.0

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

   13. Applied add-sqr-sqrt24.0

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

   14. Applied rem-sqrt-square24.0

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

   15. Simplified22.7

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

   if 2.477461544080291e-184 < n

   1. Initial program 31.9

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

   3. Applied *-un-lft-identity31.9

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

   4. Applied associate-*r*31.9

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

   5. Simplified28.6

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

   7. Applied sub-neg28.6

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

   8. Applied distribute-rgt-in28.6

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

   9. Simplified25.2

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

   11. Applied add-sqr-sqrt25.4

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

4. Final simplification24.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.5354787725804784 \cdot 10^{-50}:\\ \;\;\;\;\left|\sqrt{\left(\left(-2 \cdot \ell - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot n\right)\right) + n \cdot \left(U \cdot t\right)\right) \cdot 2}\right|\\ \mathbf{elif}\;n \le 2.477461544080291 \cdot 10^{-184}:\\ \;\;\;\;\left|\sqrt{\left(2 \cdot U\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + t \cdot n\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot U\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \cdot \sqrt{\sqrt{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot U\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(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*))))))