Error

Derivation

1. Split input into 3 regimes

2. if t < -7.1926022986855e-57

   1. Initial program 21.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied unpow321.8

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied times-frac15.6

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity15.6

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]

   7. Applied associate-*r*15.6

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 1\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]

   8. Applied simplify8.1

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{\frac{\ell}{t}}{t}} \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied div-inv8.1

       \[\leadsto \frac{2}{\left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\color{blue}{\frac{\ell}{t} \cdot \frac{1}{t}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   11. Applied *-un-lft-identity8.1

       \[\leadsto \frac{2}{\left(\frac{\frac{\sin k}{\color{blue}{1 \cdot \frac{\ell}{t}}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied *-un-lft-identity8.1

       \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{1 \cdot \sin k}}{1 \cdot \frac{\ell}{t}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   13. Applied times-frac8.1

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{\sin k}{\frac{\ell}{t}}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   14. Applied times-frac7.1

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{1}{t}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   15. Applied associate-*l*4.9

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{1}{t}} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   16. Applied simplify4.8

       \[\leadsto \frac{2}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \color{blue}{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   17. Using strategy rm

   18. Applied div-inv4.8

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]

   19. Applied simplify4.5

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}\]

   if -7.1926022986855e-57 < t < 2.24813614216676e-113

   1. Initial program 58.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied unpow358.6

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied times-frac48.7

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity48.7

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]

   7. Applied associate-*r*48.7

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 1\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]

   8. Applied simplify40.8

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{\frac{\ell}{t}}{t}} \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied div-inv40.8

       \[\leadsto \frac{2}{\left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\color{blue}{\frac{\ell}{t} \cdot \frac{1}{t}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   11. Applied *-un-lft-identity40.8

       \[\leadsto \frac{2}{\left(\frac{\frac{\sin k}{\color{blue}{1 \cdot \frac{\ell}{t}}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied *-un-lft-identity40.8

       \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{1 \cdot \sin k}}{1 \cdot \frac{\ell}{t}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   13. Applied times-frac40.8

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{\sin k}{\frac{\ell}{t}}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   14. Applied times-frac40.8

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{1}{t}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   15. Applied associate-*l*42.1

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{1}{t}} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   16. Applied simplify42.1

       \[\leadsto \frac{2}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \color{blue}{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   17. Taylor expanded around 0 38.3

       \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot \frac{{\ell}^{2} \cdot t}{{k}^{4}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - \frac{1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}\]

   18. Applied simplify36.5

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{k}^{4}} \cdot \left(\frac{2}{3} \cdot t + \frac{2}{t}\right) - \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{1}{3}}{t}\right)}\]

   if 2.24813614216676e-113 < t

   1. Initial program 23.3

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied unpow323.3

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied times-frac16.9

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity16.9

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]

   7. Applied associate-*r*16.9

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 1\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]

   8. Applied simplify9.6

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{\frac{\ell}{t}}{t}} \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied div-inv9.6

       \[\leadsto \frac{2}{\left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\color{blue}{\frac{\ell}{t} \cdot \frac{1}{t}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   11. Applied *-un-lft-identity9.6

       \[\leadsto \frac{2}{\left(\frac{\frac{\sin k}{\color{blue}{1 \cdot \frac{\ell}{t}}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied *-un-lft-identity9.6

       \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{1 \cdot \sin k}}{1 \cdot \frac{\ell}{t}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   13. Applied times-frac9.6

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{\sin k}{\frac{\ell}{t}}}}{\frac{\ell}{t} \cdot \frac{1}{t}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   14. Applied times-frac8.7

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{1}{t}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   15. Applied associate-*l*6.4

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \left(\frac{\frac{\sin k}{\frac{\ell}{t}}}{\frac{1}{t}} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   16. Applied simplify6.4

       \[\leadsto \frac{2}{\left(\frac{\frac{1}{1}}{\frac{\ell}{t}} \cdot \color{blue}{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   17. Using strategy rm

   18. Applied associate-*l/6.3

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{1}{1} \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)}{\frac{\ell}{t}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   19. Applied associate-*l/5.6

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{1}{1} \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t}}}}\]

   20. Applied associate-/r/5.5

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{1}{1} \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\tan k \cdot t\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{\ell}{t}}\]

   21. Applied simplify5.4

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\left(t \cdot \tan k\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\ell}{t}\]
3. Recombined 3 regimes into one program.
4. Removed slow pow expressions.

Runtime

Time bar (total: 6.1m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))