Error

Derivation

1. Split input into 3 regimes

2. if t < -1.2777324464472654e-176

   1. Initial program 26.4

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

   3. Applied unpow326.4

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

   4. Applied times-frac18.9

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

   5. Applied associate-*l*16.9

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

   7. Applied tan-quot16.9

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

   8. Applied associate-*l/17.2

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

   9. Applied associate-*r/17.7

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

   10. Applied frac-times18.0

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

   11. Applied associate-*l/17.9

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

   12. Simplified11.9

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}}{\ell \cdot \cos k}}\]
   13. Using strategy rm

   14. Applied associate-/l*8.0

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\frac{\ell \cdot \cos k}{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\]
   15. Using strategy rm

   16. Applied times-frac7.0

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

   17. Applied *-un-lft-identity7.0

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

   18. Applied times-frac7.2

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

   19. Simplified7.2

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

   if -1.2777324464472654e-176 < t < 5.887444405071525e-106

   1. Initial program 62.5

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

   3. Applied unpow362.5

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

   4. Applied times-frac57.2

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

   5. Applied associate-*l*57.2

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

   7. Applied tan-quot57.2

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

   8. Applied associate-*l/57.3

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

   9. Applied associate-*r/57.9

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

   10. Applied frac-times58.4

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

   11. Applied associate-*l/58.4

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

   12. Simplified43.6

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

   13. Taylor expanded around inf 21.7

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

   if 5.887444405071525e-106 < t

   1. Initial program 24.0

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

   3. Applied unpow324.0

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

   4. Applied times-frac17.3

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

   5. Applied associate-*l*14.8

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

   7. Applied tan-quot14.8

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

   8. Applied associate-*l/15.0

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

   9. Applied associate-*r/15.2

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

   10. Applied frac-times15.5

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

   11. Applied associate-*l/15.5

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

   12. Simplified10.1

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}}{\ell \cdot \cos k}}\]
   13. Using strategy rm

   14. Applied associate-/l*5.8

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\frac{\ell \cdot \cos k}{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\]
   15. Using strategy rm

   16. Applied times-frac5.1

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

   17. Applied *-un-lft-identity5.1

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

   18. Applied times-frac5.2

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

   19. Applied associate-/r*5.0

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{1}{\frac{\ell}{\sin k \cdot t}}}}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.2777324464472654 \cdot 10^{-176}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\\ \mathbf{elif}\;t \le 5.887444405071525 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(\sin k\right)}^{2} \cdot {t}^{3}}{\ell} + \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\ell}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{1}{\frac{\ell}{\sin k \cdot t}}}}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(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))))