Error

Derivation

1. Split input into 3 regimes

2. if t < -5.654210855745907e-57

   1. Initial program 23.0

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

   3. Applied unpow323.0

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

   4. Applied times-frac16.8

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

   5. Applied associate-*l*14.6

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

   7. Applied tan-quot14.6

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

   8. Applied associate-*l/14.7

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

   9. Applied associate-*r/14.8

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

   10. Applied frac-times14.6

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

   11. Applied associate-*l/14.5

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

   12. Simplified8.1

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

   14. Applied associate-/l*4.2

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

   16. Applied div-inv4.2

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

   17. Applied div-inv4.3

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

   18. Applied times-frac4.3

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

   19. Simplified4.2

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

   if -5.654210855745907e-57 < t < 3.7270847572392797e-38

   1. Initial program 54.0

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

   3. Applied unpow354.0

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

   4. Applied times-frac45.6

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

   5. Applied associate-*l*44.8

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

   7. Applied tan-quot44.8

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

   8. Applied associate-*l/45.5

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

   9. Applied associate-*r/46.4

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

   10. Applied frac-times48.0

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

   11. Applied associate-*l/47.5

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

   12. Simplified37.7

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

   13. Taylor expanded around -inf 23.1

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

   if 3.7270847572392797e-38 < t

   1. Initial program 20.9

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

   3. Applied unpow320.9

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

   4. Applied times-frac15.2

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

   5. Applied associate-*l*12.9

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

   7. Applied tan-quot13.0

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

   8. Applied associate-*l/13.0

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

   9. Applied associate-*r/13.2

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

   10. Applied frac-times13.1

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

   11. Applied associate-*l/13.2

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

   12. Simplified7.3

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

   14. Applied associate-/l*3.3

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

   16. Applied div-inv3.3

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

   17. Applied associate-/r*3.1

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

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.654210855745907 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\sin k \cdot t}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \le 3.7270847572392797 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(\sin k\right)}^{2} \cdot {t}^{3}}{\ell} + \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\ell}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\sin k \cdot t}{\ell}}{\frac{1}{t}}}{\frac{\ell \cdot \cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot t\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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))))

Details

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