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Derivation

1. Split input into 2 regimes

2. if t < -2.1315202945836788e-103 or 1.6365766351358701e-80 < t

   1. Initial program 42.2

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

   2. Initial simplification31.6

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2} + 0}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt31.6

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\sqrt{{\left(\frac{k}{t}\right)}^{2} + 0} \cdot \sqrt{{\left(\frac{k}{t}\right)}^{2} + 0}}}\]

   5. Applied times-frac31.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}}}{\sqrt{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \frac{\ell \cdot \ell}{\sqrt{{\left(\frac{k}{t}\right)}^{2} + 0}}}\]

   6. Simplified31.6

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \tan k}}{\left|\frac{k}{t}\right| \cdot \sin k}} \cdot \frac{\ell \cdot \ell}{\sqrt{{\left(\frac{k}{t}\right)}^{2} + 0}}\]

   7. Simplified26.3

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \tan k}}{\left|\frac{k}{t}\right| \cdot \sin k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\left|\frac{k}{t}\right|}}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt26.3

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \tan k}}{\left|\frac{k}{t}\right| \cdot \sin k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sqrt{\left|\frac{k}{t}\right|} \cdot \sqrt{\left|\frac{k}{t}\right|}}}\]

   10. Applied times-frac24.1

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

   11. Applied associate-*r*20.9

       \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \tan k}}{\left|\frac{k}{t}\right| \cdot \sin k} \cdot \frac{\ell}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\ell}{\sqrt{\left|\frac{k}{t}\right|}}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt21.0

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

   14. Applied times-frac20.3

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

   15. Applied associate-*l*18.7

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{2}{{t}^{3} \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{{t}^{3} \cdot \tan k}}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\sqrt[3]{\frac{2}{{t}^{3} \cdot \tan k}}}{\sin k} \cdot \frac{\ell}{\sqrt{\left|\frac{k}{t}\right|}}\right)\right)} \cdot \frac{\ell}{\sqrt{\left|\frac{k}{t}\right|}}\]
   16. Using strategy rm

   17. Applied add-cube-cbrt18.7

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

   18. Applied cbrt-prod18.7

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

   if -2.1315202945836788e-103 < t < 1.6365766351358701e-80

   1. Initial program 60.0

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

   2. Initial simplification61.2

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2} + 0}\]

   3. Taylor expanded around inf 40.3

      \[\leadsto \frac{\color{blue}{0} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2} + 0}\]
3. Recombined 2 regimes into one program.

4. Final simplification24.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.1315202945836788 \cdot 10^{-103} \lor \neg \left(t \le 1.6365766351358701 \cdot 10^{-80}\right):\\ \;\;\;\;\left(\left(\frac{\ell}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k \cdot {t}^{3}}}}{\sin k}\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{\frac{2}{\tan k \cdot {t}^{3}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{\tan k \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{2}{\tan k \cdot {t}^{3}}}}\right) \cdot \sqrt[3]{\frac{2}{\tan k \cdot {t}^{3}}}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{\sqrt{\left|\frac{k}{t}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 0}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018221 
(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))))