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Derivation

1. Split input into 3 regimes

2. if (* l l) < 0.0

   1. Initial program 45.4

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

   2. Taylor expanded around inf 20.4

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

   4. Applied associate-*r*19.6

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

   6. Applied unpow219.6

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

   7. Applied associate-*r*19.6

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

   8. Taylor expanded around 0 22.3

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{6} \cdot \frac{t \cdot {k}^{6}}{{\ell}^{2}} + \left(\frac{31}{360} \cdot \frac{t \cdot {k}^{8}}{{\ell}^{2}} + \frac{t \cdot {k}^{4}}{{\ell}^{2}}\right)}}\]

   9. Simplified18.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\frac{31}{360} \cdot {k}^{8} + \left({k}^{4} + {k}^{6} \cdot \frac{1}{6}\right)\right)}}\]

   if 0.0 < (* l l) < 2.724912876006757e+292

   1. Initial program 43.7

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

   2. Taylor expanded around inf 15.2

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

   4. Applied associate-*r*13.0

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

   6. Applied unpow213.0

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

   7. Applied associate-*r*9.2

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

   9. Applied times-frac7.5

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

   if 2.724912876006757e+292 < (* l l)

   1. Initial program 61.7

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

   3. Applied cube-mult61.7

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

   4. Applied times-frac51.7

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

   5. Applied associate-*l*51.7

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

4. Final simplification18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;\frac{2}{\left(\left({k}^{6} \cdot \frac{1}{6} + {k}^{4}\right) + {k}^{8} \cdot \frac{31}{360}\right) \cdot \frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \le 2.724912876006757 \cdot 10^{+292}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) - 1\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(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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