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Derivation

1. Split input into 2 regimes

2. if (* l l) < 1.1997133799852223e+295

   1. Initial program 44.6

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

   2. Taylor expanded around -inf 17.2

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

   4. Applied associate-*r*15.7

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

   6. Applied unpow215.7

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

   7. Applied associate-*r*13.2

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

   9. Applied clear-num13.2

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

   if 1.1997133799852223e+295 < (* l l)

   1. Initial program 61.2

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

   3. Applied add-cube-cbrt61.2

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

   4. Applied times-frac54.9

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

   5. Simplified54.9

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

   6. Simplified51.5

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

4. Final simplification19.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.1997133799852223 \cdot 10^{+295}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{{\ell}^{2} \cdot \cos k}{\left(\left(k \cdot t\right) \cdot k\right) \cdot {\left(\sin k\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(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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