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Derivation

1. Split input into 3 regimes

2. if k < -0.02496175920108493

   1. Initial program 44.7

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

   2. Taylor expanded around inf 19.7

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

   4. Applied associate-*r*19.8

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

   6. Applied unpow219.8

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

   7. Applied associate-*r*16.8

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

   9. Applied associate-*l*16.8

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

   if -0.02496175920108493 < k < -7.87339514779147e-79

   1. Initial program 56.7

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

   2. Taylor expanded around 0 20.8

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{6} \cdot \frac{t \cdot {k}^{6}}{{\ell}^{2}} + \frac{t \cdot {k}^{4}}{{\ell}^{2}}}}\]

   3. Simplified11.4

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

   if -7.87339514779147e-79 < k

   1. Initial program 48.5

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

   2. Taylor expanded around inf 27.1

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

   4. Applied associate-*r*24.7

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

   6. Applied unpow224.7

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

   7. Applied associate-*r*22.8

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

   9. Applied associate-/l*21.2

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

4. Final simplification18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -0.02496175920108493:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sin k\right)}^{2} \cdot k\right) \cdot \left(t \cdot k\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;k \le -7.87339514779147 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({k}^{4} + \frac{1}{6} \cdot {k}^{6}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot k\right) \cdot k}{\frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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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