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Derivation

1. Split input into 2 regimes

2. if l < -7.250906796774142e-162 or 8.585813453833051e-192 < l

   1. Initial program 47.2

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

   2. Taylor expanded around inf 24.6

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

   4. Applied associate-*r*23.2

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

   6. Applied unpow223.2

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

   7. Applied associate-*r*20.4

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

   9. Applied associate-/l*19.2

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

   if -7.250906796774142e-162 < l < 8.585813453833051e-192

   1. Initial program 46.0

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

   2. Taylor expanded around inf 20.9

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

   4. Applied add-cbrt-cube20.9

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

   5. Applied add-cbrt-cube23.0

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

   6. Applied cbrt-undiv23.0

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

   7. Simplified17.4

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

4. Final simplification18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -7.250906796774142 \cdot 10^{-162} \lor \neg \left(\ell \le 8.585813453833051 \cdot 10^{-192}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t}{\frac{\cos k}{k \cdot k}}\right)}^{3}}}\\ \end{array}\]

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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