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Derivation

1. Split input into 3 regimes

2. if (* l l) < 1.347905859362174e-308

   1. Initial program 45.1

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

   2. Taylor expanded around -inf 20.5

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

   4. Applied associate-*r*19.5

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

   6. Applied unpow219.5

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

   7. Applied associate-*r*19.5

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

   9. Applied add-cbrt-cube19.9

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

   10. Applied add-cbrt-cube22.5

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

   11. Applied cbrt-undiv22.5

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

   12. Simplified15.2

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

   if 1.347905859362174e-308 < (* l l) < 7.332829679811125e+307

   1. Initial program 45.0

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

   2. Taylor expanded around -inf 15.6

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

   4. Applied associate-*r*13.6

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

   6. Applied unpow213.6

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

   7. Applied associate-*r*9.3

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

   9. Applied times-frac7.7

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

   if 7.332829679811125e+307 < (* l l)

   1. Initial program 62.3

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

   2. Taylor expanded around -inf 62.0

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

   4. Applied associate-*r*62.0

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

   5. Taylor expanded around 0 62.5

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} - \frac{1}{3} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}}\]

   6. Simplified55.2

      \[\leadsto \color{blue}{\left(\frac{2}{{k}^{4}} - \frac{\frac{\frac{1}{3}}{k}}{k}\right) \cdot \left(\frac{\ell}{t} \cdot \ell\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.347905859362174 \cdot 10^{-308}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{k \cdot \left(t \cdot k\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 7.332829679811125 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{4}} - \frac{\frac{\frac{1}{3}}{k}}{k}\right) \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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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