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Derivation

1. Split input into 2 regimes

2. if l < 2.280074655019684e-248 or 8.153562515288057e-100 < l

   1. Initial program 47.8

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

   2. Initial simplification40.8

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]

   3. Taylor expanded around inf 24.9

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

   4. Taylor expanded around -inf 62.8

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

   5. Simplified21.2

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

   7. Applied frac-times14.9

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

   9. Applied *-un-lft-identity14.9

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

   10. Applied times-frac14.7

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

   11. Applied associate-*r*9.9

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

   if 2.280074655019684e-248 < l < 8.153562515288057e-100

   1. Initial program 44.8

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

   2. Initial simplification34.8

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]

   3. Taylor expanded around inf 16.7

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

   4. Taylor expanded around -inf 62.9

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

   5. Simplified13.8

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

   7. Applied frac-times7.5

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

   9. Applied associate-*l/6.1

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

4. Final simplification9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 2.280074655019684 \cdot 10^{-248} \lor \neg \left(\ell \le 8.153562515288057 \cdot 10^{-100}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{\frac{t}{\ell} \cdot {\left(\sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot k} \cdot \ell}{\frac{{\left(\sin k\right)}^{2} \cdot t}{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018221 +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))))