Error

Derivation

1. Split input into 3 regimes

2. if (pow k 4) < 4.508229574637432e-294

   1. Initial program 62.1

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

   3. Applied add-cbrt-cube62.2

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

   4. Applied simplify58.7

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
   5. Using strategy rm

   6. Applied associate-*l/59.6

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

   7. Applied associate-*l/59.5

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

   8. Applied tan-quot59.5

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

   9. Applied associate-*r/59.5

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

   10. Applied frac-times59.4

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

   11. Applied cube-div62.2

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

   12. Applied cbrt-div62.2

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

   13. Applied simplify61.3

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

   14. Applied simplify55.2

       \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]
   15. Using strategy rm

   16. Applied times-frac51.2

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}{\ell} \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\cos k}}}\]
   17. Using strategy rm

   18. Applied add-cube-cbrt51.3

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt[3]{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\right) \cdot \sqrt[3]{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}{\ell} \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\cos k}}\]

   if 4.508229574637432e-294 < (pow k 4) < 5119375.815533619

   1. Initial program 57.5

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

   3. Applied add-cbrt-cube59.4

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

   4. Applied simplify50.8

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

   5. Taylor expanded around 0 36.2

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}}^{3}}}\]

   6. Applied simplify6.7

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

   if 5119375.815533619 < (pow k 4)

   1. Initial program 43.2

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

   3. Applied add-cbrt-cube44.9

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

   4. Applied simplify33.4

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
   5. Using strategy rm

   6. Applied associate-*l/36.1

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

   7. Applied associate-*l/36.1

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

   8. Applied tan-quot36.1

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

   9. Applied associate-*r/36.1

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

   10. Applied frac-times35.8

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

   11. Applied cube-div38.7

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

   12. Applied cbrt-div38.4

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

   13. Applied simplify30.2

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

   14. Applied simplify24.0

       \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]
   15. Using strategy rm

   16. Applied times-frac23.1

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}{\ell} \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\cos k}}}\]
   17. Using strategy rm

   18. Applied add-cube-cbrt23.2

       \[\leadsto \frac{2}{\frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}{\ell} \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\left(\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}\right) \cdot \sqrt[3]{\cos k}}}}\]

   19. Applied div-inv23.2

       \[\leadsto \frac{2}{\frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}{\ell} \cdot \frac{\color{blue}{\left(\sin k \cdot t\right) \cdot \frac{1}{\frac{\ell}{t}}}}{\left(\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}\right) \cdot \sqrt[3]{\cos k}}}\]

   20. Applied times-frac23.2

       \[\leadsto \frac{2}{\frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}{\ell} \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}} \cdot \frac{\frac{1}{\frac{\ell}{t}}}{\sqrt[3]{\cos k}}\right)}}\]

   21. Applied associate-*r*20.8

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}{\ell} \cdot \frac{\sin k \cdot t}{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}\right) \cdot \frac{\frac{1}{\frac{\ell}{t}}}{\sqrt[3]{\cos k}}}}\]
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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))))