Error

Derivation

1. Split input into 3 regimes

2. if k < -0.05658464059341213

   1. Initial program 31.6

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

   2. Initial simplification19.7

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

   4. Applied times-frac19.7

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

   5. Applied associate-/l*17.4

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

   6. Taylor expanded around inf 22.5

      \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k \cdot {k}^{2}}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{{\ell}^{2} \cdot \cos k}}}\]

   7. Simplified4.9

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

   9. Applied associate-*l*4.9

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

   11. Applied div-inv5.0

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

   if -0.05658464059341213 < k < 3.152527126198032e+40

   1. Initial program 33.0

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

   2. Initial simplification31.7

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

   4. Applied times-frac16.0

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

   5. Applied associate-/l*15.4

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

   7. Applied add-cube-cbrt15.7

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

   8. Applied times-frac9.3

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

   9. Applied add-sqr-sqrt9.2

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

   10. Applied times-frac8.9

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

   11. Applied *-un-lft-identity8.9

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

   12. Applied div-inv8.9

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

   13. Applied times-frac8.9

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

   14. Applied times-frac5.9

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

   15. Simplified5.9

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

   16. Simplified4.8

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

   if 3.152527126198032e+40 < k

   1. Initial program 33.3

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

   2. Initial simplification21.0

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

   4. Applied times-frac21.0

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

   5. Applied associate-/l*18.4

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

   6. Taylor expanded around inf 25.1

      \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k \cdot {k}^{2}}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{{\ell}^{2} \cdot \cos k}}}\]

   7. Simplified5.2

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

   9. Applied div-inv5.2

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

   10. Applied associate-/l*5.2

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

4. Final simplification5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -0.05658464059341213:\\ \;\;\;\;\frac{1}{(2 \cdot \left(\frac{\frac{t}{\ell} \cdot \frac{t}{\ell}}{\frac{\cos k}{\sin k}}\right) + \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{\sin k}{\cos k}\right)\right))_*} \cdot \frac{\frac{2}{t}}{\sin k}\\ \mathbf{elif}\;k \le 3.152527126198032 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot 2}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\frac{\ell}{t}}{\sin k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}{t \cdot \sqrt[3]{\tan k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{(2 \cdot \left(\frac{\frac{t}{\ell} \cdot \frac{t}{\ell}}{\frac{\cos k}{\sin k}}\right) + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}{\frac{1}{\sin k}}}\\ \end{array}\]

Runtime

Time bar (total: 7.6m)Debug logProfile

herbie shell --seed 2018249 +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))))