Average Error: 46.8 → 1.4
Time: 3.8m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -4.2003454318958233 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(\frac{\ell}{\sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{k}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\frac{k}{\ell}}}{t \cdot \sqrt[3]{\tan k}}\\ \mathbf{elif}\;k \le 2.9353542974333805 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\left(\frac{\ell}{k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(\frac{\ell}{k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\right)}{\sin k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{2}}}{\frac{t \cdot \sqrt[3]{\tan k}}{\frac{\sqrt{\sqrt[3]{2}}}{\frac{k}{\ell}}}} \cdot \left(\left(\frac{\ell}{\sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{k}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if k < -4.2003454318958233e-150

    1. Initial program 46.2

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

    2. Initial simplification28.7

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
    3. Using strategy rm

    4. Applied times-frac28.1

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    5. Applied add-cube-cbrt28.2

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

    6. Applied times-frac27.8

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    7. Applied times-frac16.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

    8. Simplified8.8

      \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity8.8

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

    11. Applied associate-/r/8.8

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

    12. Applied times-frac9.0

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

    13. Applied times-frac9.0

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

    14. Simplified9.0

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

    15. Simplified4.9

      \[\leadsto \left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\ell}{\sin k}\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{k}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
    16. Using strategy rm

    17. Applied cbrt-div5.0

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

    18. Applied associate-*r/1.3

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

    19. Applied frac-times1.2

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

    20. Simplified1.2

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

    if -4.2003454318958233e-150 < k < 2.9353542974333805e-103

    1. Initial program 62.2

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

    2. Initial simplification59.8

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
    3. Using strategy rm

    4. Applied times-frac55.5

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    5. Applied add-cube-cbrt55.6

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

    6. Applied times-frac54.9

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    7. Applied times-frac49.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

    8. Simplified43.0

      \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity43.0

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

    11. Applied associate-/r/42.9

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

    12. Applied times-frac43.7

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

    13. Applied times-frac43.7

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

    14. Simplified43.7

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

    15. Simplified40.9

      \[\leadsto \left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\ell}{\sin k}\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{k}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
    16. Using strategy rm

    17. Applied associate-*r/21.8

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

    18. Applied associate-*l/17.2

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

    19. Applied associate-*r/17.2

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

    20. Applied associate-*l/17.2

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

    21. Applied frac-times11.2

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

    22. Simplified7.6

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

    if 2.9353542974333805e-103 < k

    1. Initial program 45.5

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

    2. Initial simplification28.0

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
    3. Using strategy rm

    4. Applied times-frac27.7

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    5. Applied add-cube-cbrt27.8

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

    6. Applied times-frac27.5

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    7. Applied times-frac15.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

    8. Simplified8.7

      \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity8.7

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

    11. Applied associate-/r/8.7

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

    12. Applied times-frac8.8

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

    13. Applied times-frac8.8

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

    14. Simplified8.8

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

    15. Simplified4.9

      \[\leadsto \left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\ell}{\sin k}\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{k}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
    16. Using strategy rm

    17. Applied cbrt-div4.9

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

    18. Applied associate-*r/1.0

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

    19. Applied frac-times0.9

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

    20. Simplified0.9

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

    22. Applied *-un-lft-identity0.9

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

    23. Applied add-sqr-sqrt0.8

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

    24. Applied times-frac0.8

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

    25. Applied associate-/l*0.8

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

    26. Simplified0.8

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.2003454318958233 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(\frac{\ell}{\sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{k}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\frac{k}{\ell}}}{t \cdot \sqrt[3]{\tan k}}\\ \mathbf{elif}\;k \le 2.9353542974333805 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\left(\frac{\ell}{k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(\frac{\ell}{k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\right)}{\sin k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{2}}}{\frac{t \cdot \sqrt[3]{\tan k}}{\frac{\sqrt{\sqrt[3]{2}}}{\frac{k}{\ell}}}} \cdot \left(\left(\frac{\ell}{\sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{k}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.01.40.11.932.9%
herbie shell --seed 2018290 
(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))))