Average Error: 47.1 → 7.8
Time: 6.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{k} \cdot \left(\left(\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \left(\frac{\frac{\frac{\sqrt[3]{\sqrt{\sqrt[3]{2}}}}{\sqrt[3]{t}}}{\tan k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\sqrt[3]{t}}}} \cdot \frac{\sqrt[3]{\sqrt{\sqrt[3]{2}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{2}}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\right)\right) \cdot \frac{\frac{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}{\sqrt[3]{\sin k}}}{\frac{1}{t}}\right)\]
\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)}
↓
\frac{\frac{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{k} \cdot \left(\left(\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \left(\frac{\frac{\frac{\sqrt[3]{\sqrt{\sqrt[3]{2}}}}{\sqrt[3]{t}}}{\tan k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\sqrt[3]{t}}}} \cdot \frac{\sqrt[3]{\sqrt{\sqrt[3]{2}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{2}}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\right)\right) \cdot \frac{\frac{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}{\sqrt[3]{\sin k}}}{\frac{1}{t}}\right)
double f(double t, double l, double k) {
        double r14479868 = 2.0;
        double r14479869 = t;
        double r14479870 = 3.0;
        double r14479871 = pow(r14479869, r14479870);
        double r14479872 = l;
        double r14479873 = r14479872 * r14479872;
        double r14479874 = r14479871 / r14479873;
        double r14479875 = k;
        double r14479876 = sin(r14479875);
        double r14479877 = r14479874 * r14479876;
        double r14479878 = tan(r14479875);
        double r14479879 = r14479877 * r14479878;
        double r14479880 = 1.0;
        double r14479881 = r14479875 / r14479869;
        double r14479882 = pow(r14479881, r14479868);
        double r14479883 = r14479880 + r14479882;
        double r14479884 = r14479883 - r14479880;
        double r14479885 = r14479879 * r14479884;
        double r14479886 = r14479868 / r14479885;
        return r14479886;
}
↓
double f(double t, double l, double k) {
        double r14479887 = 2.0;
        double r14479888 = sqrt(r14479887);
        double r14479889 = sqrt(r14479888);
        double r14479890 = t;
        double r14479891 = l;
        double r14479892 = r14479890 / r14479891;
        double r14479893 = r14479889 / r14479892;
        double r14479894 = k;
        double r14479895 = sin(r14479894);
        double r14479896 = cbrt(r14479895);
        double r14479897 = r14479896 * r14479896;
        double r14479898 = r14479893 / r14479897;
        double r14479899 = r14479898 / r14479894;
        double r14479900 = cbrt(r14479887);
        double r14479901 = r14479900 * r14479900;
        double r14479902 = sqrt(r14479901);
        double r14479903 = cbrt(r14479890);
        double r14479904 = r14479903 * r14479903;
        double r14479905 = r14479902 / r14479904;
        double r14479906 = 1.0;
        double r14479907 = r14479906 / r14479904;
        double r14479908 = r14479905 / r14479907;
        double r14479909 = sqrt(r14479900);
        double r14479910 = cbrt(r14479909);
        double r14479911 = r14479910 / r14479903;
        double r14479912 = tan(r14479894);
        double r14479913 = r14479911 / r14479912;
        double r14479914 = cbrt(r14479894);
        double r14479915 = cbrt(r14479903);
        double r14479916 = r14479914 / r14479915;
        double r14479917 = r14479913 / r14479916;
        double r14479918 = r14479910 * r14479910;
        double r14479919 = r14479914 * r14479914;
        double r14479920 = r14479915 * r14479915;
        double r14479921 = r14479919 / r14479920;
        double r14479922 = r14479918 / r14479921;
        double r14479923 = r14479917 * r14479922;
        double r14479924 = r14479908 * r14479923;
        double r14479925 = r14479893 / r14479896;
        double r14479926 = r14479906 / r14479890;
        double r14479927 = r14479925 / r14479926;
        double r14479928 = r14479924 * r14479927;
        double r14479929 = r14479899 * r14479928;
        return r14479929;
}

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. Initial program 47.1

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

  2. Simplified31.1

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

  4. Applied add-sqr-sqrt31.2

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

  5. Applied times-frac31.1

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

  6. Applied times-frac31.1

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

  7. Applied times-frac20.6

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\sqrt{2}}{t}}{\tan k}}{\frac{k}{t}}}\]
  8. Using strategy rm

  9. Applied div-inv20.6

    \[\leadsto \frac{\frac{\frac{\sqrt{2}}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\sin k}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \frac{\frac{\frac{\sqrt{2}}{t}}{\tan k}}{\frac{k}{t}}\]

  10. Applied add-cube-cbrt20.8

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

  11. Applied add-sqr-sqrt20.8

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

  12. Applied sqrt-prod20.7

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

  13. Applied times-frac20.6

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

  14. Applied times-frac19.8

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

  15. Applied times-frac13.9

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

  16. Applied associate-*l*12.3

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

  18. Applied add-cube-cbrt12.4

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

  19. Applied *-un-lft-identity12.4

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

  20. Applied times-frac12.4

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

  21. Applied *-un-lft-identity12.4

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

  22. Applied add-cube-cbrt12.3

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

  23. Applied add-cube-cbrt12.4

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

  24. Applied sqrt-prod12.4

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

  25. Applied times-frac12.4

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

  26. Applied times-frac12.4

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

  27. Applied times-frac8.3

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

  29. Applied add-cube-cbrt8.4

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

  30. Applied add-cube-cbrt8.5

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

  31. Applied times-frac8.5

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

  32. Applied *-un-lft-identity8.5

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

  33. Applied *-un-lft-identity8.5

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

  34. Applied add-cube-cbrt8.5

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

  35. Applied times-frac8.4

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

  36. Applied times-frac8.4

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

  37. Applied times-frac7.8

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

  38. Final simplification7.8

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

Reproduce

herbie shell --seed 2019153 
(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))))