Average Error: 46.9 → 4.6
Time: 7.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(\frac{\frac{\sqrt[3]{\ell}}{t}}{\tan k} \cdot \left(\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\frac{k}{\sqrt[3]{t}}} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right) \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\frac{\frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}}}{\sin k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{t}}} \cdot \frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)\]
\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)}
↓
\left(\left(\frac{\frac{\sqrt[3]{\ell}}{t}}{\tan k} \cdot \left(\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\frac{k}{\sqrt[3]{t}}} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right) \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\frac{\frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}}}{\sin k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{t}}} \cdot \frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)
double f(double t, double l, double k) {
        double r15849416 = 2.0;
        double r15849417 = t;
        double r15849418 = 3.0;
        double r15849419 = pow(r15849417, r15849418);
        double r15849420 = l;
        double r15849421 = r15849420 * r15849420;
        double r15849422 = r15849419 / r15849421;
        double r15849423 = k;
        double r15849424 = sin(r15849423);
        double r15849425 = r15849422 * r15849424;
        double r15849426 = tan(r15849423);
        double r15849427 = r15849425 * r15849426;
        double r15849428 = 1.0;
        double r15849429 = r15849423 / r15849417;
        double r15849430 = pow(r15849429, r15849416);
        double r15849431 = r15849428 + r15849430;
        double r15849432 = r15849431 - r15849428;
        double r15849433 = r15849427 * r15849432;
        double r15849434 = r15849416 / r15849433;
        return r15849434;
}
↓
double f(double t, double l, double k) {
        double r15849435 = l;
        double r15849436 = cbrt(r15849435);
        double r15849437 = t;
        double r15849438 = r15849436 / r15849437;
        double r15849439 = k;
        double r15849440 = tan(r15849439);
        double r15849441 = r15849438 / r15849440;
        double r15849442 = 2.0;
        double r15849443 = cbrt(r15849442);
        double r15849444 = cbrt(r15849437);
        double r15849445 = r15849443 / r15849444;
        double r15849446 = r15849439 / r15849444;
        double r15849447 = r15849445 / r15849446;
        double r15849448 = r15849436 * r15849436;
        double r15849449 = r15849447 * r15849448;
        double r15849450 = r15849441 * r15849449;
        double r15849451 = 1.0;
        double r15849452 = r15849444 * r15849444;
        double r15849453 = r15849451 / r15849452;
        double r15849454 = r15849445 / r15849453;
        double r15849455 = r15849450 * r15849454;
        double r15849456 = sqrt(r15849443);
        double r15849457 = r15849436 / r15849444;
        double r15849458 = r15849444 / r15849457;
        double r15849459 = r15849456 / r15849458;
        double r15849460 = sin(r15849439);
        double r15849461 = r15849459 / r15849460;
        double r15849462 = cbrt(r15849439);
        double r15849463 = r15849462 / r15849444;
        double r15849464 = r15849461 / r15849463;
        double r15849465 = r15849448 / r15849452;
        double r15849466 = r15849451 / r15849465;
        double r15849467 = r15849456 / r15849466;
        double r15849468 = r15849462 * r15849462;
        double r15849469 = r15849468 / r15849452;
        double r15849470 = r15849467 / r15849469;
        double r15849471 = r15849464 * r15849470;
        double r15849472 = r15849455 * r15849471;
        return r15849472;
}

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 46.9

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

  2. Simplified30.3

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

  4. Applied add-cube-cbrt30.4

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

  5. Applied times-frac29.8

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

  6. Applied add-cube-cbrt29.9

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

  7. Applied times-frac29.8

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

  8. Applied times-frac29.0

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

  9. Applied times-frac15.5

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

  10. Simplified13.2

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

  12. Applied add-cube-cbrt13.2

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

  13. Applied add-cube-cbrt13.1

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

  14. Applied times-frac13.1

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

  15. Applied *-un-lft-identity13.1

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

  16. Applied add-cube-cbrt13.2

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

  17. Applied add-cube-cbrt13.2

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

  18. Applied times-frac13.2

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

  19. Applied *-un-lft-identity13.2

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

  20. Applied times-frac13.2

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

  21. Applied add-sqr-sqrt13.1

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

  22. Applied times-frac12.9

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

  23. Applied times-frac12.3

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

  24. Applied times-frac11.6

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

  26. Applied add-cube-cbrt11.5

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

  27. Applied *-un-lft-identity11.5

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

  28. Applied times-frac11.5

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

  29. Applied times-frac7.9

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

  30. Applied associate-*l*7.9

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

  32. Applied *-un-lft-identity7.9

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

  33. Applied *-un-lft-identity7.9

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

  34. Applied add-cube-cbrt8.0

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

  35. Applied times-frac8.0

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

  36. Applied times-frac6.2

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

  37. Applied associate-*r*4.6

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

  38. Final simplification4.6

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

Reproduce

herbie shell --seed 2019163 
(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))))