Average Error: 47.9 → 8.3
Time: 3.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(\frac{\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\tan k}}{\frac{k}{\sqrt[3]{t}}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}{\sin k}}{\frac{k}{\sqrt[3]{t}}}\right)\]
\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{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\tan k}}{\frac{k}{\sqrt[3]{t}}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}{\sin k}}{\frac{k}{\sqrt[3]{t}}}\right)
double f(double t, double l, double k) {
        double r4300426 = 2.0;
        double r4300427 = t;
        double r4300428 = 3.0;
        double r4300429 = pow(r4300427, r4300428);
        double r4300430 = l;
        double r4300431 = r4300430 * r4300430;
        double r4300432 = r4300429 / r4300431;
        double r4300433 = k;
        double r4300434 = sin(r4300433);
        double r4300435 = r4300432 * r4300434;
        double r4300436 = tan(r4300433);
        double r4300437 = r4300435 * r4300436;
        double r4300438 = 1.0;
        double r4300439 = r4300433 / r4300427;
        double r4300440 = pow(r4300439, r4300426);
        double r4300441 = r4300438 + r4300440;
        double r4300442 = r4300441 - r4300438;
        double r4300443 = r4300437 * r4300442;
        double r4300444 = r4300426 / r4300443;
        return r4300444;
}
↓
double f(double t, double l, double k) {
        double r4300445 = 2.0;
        double r4300446 = sqrt(r4300445);
        double r4300447 = t;
        double r4300448 = cbrt(r4300447);
        double r4300449 = r4300446 / r4300448;
        double r4300450 = k;
        double r4300451 = tan(r4300450);
        double r4300452 = r4300449 / r4300451;
        double r4300453 = r4300450 / r4300448;
        double r4300454 = r4300452 / r4300453;
        double r4300455 = l;
        double r4300456 = r4300455 / r4300447;
        double r4300457 = r4300454 * r4300456;
        double r4300458 = r4300448 * r4300448;
        double r4300459 = r4300446 / r4300458;
        double r4300460 = 1.0;
        double r4300461 = r4300460 / r4300458;
        double r4300462 = r4300459 / r4300461;
        double r4300463 = r4300457 * r4300462;
        double r4300464 = cbrt(r4300455);
        double r4300465 = r4300464 * r4300464;
        double r4300466 = r4300465 / r4300458;
        double r4300467 = r4300466 / r4300461;
        double r4300468 = r4300464 / r4300448;
        double r4300469 = sin(r4300450);
        double r4300470 = r4300468 / r4300469;
        double r4300471 = r4300470 / r4300453;
        double r4300472 = r4300467 * r4300471;
        double r4300473 = r4300463 * r4300472;
        return r4300473;
}

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 47.9

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

  2. Simplified31.1

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

  4. Applied times-frac20.2

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

  6. Applied *-un-lft-identity20.2

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

  7. Applied *-un-lft-identity20.2

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

  8. Applied times-frac19.4

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

  9. Applied times-frac14.0

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

  10. Applied associate-*r*12.6

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

  12. Applied add-cube-cbrt12.8

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

  13. Applied *-un-lft-identity12.8

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

  14. Applied times-frac12.8

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

  15. Applied *-un-lft-identity12.8

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

  16. Applied add-cube-cbrt12.6

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

  17. Applied add-sqr-sqrt12.7

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

  18. Applied times-frac12.6

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

  19. Applied times-frac12.6

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

  20. Applied times-frac11.9

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

  21. Applied associate-*l*11.9

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

  23. Applied add-cube-cbrt12.0

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

  24. Applied *-un-lft-identity12.0

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

  25. Applied times-frac12.0

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

  26. Applied *-un-lft-identity12.0

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

  27. Applied add-cube-cbrt11.9

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

  28. Applied add-cube-cbrt12.0

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

  29. Applied times-frac12.0

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

  30. Applied times-frac12.0

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

  31. Applied times-frac8.3

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

  32. Final simplification8.3

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

Reproduce

herbie shell --seed 2019154 +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))))