Average Error: 47.4 → 7.7
Time: 2.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{\frac{\ell}{\sqrt[3]{t}}}{\sin k}}{\frac{k}{\sqrt[3]{t}}} \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\frac{\frac{2}{\sqrt[3]{t}}}{\tan k}}{\frac{k}{\sqrt[3]{t}}}\right) \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)\]
\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(\frac{\frac{\frac{\ell}{\sqrt[3]{t}}}{\sin k}}{\frac{k}{\sqrt[3]{t}}} \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\frac{\frac{2}{\sqrt[3]{t}}}{\tan k}}{\frac{k}{\sqrt[3]{t}}}\right) \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)
double f(double t, double l, double k) {
        double r3552401 = 2.0;
        double r3552402 = t;
        double r3552403 = 3.0;
        double r3552404 = pow(r3552402, r3552403);
        double r3552405 = l;
        double r3552406 = r3552405 * r3552405;
        double r3552407 = r3552404 / r3552406;
        double r3552408 = k;
        double r3552409 = sin(r3552408);
        double r3552410 = r3552407 * r3552409;
        double r3552411 = tan(r3552408);
        double r3552412 = r3552410 * r3552411;
        double r3552413 = 1.0;
        double r3552414 = r3552408 / r3552402;
        double r3552415 = pow(r3552414, r3552401);
        double r3552416 = r3552413 + r3552415;
        double r3552417 = r3552416 - r3552413;
        double r3552418 = r3552412 * r3552417;
        double r3552419 = r3552401 / r3552418;
        return r3552419;
}
↓
double f(double t, double l, double k) {
        double r3552420 = l;
        double r3552421 = t;
        double r3552422 = cbrt(r3552421);
        double r3552423 = r3552420 / r3552422;
        double r3552424 = k;
        double r3552425 = sin(r3552424);
        double r3552426 = r3552423 / r3552425;
        double r3552427 = r3552424 / r3552422;
        double r3552428 = r3552426 / r3552427;
        double r3552429 = 1.0;
        double r3552430 = r3552422 * r3552422;
        double r3552431 = r3552429 / r3552430;
        double r3552432 = r3552431 / r3552431;
        double r3552433 = r3552428 * r3552432;
        double r3552434 = r3552420 / r3552421;
        double r3552435 = 2.0;
        double r3552436 = r3552435 / r3552422;
        double r3552437 = tan(r3552424);
        double r3552438 = r3552436 / r3552437;
        double r3552439 = r3552438 / r3552427;
        double r3552440 = r3552434 * r3552439;
        double r3552441 = r3552440 * r3552432;
        double r3552442 = r3552433 * r3552441;
        return r3552442;
}

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 47.4

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

  2. Simplified30.7

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

  4. Applied times-frac19.9

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

  6. Applied *-un-lft-identity19.9

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

  7. Applied *-un-lft-identity19.9

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

  8. Applied times-frac19.1

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

  9. Applied times-frac13.4

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

  10. Applied associate-*r*11.9

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

  12. Applied add-cube-cbrt12.1

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

  13. Applied *-un-lft-identity12.1

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

  14. Applied times-frac12.1

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

  15. Applied *-un-lft-identity12.1

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

  16. Applied add-cube-cbrt11.9

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

  17. Applied *-un-lft-identity11.9

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

  18. Applied times-frac11.9

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

  19. Applied times-frac11.9

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

  20. Applied times-frac11.3

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

  21. Applied associate-*l*11.3

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

  23. Applied add-cube-cbrt11.5

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

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

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

  25. Applied times-frac11.5

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

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

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

  27. Applied add-cube-cbrt11.3

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

  28. Applied *-un-lft-identity11.3

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

  29. Applied times-frac11.3

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

  30. Applied times-frac11.3

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

  31. Applied times-frac7.7

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

  32. Final simplification7.7

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

Reproduce

herbie shell --seed 2019156 +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))))