Average Error: 47.1 → 1.1
Time: 1.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\left(\frac{\ell}{\sin k} \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \tan k}\]
\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{\left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\left(\frac{\ell}{\sin k} \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \tan k}
double f(double t, double l, double k) {
        double r3384440 = 2.0;
        double r3384441 = t;
        double r3384442 = 3.0;
        double r3384443 = pow(r3384441, r3384442);
        double r3384444 = l;
        double r3384445 = r3384444 * r3384444;
        double r3384446 = r3384443 / r3384445;
        double r3384447 = k;
        double r3384448 = sin(r3384447);
        double r3384449 = r3384446 * r3384448;
        double r3384450 = tan(r3384447);
        double r3384451 = r3384449 * r3384450;
        double r3384452 = 1.0;
        double r3384453 = r3384447 / r3384441;
        double r3384454 = pow(r3384453, r3384440);
        double r3384455 = r3384452 + r3384454;
        double r3384456 = r3384455 - r3384452;
        double r3384457 = r3384451 * r3384456;
        double r3384458 = r3384440 / r3384457;
        return r3384458;
}
↓
double f(double t, double l, double k) {
        double r3384459 = 2.0;
        double r3384460 = sqrt(r3384459);
        double r3384461 = cbrt(r3384460);
        double r3384462 = k;
        double r3384463 = r3384461 / r3384462;
        double r3384464 = l;
        double r3384465 = sin(r3384462);
        double r3384466 = r3384464 / r3384465;
        double r3384467 = sqrt(r3384460);
        double r3384468 = r3384466 * r3384467;
        double r3384469 = r3384468 * r3384467;
        double r3384470 = r3384463 * r3384469;
        double r3384471 = r3384461 * r3384461;
        double r3384472 = r3384470 * r3384471;
        double r3384473 = t;
        double r3384474 = r3384464 / r3384462;
        double r3384475 = r3384473 / r3384474;
        double r3384476 = tan(r3384462);
        double r3384477 = r3384475 * r3384476;
        double r3384478 = r3384472 / r3384477;
        return r3384478;
}

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. Simplified17.4

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

  4. Applied div-inv17.4

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

  5. Applied add-sqr-sqrt17.5

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

  6. Applied times-frac17.3

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

  7. Applied times-frac15.7

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

  8. Simplified7.3

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

  9. Simplified5.3

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

  11. Applied associate-/r/5.3

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

  12. Applied times-frac4.5

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

  13. Applied add-cube-cbrt4.5

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

  14. Applied times-frac4.1

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

  15. Applied associate-*l*3.0

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

  16. Simplified3.0

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

  18. Applied associate-*l/3.0

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

  19. Applied associate-*r/3.0

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

  20. Applied frac-times1.2

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

  22. Applied add-sqr-sqrt1.2

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

  23. Applied sqrt-prod1.1

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

  24. Applied associate-*l*1.1

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

  25. Final simplification1.1

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

Reproduce

herbie shell --seed 2019146 
(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))))