Average Error: 31.9 → 11.4
Time: 5.7m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt[3]{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sqrt[3]{\tan k}}{\frac{1}{t}}} \cdot \frac{\frac{\ell}{t} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}\right) \cdot \frac{\frac{\ell}{t}}{\sin k}\]
\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)}
↓
\left(\frac{\frac{\sqrt[3]{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sqrt[3]{\tan k}}{\frac{1}{t}}} \cdot \frac{\frac{\ell}{t} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}\right) \cdot \frac{\frac{\ell}{t}}{\sin k}
double f(double t, double l, double k) {
        double r11793327 = 2.0;
        double r11793328 = t;
        double r11793329 = 3.0;
        double r11793330 = pow(r11793328, r11793329);
        double r11793331 = l;
        double r11793332 = r11793331 * r11793331;
        double r11793333 = r11793330 / r11793332;
        double r11793334 = k;
        double r11793335 = sin(r11793334);
        double r11793336 = r11793333 * r11793335;
        double r11793337 = tan(r11793334);
        double r11793338 = r11793336 * r11793337;
        double r11793339 = 1.0;
        double r11793340 = r11793334 / r11793328;
        double r11793341 = pow(r11793340, r11793327);
        double r11793342 = r11793339 + r11793341;
        double r11793343 = r11793342 + r11793339;
        double r11793344 = r11793338 * r11793343;
        double r11793345 = r11793327 / r11793344;
        return r11793345;
}
↓
double f(double t, double l, double k) {
        double r11793346 = 2.0;
        double r11793347 = cbrt(r11793346);
        double r11793348 = k;
        double r11793349 = t;
        double r11793350 = r11793348 / r11793349;
        double r11793351 = r11793350 * r11793350;
        double r11793352 = r11793351 + r11793346;
        double r11793353 = sqrt(r11793352);
        double r11793354 = r11793347 / r11793353;
        double r11793355 = tan(r11793348);
        double r11793356 = cbrt(r11793355);
        double r11793357 = 1.0;
        double r11793358 = r11793357 / r11793349;
        double r11793359 = r11793356 / r11793358;
        double r11793360 = r11793354 / r11793359;
        double r11793361 = l;
        double r11793362 = r11793361 / r11793349;
        double r11793363 = r11793347 * r11793347;
        double r11793364 = r11793363 / r11793353;
        double r11793365 = r11793362 * r11793364;
        double r11793366 = r11793356 * r11793356;
        double r11793367 = r11793365 / r11793366;
        double r11793368 = r11793360 * r11793367;
        double r11793369 = sin(r11793348);
        double r11793370 = r11793362 / r11793369;
        double r11793371 = r11793368 * r11793370;
        return r11793371;
}

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 31.9

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

  2. Simplified18.3

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

  4. Applied times-frac16.8

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

  5. Applied *-un-lft-identity16.8

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

  6. Applied times-frac16.1

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

  7. Applied *-un-lft-identity16.1

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

  8. Applied times-frac14.7

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

  9. Simplified14.7

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

  11. Applied div-inv14.7

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

  12. Applied *-un-lft-identity14.7

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

  13. Applied times-frac12.5

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

  14. Applied *-un-lft-identity12.5

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

  15. Applied times-frac12.0

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

  16. Simplified11.9

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

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

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

  19. Applied add-cube-cbrt11.9

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

  20. Applied times-frac11.9

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

  21. Applied add-cube-cbrt12.1

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

  22. Applied times-frac12.1

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

  23. Applied add-sqr-sqrt12.1

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

  24. Applied add-cube-cbrt12.2

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

  25. Applied times-frac12.2

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

  26. Applied times-frac12.1

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

  27. Applied associate-*r*11.4

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

  28. Simplified11.4

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

  29. Final simplification11.4

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

Reproduce

herbie shell --seed 2019130 
(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))))