Average Error: 47.1 → 11.3
Time: 1.7m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}} \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}} \cdot \left(\frac{\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\tan k}}{\sqrt[3]{\frac{k}{t}}} \cdot \frac{\ell}{t}\right)\right)\]
\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)}
↓
\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}} \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}} \cdot \left(\frac{\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\tan k}}{\sqrt[3]{\frac{k}{t}}} \cdot \frac{\ell}{t}\right)\right)
double f(double t, double l, double k) {
        double r2277895 = 2.0;
        double r2277896 = t;
        double r2277897 = 3.0;
        double r2277898 = pow(r2277896, r2277897);
        double r2277899 = l;
        double r2277900 = r2277899 * r2277899;
        double r2277901 = r2277898 / r2277900;
        double r2277902 = k;
        double r2277903 = sin(r2277902);
        double r2277904 = r2277901 * r2277903;
        double r2277905 = tan(r2277902);
        double r2277906 = r2277904 * r2277905;
        double r2277907 = 1.0;
        double r2277908 = r2277902 / r2277896;
        double r2277909 = pow(r2277908, r2277895);
        double r2277910 = r2277907 + r2277909;
        double r2277911 = r2277910 - r2277907;
        double r2277912 = r2277906 * r2277911;
        double r2277913 = r2277895 / r2277912;
        return r2277913;
}
↓
double f(double t, double l, double k) {
        double r2277914 = l;
        double r2277915 = t;
        double r2277916 = r2277914 / r2277915;
        double r2277917 = k;
        double r2277918 = sin(r2277917);
        double r2277919 = r2277916 / r2277918;
        double r2277920 = r2277917 / r2277915;
        double r2277921 = r2277919 / r2277920;
        double r2277922 = 2.0;
        double r2277923 = sqrt(r2277922);
        double r2277924 = cbrt(r2277915);
        double r2277925 = r2277924 * r2277924;
        double r2277926 = r2277923 / r2277925;
        double r2277927 = cbrt(r2277920);
        double r2277928 = r2277927 * r2277927;
        double r2277929 = r2277926 / r2277928;
        double r2277930 = r2277923 / r2277924;
        double r2277931 = tan(r2277917);
        double r2277932 = r2277930 / r2277931;
        double r2277933 = r2277932 / r2277927;
        double r2277934 = r2277933 * r2277916;
        double r2277935 = r2277929 * r2277934;
        double r2277936 = r2277921 * r2277935;
        return r2277936;
}

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. Simplified31.0

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

  4. Applied times-frac20.5

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

  6. Applied *-un-lft-identity20.5

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

  7. Applied *-un-lft-identity20.5

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

  8. Applied times-frac19.7

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

  9. Applied times-frac13.6

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

  10. Applied associate-*r*12.0

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

  12. Applied add-cube-cbrt12.2

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

  13. Applied *-un-lft-identity12.2

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

  14. Applied add-cube-cbrt12.3

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

  15. Applied add-sqr-sqrt12.3

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

  16. Applied times-frac12.3

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

  17. Applied times-frac12.3

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

  18. Applied times-frac11.7

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

  19. Applied associate-*l*11.3

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

  20. Final simplification11.3

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

Reproduce

herbie shell --seed 2019153 +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))))