Average Error: 32.7 → 13.7
Time: 37.2s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}\right)}}{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\tan k}}{t}}{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}}\]
\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{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}\right)}}{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\tan k}}{t}}{\sqrt[3]{2 + \frac{k}{t} \cdot \frac{k}{t}}}
double f(double t, double l, double k) {
        double r1887876 = 2.0;
        double r1887877 = t;
        double r1887878 = 3.0;
        double r1887879 = pow(r1887877, r1887878);
        double r1887880 = l;
        double r1887881 = r1887880 * r1887880;
        double r1887882 = r1887879 / r1887881;
        double r1887883 = k;
        double r1887884 = sin(r1887883);
        double r1887885 = r1887882 * r1887884;
        double r1887886 = tan(r1887883);
        double r1887887 = r1887885 * r1887886;
        double r1887888 = 1.0;
        double r1887889 = r1887883 / r1887877;
        double r1887890 = pow(r1887889, r1887876);
        double r1887891 = r1887888 + r1887890;
        double r1887892 = r1887891 + r1887888;
        double r1887893 = r1887887 * r1887892;
        double r1887894 = r1887876 / r1887893;
        return r1887894;
}
↓
double f(double t, double l, double k) {
        double r1887895 = 2.0;
        double r1887896 = cbrt(r1887895);
        double r1887897 = r1887896 * r1887896;
        double r1887898 = t;
        double r1887899 = l;
        double r1887900 = r1887898 / r1887899;
        double r1887901 = k;
        double r1887902 = sin(r1887901);
        double r1887903 = r1887900 * r1887902;
        double r1887904 = r1887901 / r1887898;
        double r1887905 = r1887904 * r1887904;
        double r1887906 = r1887895 + r1887905;
        double r1887907 = cbrt(r1887906);
        double r1887908 = r1887903 * r1887907;
        double r1887909 = r1887900 * r1887908;
        double r1887910 = r1887897 / r1887909;
        double r1887911 = r1887910 / r1887907;
        double r1887912 = tan(r1887901);
        double r1887913 = r1887896 / r1887912;
        double r1887914 = r1887913 / r1887898;
        double r1887915 = r1887914 / r1887907;
        double r1887916 = r1887911 * r1887915;
        return r1887916;
}

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 32.7

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

  2. Simplified21.4

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

  4. Applied associate-*r*19.7

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

  6. Applied add-cube-cbrt19.9

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

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

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

  8. Applied add-cube-cbrt19.8

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

  9. Applied times-frac19.8

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

  10. Applied times-frac19.5

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

  11. Applied times-frac17.5

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

  12. Simplified14.6

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

  14. Applied div-inv14.6

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

  15. Applied associate-/l*14.6

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

  16. Simplified13.7

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

  17. Final simplification13.7

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

Reproduce

herbie shell --seed 2019156 
(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))))