Average Error: 47.4 → 7.4
Time: 4.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \left(\frac{\frac{\frac{2}{\sqrt[3]{t}}}{\tan k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{t}}} \cdot \frac{\ell}{t}\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\sin k}}}{\frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\sin k}} \cdot \sqrt[3]{\frac{\frac{\ell}{t}}{\sin k}}}{k}\right)\]
\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{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \left(\frac{\frac{\frac{2}{\sqrt[3]{t}}}{\tan k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{t}}} \cdot \frac{\ell}{t}\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\sin k}}}{\frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\sin k}} \cdot \sqrt[3]{\frac{\frac{\ell}{t}}{\sin k}}}{k}\right)
double f(double t, double l, double k) {
        double r4035149 = 2.0;
        double r4035150 = t;
        double r4035151 = 3.0;
        double r4035152 = pow(r4035150, r4035151);
        double r4035153 = l;
        double r4035154 = r4035153 * r4035153;
        double r4035155 = r4035152 / r4035154;
        double r4035156 = k;
        double r4035157 = sin(r4035156);
        double r4035158 = r4035155 * r4035157;
        double r4035159 = tan(r4035156);
        double r4035160 = r4035158 * r4035159;
        double r4035161 = 1.0;
        double r4035162 = r4035156 / r4035150;
        double r4035163 = pow(r4035162, r4035149);
        double r4035164 = r4035161 + r4035163;
        double r4035165 = r4035164 - r4035161;
        double r4035166 = r4035160 * r4035165;
        double r4035167 = r4035149 / r4035166;
        return r4035167;
}
↓
double f(double t, double l, double k) {
        double r4035168 = 1.0;
        double r4035169 = t;
        double r4035170 = cbrt(r4035169);
        double r4035171 = r4035170 * r4035170;
        double r4035172 = r4035168 / r4035171;
        double r4035173 = k;
        double r4035174 = cbrt(r4035173);
        double r4035175 = r4035174 * r4035174;
        double r4035176 = r4035175 / r4035171;
        double r4035177 = r4035172 / r4035176;
        double r4035178 = 2.0;
        double r4035179 = r4035178 / r4035170;
        double r4035180 = tan(r4035173);
        double r4035181 = r4035179 / r4035180;
        double r4035182 = r4035174 / r4035170;
        double r4035183 = r4035181 / r4035182;
        double r4035184 = l;
        double r4035185 = r4035184 / r4035169;
        double r4035186 = r4035183 * r4035185;
        double r4035187 = r4035177 * r4035186;
        double r4035188 = sin(r4035173);
        double r4035189 = r4035185 / r4035188;
        double r4035190 = cbrt(r4035189);
        double r4035191 = r4035168 / r4035169;
        double r4035192 = r4035190 / r4035191;
        double r4035193 = r4035190 * r4035190;
        double r4035194 = r4035193 / r4035173;
        double r4035195 = r4035192 * r4035194;
        double r4035196 = r4035187 * r4035195;
        return r4035196;
}

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. Simplified30.8

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\tan k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t}}}\]
  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-frac13.8

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

  10. Applied associate-*r*12.2

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

  12. Applied add-cube-cbrt12.4

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

  13. Applied add-cube-cbrt12.5

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

  14. Applied times-frac12.5

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

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

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

  16. Applied add-cube-cbrt12.4

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

  17. Applied *-un-lft-identity12.4

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

  18. Applied times-frac12.4

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

  19. Applied times-frac12.4

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

  20. Applied times-frac11.7

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

  21. Applied associate-*l*11.2

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

  23. Applied div-inv11.2

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

  24. Applied add-cube-cbrt11.3

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

  25. Applied times-frac7.4

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

  26. Final simplification7.4

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

Reproduce

herbie shell --seed 2019151 +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))))