Average Error: 31.4 → 10.8
Time: 4.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\sqrt{\sqrt{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}}{\frac{\tan k}{\frac{\ell}{t}}} \cdot \frac{\sqrt{\sqrt{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}}{t}\right) \cdot \frac{\sqrt{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sin k}}}\]
\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{\sqrt{\sqrt{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}}{\frac{\tan k}{\frac{\ell}{t}}} \cdot \frac{\sqrt{\sqrt{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}}{t}\right) \cdot \frac{\sqrt{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sin k}}}
double f(double t, double l, double k) {
        double r8223844 = 2.0;
        double r8223845 = t;
        double r8223846 = 3.0;
        double r8223847 = pow(r8223845, r8223846);
        double r8223848 = l;
        double r8223849 = r8223848 * r8223848;
        double r8223850 = r8223847 / r8223849;
        double r8223851 = k;
        double r8223852 = sin(r8223851);
        double r8223853 = r8223850 * r8223852;
        double r8223854 = tan(r8223851);
        double r8223855 = r8223853 * r8223854;
        double r8223856 = 1.0;
        double r8223857 = r8223851 / r8223845;
        double r8223858 = pow(r8223857, r8223844);
        double r8223859 = r8223856 + r8223858;
        double r8223860 = r8223859 + r8223856;
        double r8223861 = r8223855 * r8223860;
        double r8223862 = r8223844 / r8223861;
        return r8223862;
}
↓
double f(double t, double l, double k) {
        double r8223863 = 2.0;
        double r8223864 = k;
        double r8223865 = t;
        double r8223866 = r8223864 / r8223865;
        double r8223867 = r8223866 * r8223866;
        double r8223868 = r8223863 + r8223867;
        double r8223869 = r8223863 / r8223868;
        double r8223870 = sqrt(r8223869);
        double r8223871 = sqrt(r8223870);
        double r8223872 = tan(r8223864);
        double r8223873 = l;
        double r8223874 = r8223873 / r8223865;
        double r8223875 = r8223872 / r8223874;
        double r8223876 = r8223871 / r8223875;
        double r8223877 = r8223871 / r8223865;
        double r8223878 = r8223876 * r8223877;
        double r8223879 = 1.0;
        double r8223880 = sin(r8223864);
        double r8223881 = r8223874 / r8223880;
        double r8223882 = r8223879 / r8223881;
        double r8223883 = r8223870 / r8223882;
        double r8223884 = r8223878 * r8223883;
        return r8223884;
}

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 31.4

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

  2. Simplified18.4

    \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\tan k}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k \cdot t}}}}\]
  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-frac16.2

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

  7. Applied add-sqr-sqrt16.2

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

  8. Applied times-frac14.7

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sin k}}} \cdot \frac{\sqrt{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\tan k}{\frac{\frac{\ell}{t}}{t}}}}\]
  9. Using strategy rm

  10. Applied associate-/r/12.1

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

  11. Applied add-sqr-sqrt12.1

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

  12. Applied times-frac10.8

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

  13. Final simplification10.8

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

Reproduce

herbie shell --seed 2019133 
(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))))