Average Error: 48.2 → 4.1
Time: 2.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\cos k}}{\ell}} \cdot \left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{1}{\frac{\sin k}{\ell}}\right)\right) \cdot 2\]
\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{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\cos k}}{\ell}} \cdot \left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{1}{\frac{\sin k}{\ell}}\right)\right) \cdot 2
double f(double t, double l, double k) {
        double r10033852 = 2.0;
        double r10033853 = t;
        double r10033854 = 3.0;
        double r10033855 = pow(r10033853, r10033854);
        double r10033856 = l;
        double r10033857 = r10033856 * r10033856;
        double r10033858 = r10033855 / r10033857;
        double r10033859 = k;
        double r10033860 = sin(r10033859);
        double r10033861 = r10033858 * r10033860;
        double r10033862 = tan(r10033859);
        double r10033863 = r10033861 * r10033862;
        double r10033864 = 1.0;
        double r10033865 = r10033859 / r10033853;
        double r10033866 = pow(r10033865, r10033852);
        double r10033867 = r10033864 + r10033866;
        double r10033868 = r10033867 - r10033864;
        double r10033869 = r10033863 * r10033868;
        double r10033870 = r10033852 / r10033869;
        return r10033870;
}
↓
double f(double t, double l, double k) {
        double r10033871 = 1.0;
        double r10033872 = k;
        double r10033873 = 2.0;
        double r10033874 = 2.0;
        double r10033875 = r10033873 / r10033874;
        double r10033876 = pow(r10033872, r10033875);
        double r10033877 = r10033871 / r10033876;
        double r10033878 = 1.0;
        double r10033879 = pow(r10033877, r10033878);
        double r10033880 = sin(r10033872);
        double r10033881 = cos(r10033872);
        double r10033882 = r10033880 / r10033881;
        double r10033883 = l;
        double r10033884 = r10033882 / r10033883;
        double r10033885 = r10033879 / r10033884;
        double r10033886 = t;
        double r10033887 = pow(r10033886, r10033878);
        double r10033888 = r10033871 / r10033887;
        double r10033889 = r10033888 / r10033876;
        double r10033890 = pow(r10033889, r10033878);
        double r10033891 = r10033880 / r10033883;
        double r10033892 = r10033871 / r10033891;
        double r10033893 = r10033890 * r10033892;
        double r10033894 = r10033885 * r10033893;
        double r10033895 = r10033894 * r10033873;
        return r10033895;
}

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 48.2

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

  2. Simplified40.3

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

  3. Taylor expanded around inf 21.9

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

  5. Applied sqr-pow21.9

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

  6. Applied associate-*r*19.9

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

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

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

  9. Applied times-frac19.7

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

  10. Applied unpow-prod-down19.7

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

  11. Applied associate-*l*18.3

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

  12. Simplified18.2

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

  14. Applied *-un-lft-identity18.2

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

  15. Applied times-frac18.2

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

  16. Applied times-frac13.9

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

  17. Applied *-un-lft-identity13.9

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

  18. Applied *-un-lft-identity13.9

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

  19. Applied times-frac13.9

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

  20. Applied unpow-prod-down13.9

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

  21. Applied times-frac9.0

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

  22. Applied associate-*r*4.4

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

  23. Simplified4.1

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

  24. Final simplification4.1

    \[\leadsto \left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\cos k}}{\ell}} \cdot \left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{1}{\frac{\sin k}{\ell}}\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))