Average Error: 48.5 → 1.5
Time: 6.8m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{{\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right) \cdot \left(\frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)\right)\]
\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)}
↓
2 \cdot \left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{{\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right) \cdot \left(\frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)\right)
double f(double t, double l, double k) {
        double r16254895 = 2.0;
        double r16254896 = t;
        double r16254897 = 3.0;
        double r16254898 = pow(r16254896, r16254897);
        double r16254899 = l;
        double r16254900 = r16254899 * r16254899;
        double r16254901 = r16254898 / r16254900;
        double r16254902 = k;
        double r16254903 = sin(r16254902);
        double r16254904 = r16254901 * r16254903;
        double r16254905 = tan(r16254902);
        double r16254906 = r16254904 * r16254905;
        double r16254907 = 1.0;
        double r16254908 = r16254902 / r16254896;
        double r16254909 = pow(r16254908, r16254895);
        double r16254910 = r16254907 + r16254909;
        double r16254911 = r16254910 - r16254907;
        double r16254912 = r16254906 * r16254911;
        double r16254913 = r16254895 / r16254912;
        return r16254913;
}
↓
double f(double t, double l, double k) {
        double r16254914 = 2.0;
        double r16254915 = k;
        double r16254916 = cos(r16254915);
        double r16254917 = sin(r16254915);
        double r16254918 = l;
        double r16254919 = r16254917 / r16254918;
        double r16254920 = r16254916 / r16254919;
        double r16254921 = 1.0;
        double r16254922 = 2.0;
        double r16254923 = r16254914 / r16254922;
        double r16254924 = pow(r16254915, r16254923);
        double r16254925 = r16254921 / r16254924;
        double r16254926 = t;
        double r16254927 = 1.0;
        double r16254928 = pow(r16254926, r16254927);
        double r16254929 = cbrt(r16254928);
        double r16254930 = r16254925 / r16254929;
        double r16254931 = pow(r16254930, r16254927);
        double r16254932 = cbrt(r16254919);
        double r16254933 = r16254931 / r16254932;
        double r16254934 = r16254920 * r16254933;
        double r16254935 = cbrt(r16254924);
        double r16254936 = r16254935 * r16254935;
        double r16254937 = r16254921 / r16254936;
        double r16254938 = r16254937 / r16254929;
        double r16254939 = pow(r16254938, r16254927);
        double r16254940 = r16254939 / r16254932;
        double r16254941 = r16254921 / r16254935;
        double r16254942 = r16254941 / r16254929;
        double r16254943 = pow(r16254942, r16254927);
        double r16254944 = r16254943 / r16254932;
        double r16254945 = r16254940 * r16254944;
        double r16254946 = r16254934 * r16254945;
        double r16254947 = r16254914 * r16254946;
        return r16254947;
}

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 48.5

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

  2. Simplified41.1

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

  3. Taylor expanded around inf 22.3

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

  4. Simplified20.3

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

  6. Applied times-frac15.6

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

  8. Applied add-cube-cbrt15.8

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

  9. Applied add-cube-cbrt15.9

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

  10. Applied sqr-pow15.9

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

  11. Applied *-un-lft-identity15.9

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

  12. Applied times-frac15.7

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

  13. Applied times-frac11.1

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

  14. Applied unpow-prod-down11.1

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

  15. Applied times-frac4.5

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

  16. Applied associate-*l*2.4

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

  18. Applied add-cube-cbrt2.4

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

  19. Applied *-un-lft-identity2.4

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

  20. Applied times-frac2.4

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

  21. Applied times-frac2.4

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

  22. Applied unpow-prod-down2.4

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

  23. Applied times-frac1.5

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

  24. Final simplification1.5

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

Reproduce

herbie shell --seed 2019200 +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))))