Average Error: 47.5 → 1.6
Time: 1.7m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -2.8470394918342432 \cdot 10^{+209}:\\ \;\;\;\;\frac{\left(\left(\frac{\sqrt{2}}{\tan k} \cdot \ell\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{k}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\\ \mathbf{elif}\;k \le 3.9776562910929405 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{\sqrt{2}}}{k}\right) \cdot \frac{\sqrt{\sqrt{2}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\sqrt{2}}{\tan k} \cdot \ell\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{k}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;k \le -2.8470394918342432 \cdot 10^{+209}:\\
\;\;\;\;\frac{\left(\left(\frac{\sqrt{2}}{\tan k} \cdot \ell\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{k}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\\

\mathbf{elif}\;k \le 3.9776562910929405 \cdot 10^{-16}:\\
\;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{\sqrt{2}}}{k}\right) \cdot \frac{\sqrt{\sqrt{2}}}{t}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(\frac{\sqrt{2}}{\tan k} \cdot \ell\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{k}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\\

\end{array}
double f(double t, double l, double k) {
        double r3383871 = 2.0;
        double r3383872 = t;
        double r3383873 = 3.0;
        double r3383874 = pow(r3383872, r3383873);
        double r3383875 = l;
        double r3383876 = r3383875 * r3383875;
        double r3383877 = r3383874 / r3383876;
        double r3383878 = k;
        double r3383879 = sin(r3383878);
        double r3383880 = r3383877 * r3383879;
        double r3383881 = tan(r3383878);
        double r3383882 = r3383880 * r3383881;
        double r3383883 = 1.0;
        double r3383884 = r3383878 / r3383872;
        double r3383885 = pow(r3383884, r3383871);
        double r3383886 = r3383883 + r3383885;
        double r3383887 = r3383886 - r3383883;
        double r3383888 = r3383882 * r3383887;
        double r3383889 = r3383871 / r3383888;
        return r3383889;
}
↓
double f(double t, double l, double k) {
        double r3383890 = k;
        double r3383891 = -2.8470394918342432e+209;
        bool r3383892 = r3383890 <= r3383891;
        double r3383893 = 2.0;
        double r3383894 = sqrt(r3383893);
        double r3383895 = tan(r3383890);
        double r3383896 = r3383894 / r3383895;
        double r3383897 = l;
        double r3383898 = r3383896 * r3383897;
        double r3383899 = cbrt(r3383894);
        double r3383900 = r3383899 / r3383890;
        double r3383901 = r3383898 * r3383900;
        double r3383902 = r3383899 * r3383899;
        double r3383903 = r3383901 * r3383902;
        double r3383904 = t;
        double r3383905 = r3383897 / r3383890;
        double r3383906 = r3383904 / r3383905;
        double r3383907 = sin(r3383890);
        double r3383908 = r3383906 * r3383907;
        double r3383909 = r3383903 / r3383908;
        double r3383910 = 3.9776562910929405e-16;
        bool r3383911 = r3383890 <= r3383910;
        double r3383912 = r3383897 / r3383907;
        double r3383913 = r3383912 * r3383896;
        double r3383914 = sqrt(r3383894);
        double r3383915 = r3383914 / r3383890;
        double r3383916 = r3383905 * r3383915;
        double r3383917 = r3383914 / r3383904;
        double r3383918 = r3383916 * r3383917;
        double r3383919 = r3383913 * r3383918;
        double r3383920 = r3383911 ? r3383919 : r3383909;
        double r3383921 = r3383892 ? r3383909 : r3383920;
        return r3383921;
}

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. Split input into 2 regimes

  2. if k < -2.8470394918342432e+209 or 3.9776562910929405e-16 < k

    1. Initial program 41.4

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

    2. Simplified15.5

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

    4. Applied div-inv15.5

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

    5. Applied add-sqr-sqrt15.5

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

    6. Applied times-frac15.5

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

    7. Applied times-frac15.7

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

    8. Simplified6.4

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

    9. Simplified6.4

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

    11. Applied associate-/r/6.4

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

    12. Applied times-frac5.4

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

    13. Applied add-cube-cbrt5.4

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

    14. Applied times-frac5.0

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

    15. Applied associate-*l*0.4

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

    16. Simplified0.4

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

    18. Applied associate-*r/0.4

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

    19. Applied associate-*r/0.4

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

    20. Applied frac-times0.4

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

    if -2.8470394918342432e+209 < k < 3.9776562910929405e-16

    1. Initial program 54.1

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

    2. Simplified19.2

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

    4. Applied div-inv19.3

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

    5. Applied add-sqr-sqrt19.4

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

    6. Applied times-frac19.2

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

    7. Applied times-frac15.2

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

    8. Simplified7.9

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

    9. Simplified4.0

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

    11. Applied *-un-lft-identity4.0

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

    12. Applied times-frac3.4

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

    13. Applied add-sqr-sqrt3.2

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

    14. Applied times-frac3.1

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

    15. Simplified3.1

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

    16. Simplified3.0

      \[\leadsto \left(\frac{\sqrt{\sqrt{2}}}{t} \cdot \color{blue}{\left(\frac{\sqrt{\sqrt{2}}}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.8470394918342432 \cdot 10^{+209}:\\ \;\;\;\;\frac{\left(\left(\frac{\sqrt{2}}{\tan k} \cdot \ell\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{k}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\\ \mathbf{elif}\;k \le 3.9776562910929405 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\sqrt{2}}{\tan k}\right) \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{\sqrt{2}}}{k}\right) \cdot \frac{\sqrt{\sqrt{2}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\sqrt{2}}{\tan k} \cdot \ell\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{k}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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))))