Average Error: 32.1 → 12.8
Time: 3.7m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.219905824522185 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{t}}\right)\right)} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;t \le 2.4208448371171848 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt[3]{2}}{\frac{\sin k}{\sqrt[3]{2}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{t}}\right)\right)} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\\ \end{array}\]
\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}\;t \le -7.219905824522185 \cdot 10^{-174}:\\
\;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{t}}\right)\right)} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\\

\mathbf{elif}\;t \le 2.4208448371171848 \cdot 10^{-179}:\\
\;\;\;\;\frac{\sqrt[3]{2}}{\frac{\sin k}{\sqrt[3]{2}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{t}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{t}}\right)\right)} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\\

\end{array}
double f(double t, double l, double k) {
        double r27590848 = 2.0;
        double r27590849 = t;
        double r27590850 = 3.0;
        double r27590851 = pow(r27590849, r27590850);
        double r27590852 = l;
        double r27590853 = r27590852 * r27590852;
        double r27590854 = r27590851 / r27590853;
        double r27590855 = k;
        double r27590856 = sin(r27590855);
        double r27590857 = r27590854 * r27590856;
        double r27590858 = tan(r27590855);
        double r27590859 = r27590857 * r27590858;
        double r27590860 = 1.0;
        double r27590861 = r27590855 / r27590849;
        double r27590862 = pow(r27590861, r27590848);
        double r27590863 = r27590860 + r27590862;
        double r27590864 = r27590863 + r27590860;
        double r27590865 = r27590859 * r27590864;
        double r27590866 = r27590848 / r27590865;
        return r27590866;
}
↓
double f(double t, double l, double k) {
        double r27590867 = t;
        double r27590868 = -7.219905824522185e-174;
        bool r27590869 = r27590867 <= r27590868;
        double r27590870 = 2.0;
        double r27590871 = k;
        double r27590872 = r27590871 / r27590867;
        double r27590873 = r27590872 * r27590872;
        double r27590874 = r27590870 + r27590873;
        double r27590875 = r27590870 / r27590874;
        double r27590876 = sin(r27590871);
        double r27590877 = 1.0;
        double r27590878 = l;
        double r27590879 = r27590878 / r27590867;
        double r27590880 = r27590877 / r27590879;
        double r27590881 = r27590876 * r27590880;
        double r27590882 = r27590867 * r27590881;
        double r27590883 = r27590876 * r27590882;
        double r27590884 = r27590875 / r27590883;
        double r27590885 = cos(r27590871);
        double r27590886 = r27590885 * r27590879;
        double r27590887 = r27590884 * r27590886;
        double r27590888 = 2.4208448371171848e-179;
        bool r27590889 = r27590867 <= r27590888;
        double r27590890 = cbrt(r27590870);
        double r27590891 = r27590876 / r27590890;
        double r27590892 = r27590890 / r27590891;
        double r27590893 = tan(r27590871);
        double r27590894 = r27590893 * r27590874;
        double r27590895 = r27590890 / r27590894;
        double r27590896 = r27590867 / r27590878;
        double r27590897 = r27590896 * r27590896;
        double r27590898 = r27590895 / r27590897;
        double r27590899 = r27590898 / r27590867;
        double r27590900 = r27590892 * r27590899;
        double r27590901 = r27590889 ? r27590900 : r27590887;
        double r27590902 = r27590869 ? r27590887 : r27590901;
        return r27590902;
}

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 t < -7.219905824522185e-174 or 2.4208448371171848e-179 < t

    1. Initial program 27.1

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

    2. Simplified20.3

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

    4. Applied associate-*l*15.7

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

    6. Applied *-un-lft-identity15.7

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

    7. Applied times-frac14.9

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

    8. Applied associate-*r*12.1

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

    10. Applied tan-quot12.1

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

    11. Applied associate-*l/12.1

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

    12. Applied frac-times11.1

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

    13. Applied associate-*r/9.1

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

    14. Applied associate-/r/8.0

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

    if -7.219905824522185e-174 < t < 2.4208448371171848e-179

    1. Initial program 62.7

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

    2. Simplified50.9

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

    4. Applied associate-*l*50.9

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

    6. Applied *-un-lft-identity50.9

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

    7. Applied times-frac50.9

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

    8. Applied associate-*r*50.9

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

    10. Applied *-un-lft-identity50.9

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

    11. Applied add-cube-cbrt50.9

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

    12. Applied times-frac50.9

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

    13. Applied times-frac49.7

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

    14. Simplified49.7

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

    15. Simplified41.5

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.219905824522185 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{t}}\right)\right)} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;t \le 2.4208448371171848 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt[3]{2}}{\frac{\sin k}{\sqrt[3]{2}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{t}}\right)\right)} \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(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))))