Average Error: 32.9 → 12.7
Time: 53.1s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.86000581215833313418115808157717702245 \cdot 10^{-239} \lor \neg \left(t \le 4.402616231627377082170896165667420584389 \cdot 10^{-246}\right):\\ \;\;\;\;\frac{2}{\left(\frac{\tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)\right) \cdot \tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]
\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 -4.86000581215833313418115808157717702245 \cdot 10^{-239} \lor \neg \left(t \le 4.402616231627377082170896165667420584389 \cdot 10^{-246}\right):\\
\;\;\;\;\frac{2}{\left(\frac{\tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)\right) \cdot \tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\

\end{array}
double f(double t, double l, double k) {
        double r130773 = 2.0;
        double r130774 = t;
        double r130775 = 3.0;
        double r130776 = pow(r130774, r130775);
        double r130777 = l;
        double r130778 = r130777 * r130777;
        double r130779 = r130776 / r130778;
        double r130780 = k;
        double r130781 = sin(r130780);
        double r130782 = r130779 * r130781;
        double r130783 = tan(r130780);
        double r130784 = r130782 * r130783;
        double r130785 = 1.0;
        double r130786 = r130780 / r130774;
        double r130787 = pow(r130786, r130773);
        double r130788 = r130785 + r130787;
        double r130789 = r130788 + r130785;
        double r130790 = r130784 * r130789;
        double r130791 = r130773 / r130790;
        return r130791;
}
↓
double f(double t, double l, double k) {
        double r130792 = t;
        double r130793 = -4.860005812158333e-239;
        bool r130794 = r130792 <= r130793;
        double r130795 = 4.402616231627377e-246;
        bool r130796 = r130792 <= r130795;
        double r130797 = !r130796;
        bool r130798 = r130794 || r130797;
        double r130799 = 2.0;
        double r130800 = k;
        double r130801 = tan(r130800);
        double r130802 = l;
        double r130803 = cbrt(r130792);
        double r130804 = 3.0;
        double r130805 = pow(r130803, r130804);
        double r130806 = r130802 / r130805;
        double r130807 = r130801 / r130806;
        double r130808 = 1.0;
        double r130809 = r130800 / r130792;
        double r130810 = pow(r130809, r130799);
        double r130811 = r130808 + r130810;
        double r130812 = r130811 + r130808;
        double r130813 = r130807 * r130812;
        double r130814 = r130805 / r130802;
        double r130815 = sin(r130800);
        double r130816 = r130814 * r130815;
        double r130817 = r130805 * r130816;
        double r130818 = r130813 * r130817;
        double r130819 = r130799 / r130818;
        double r130820 = r130800 * r130792;
        double r130821 = r130820 / r130802;
        double r130822 = 0.16666666666666666;
        double r130823 = 3.0;
        double r130824 = pow(r130800, r130823);
        double r130825 = r130824 * r130792;
        double r130826 = r130825 / r130802;
        double r130827 = r130822 * r130826;
        double r130828 = r130821 - r130827;
        double r130829 = r130805 * r130828;
        double r130830 = r130829 * r130801;
        double r130831 = r130830 / r130806;
        double r130832 = r130831 * r130812;
        double r130833 = r130799 / r130832;
        double r130834 = r130798 ? r130819 : r130833;
        return r130834;
}

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 < -4.860005812158333e-239 or 4.402616231627377e-246 < t

    1. Initial program 30.6

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

    3. Applied add-cube-cbrt30.8

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

    4. Applied unpow-prod-down30.8

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

    5. Applied times-frac23.1

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

    6. Applied associate-*l*21.2

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

    8. Applied unpow-prod-down21.2

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

    9. Applied associate-/l*15.5

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

    11. Applied associate-*l/14.5

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

    12. Applied associate-*l/13.0

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

    14. Applied *-un-lft-identity13.0

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

    15. Applied times-frac12.1

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

    16. Applied associate-*l*10.4

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

    if -4.860005812158333e-239 < t < 4.402616231627377e-246

    1. Initial program 64.0

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

    3. Applied add-cube-cbrt64.0

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

    4. Applied unpow-prod-down64.0

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

    5. Applied times-frac64.0

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

    6. Applied associate-*l*64.0

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

    8. Applied unpow-prod-down64.0

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

    9. Applied associate-/l*58.5

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

    11. Applied associate-*l/58.5

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

    12. Applied associate-*l/58.5

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

    13. Taylor expanded around 0 43.6

      \[\leadsto \frac{2}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \color{blue}{\left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)}\right) \cdot \tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.86000581215833313418115808157717702245 \cdot 10^{-239} \lor \neg \left(t \le 4.402616231627377082170896165667420584389 \cdot 10^{-246}\right):\\ \;\;\;\;\frac{2}{\left(\frac{\tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)\right) \cdot \tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))