Average Error: 47.4 → 0.7
Time: 2.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1928263201893.6716:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}\\ \mathbf{elif}\;k \le 8.256232424735127 \cdot 10^{-117}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\frac{\ell}{k}}{k}}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}\\ \end{array}\]
\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 -1928263201893.6716:\\
\;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}\\

\mathbf{elif}\;k \le 8.256232424735127 \cdot 10^{-117}:\\
\;\;\;\;\left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\frac{\ell}{k}}{k}}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}\\

\end{array}
double f(double t, double l, double k) {
        double r3803785 = 2.0;
        double r3803786 = t;
        double r3803787 = 3.0;
        double r3803788 = pow(r3803786, r3803787);
        double r3803789 = l;
        double r3803790 = r3803789 * r3803789;
        double r3803791 = r3803788 / r3803790;
        double r3803792 = k;
        double r3803793 = sin(r3803792);
        double r3803794 = r3803791 * r3803793;
        double r3803795 = tan(r3803792);
        double r3803796 = r3803794 * r3803795;
        double r3803797 = 1.0;
        double r3803798 = r3803792 / r3803786;
        double r3803799 = pow(r3803798, r3803785);
        double r3803800 = r3803797 + r3803799;
        double r3803801 = r3803800 - r3803797;
        double r3803802 = r3803796 * r3803801;
        double r3803803 = r3803785 / r3803802;
        return r3803803;
}
↓
double f(double t, double l, double k) {
        double r3803804 = k;
        double r3803805 = -1928263201893.6716;
        bool r3803806 = r3803804 <= r3803805;
        double r3803807 = 2.0;
        double r3803808 = sqrt(r3803807);
        double r3803809 = sqrt(r3803808);
        double r3803810 = r3803809 * r3803809;
        double r3803811 = t;
        double r3803812 = l;
        double r3803813 = r3803812 / r3803804;
        double r3803814 = r3803811 / r3803813;
        double r3803815 = r3803810 / r3803814;
        double r3803816 = r3803808 * r3803812;
        double r3803817 = tan(r3803804);
        double r3803818 = r3803816 / r3803817;
        double r3803819 = sin(r3803804);
        double r3803820 = r3803818 / r3803819;
        double r3803821 = r3803820 / r3803804;
        double r3803822 = r3803815 * r3803821;
        double r3803823 = 8.256232424735127e-117;
        bool r3803824 = r3803804 <= r3803823;
        double r3803825 = cbrt(r3803808);
        double r3803826 = r3803813 / r3803804;
        double r3803827 = r3803811 / r3803826;
        double r3803828 = r3803825 / r3803827;
        double r3803829 = r3803820 * r3803828;
        double r3803830 = r3803825 * r3803825;
        double r3803831 = r3803829 * r3803830;
        double r3803832 = r3803824 ? r3803831 : r3803822;
        double r3803833 = r3803806 ? r3803822 : r3803832;
        return r3803833;
}

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 < -1928263201893.6716 or 8.256232424735127e-117 < k

    1. Initial program 44.9

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

    2. Simplified14.8

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

    4. Applied div-inv14.8

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

    5. Applied add-sqr-sqrt14.9

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

    6. Applied times-frac14.8

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

    7. Applied times-frac14.8

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

    8. Simplified5.7

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

    9. Simplified5.7

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

    11. Applied associate-/r/5.7

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

    12. Applied times-frac4.3

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

    13. Applied add-sqr-sqrt4.3

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

    14. Applied sqrt-prod4.2

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

    15. Applied times-frac3.9

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

    16. Applied associate-*l*0.6

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

    17. Simplified0.6

      \[\leadsto \frac{\sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \color{blue}{\frac{\sqrt{\sqrt{2}} \cdot \frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}}\]
    18. Using strategy rm

    19. Applied *-un-lft-identity0.6

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

    20. Applied times-frac0.6

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

    21. Applied associate-*r*0.6

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

    22. Simplified0.6

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

    if -1928263201893.6716 < k < 8.256232424735127e-117

    1. Initial program 60.1

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

    2. Simplified28.6

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

    4. Applied div-inv28.6

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

    5. Applied add-sqr-sqrt28.8

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

    6. Applied times-frac28.4

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

    7. Applied times-frac20.9

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

    8. Simplified14.5

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

    9. Simplified3.1

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

    11. Applied *-un-lft-identity3.1

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

    12. Applied add-cube-cbrt3.1

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

    13. Applied times-frac3.0

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

    14. Applied associate-*l*3.0

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

    15. Simplified0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1928263201893.6716:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}\\ \mathbf{elif}\;k \le 8.256232424735127 \cdot 10^{-117}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\frac{\ell}{k}}{k}}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k}}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))))