Average Error: 32.6 → 7.6
Time: 42.9s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -2.058943866552354481315941633676695654592 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \left(\frac{\cos k}{\sin k \cdot \sin k} \cdot \frac{\ell}{k}\right)} + \left(\frac{1}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot 2\right)\right) \cdot \frac{1}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \le 3.021208706573349883409932381568387982727 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \left(\frac{\cos k}{\sin k \cdot \sin k} \cdot \frac{\ell}{k}\right)} + \left(\frac{1}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot 2\right)\right) \cdot \frac{1}{\frac{\ell}{t}}}\\ \end{array}\]
\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 -2.058943866552354481315941633676695654592 \cdot 10^{-157}:\\
\;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \left(\frac{\cos k}{\sin k \cdot \sin k} \cdot \frac{\ell}{k}\right)} + \left(\frac{1}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot 2\right)\right) \cdot \frac{1}{\frac{\ell}{t}}}\\

\mathbf{elif}\;k \le 3.021208706573349883409932381568387982727 \cdot 10^{-158}:\\
\;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\

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

\end{array}
double f(double t, double l, double k) {
        double r4189791 = 2.0;
        double r4189792 = t;
        double r4189793 = 3.0;
        double r4189794 = pow(r4189792, r4189793);
        double r4189795 = l;
        double r4189796 = r4189795 * r4189795;
        double r4189797 = r4189794 / r4189796;
        double r4189798 = k;
        double r4189799 = sin(r4189798);
        double r4189800 = r4189797 * r4189799;
        double r4189801 = tan(r4189798);
        double r4189802 = r4189800 * r4189801;
        double r4189803 = 1.0;
        double r4189804 = r4189798 / r4189792;
        double r4189805 = pow(r4189804, r4189791);
        double r4189806 = r4189803 + r4189805;
        double r4189807 = r4189806 + r4189803;
        double r4189808 = r4189802 * r4189807;
        double r4189809 = r4189791 / r4189808;
        return r4189809;
}
↓
double f(double t, double l, double k) {
        double r4189810 = k;
        double r4189811 = -2.0589438665523545e-157;
        bool r4189812 = r4189810 <= r4189811;
        double r4189813 = 2.0;
        double r4189814 = t;
        double r4189815 = l;
        double r4189816 = r4189815 / r4189810;
        double r4189817 = cos(r4189810);
        double r4189818 = sin(r4189810);
        double r4189819 = r4189818 * r4189818;
        double r4189820 = r4189817 / r4189819;
        double r4189821 = r4189820 * r4189816;
        double r4189822 = r4189816 * r4189821;
        double r4189823 = r4189814 / r4189822;
        double r4189824 = 1.0;
        double r4189825 = r4189815 / r4189814;
        double r4189826 = r4189825 / r4189814;
        double r4189827 = r4189824 / r4189826;
        double r4189828 = r4189819 / r4189817;
        double r4189829 = r4189828 * r4189813;
        double r4189830 = r4189827 * r4189829;
        double r4189831 = r4189824 / r4189825;
        double r4189832 = r4189830 * r4189831;
        double r4189833 = r4189823 + r4189832;
        double r4189834 = r4189813 / r4189833;
        double r4189835 = 3.02120870657335e-158;
        bool r4189836 = r4189810 <= r4189835;
        double r4189837 = cbrt(r4189814);
        double r4189838 = 3.0;
        double r4189839 = pow(r4189837, r4189838);
        double r4189840 = r4189839 / r4189815;
        double r4189841 = r4189840 * r4189818;
        double r4189842 = r4189837 * r4189837;
        double r4189843 = pow(r4189842, r4189838);
        double r4189844 = r4189843 / r4189815;
        double r4189845 = r4189841 * r4189844;
        double r4189846 = tan(r4189810);
        double r4189847 = r4189845 * r4189846;
        double r4189848 = 1.0;
        double r4189849 = r4189810 / r4189814;
        double r4189850 = pow(r4189849, r4189813);
        double r4189851 = r4189848 + r4189850;
        double r4189852 = r4189848 + r4189851;
        double r4189853 = r4189847 * r4189852;
        double r4189854 = r4189813 / r4189853;
        double r4189855 = r4189836 ? r4189854 : r4189834;
        double r4189856 = r4189812 ? r4189834 : r4189855;
        return r4189856;
}

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 < -2.0589438665523545e-157 or 3.02120870657335e-158 < k

    1. Initial program 31.5

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

    2. Taylor expanded around inf 26.4

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

    3. Simplified18.7

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

    5. Applied clear-num18.7

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

    6. Simplified6.8

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

    8. Applied *-un-lft-identity6.8

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

    9. Applied times-frac6.5

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

    10. Applied *-un-lft-identity6.5

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

    11. Applied times-frac6.5

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

    12. Applied associate-*l*5.2

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

    14. Applied associate-*r*4.5

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

    if -2.0589438665523545e-157 < k < 3.02120870657335e-158

    1. Initial program 39.3

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

    3. Applied add-cube-cbrt39.4

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

    4. Applied unpow-prod-down39.4

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

    5. Applied times-frac35.0

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

    6. Applied associate-*l*25.6

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.058943866552354481315941633676695654592 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \left(\frac{\cos k}{\sin k \cdot \sin k} \cdot \frac{\ell}{k}\right)} + \left(\frac{1}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot 2\right)\right) \cdot \frac{1}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \le 3.021208706573349883409932381568387982727 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \left(\frac{\cos k}{\sin k \cdot \sin k} \cdot \frac{\ell}{k}\right)} + \left(\frac{1}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot 2\right)\right) \cdot \frac{1}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))))