Average Error: 31.8 → 16.7
Time: 21.6s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.3219154401390226 \cdot 10^{-234}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t \le 9.31487464079908344 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]
\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 -7.3219154401390226 \cdot 10^{-234}:\\
\;\;\;\;\frac{2}{\left(\left(\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)}\\

\mathbf{elif}\;t \le 9.31487464079908344 \cdot 10^{-123}:\\
\;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r124745 = 2.0;
        double r124746 = t;
        double r124747 = 3.0;
        double r124748 = pow(r124746, r124747);
        double r124749 = l;
        double r124750 = r124749 * r124749;
        double r124751 = r124748 / r124750;
        double r124752 = k;
        double r124753 = sin(r124752);
        double r124754 = r124751 * r124753;
        double r124755 = tan(r124752);
        double r124756 = r124754 * r124755;
        double r124757 = 1.0;
        double r124758 = r124752 / r124746;
        double r124759 = pow(r124758, r124745);
        double r124760 = r124757 + r124759;
        double r124761 = r124760 + r124757;
        double r124762 = r124756 * r124761;
        double r124763 = r124745 / r124762;
        return r124763;
}
↓
double f(double t, double l, double k) {
        double r124764 = t;
        double r124765 = -7.321915440139023e-234;
        bool r124766 = r124764 <= r124765;
        double r124767 = 2.0;
        double r124768 = cbrt(r124764);
        double r124769 = 3.0;
        double r124770 = pow(r124768, r124769);
        double r124771 = l;
        double r124772 = r124771 / r124770;
        double r124773 = r124770 / r124772;
        double r124774 = r124770 / r124771;
        double r124775 = k;
        double r124776 = sin(r124775);
        double r124777 = r124774 * r124776;
        double r124778 = r124773 * r124777;
        double r124779 = tan(r124775);
        double r124780 = r124778 * r124779;
        double r124781 = 1.0;
        double r124782 = r124775 / r124764;
        double r124783 = pow(r124782, r124767);
        double r124784 = r124781 + r124783;
        double r124785 = r124784 + r124781;
        double r124786 = r124780 * r124785;
        double r124787 = r124767 / r124786;
        double r124788 = 9.314874640799083e-123;
        bool r124789 = r124764 <= r124788;
        double r124790 = 3.0;
        double r124791 = pow(r124764, r124790);
        double r124792 = 2.0;
        double r124793 = pow(r124776, r124792);
        double r124794 = r124791 * r124793;
        double r124795 = cos(r124775);
        double r124796 = pow(r124771, r124792);
        double r124797 = r124795 * r124796;
        double r124798 = r124794 / r124797;
        double r124799 = r124767 * r124798;
        double r124800 = 1.0;
        double r124801 = -1.0;
        double r124802 = pow(r124801, r124769);
        double r124803 = r124800 / r124802;
        double r124804 = pow(r124803, r124781);
        double r124805 = pow(r124775, r124792);
        double r124806 = r124805 * r124793;
        double r124807 = r124764 * r124806;
        double r124808 = r124807 / r124797;
        double r124809 = r124804 * r124808;
        double r124810 = r124799 - r124809;
        double r124811 = r124767 / r124810;
        double r124812 = r124768 * r124768;
        double r124813 = r124769 / r124792;
        double r124814 = pow(r124812, r124813);
        double r124815 = r124814 / r124771;
        double r124816 = r124815 * r124777;
        double r124817 = r124816 * r124779;
        double r124818 = r124814 * r124817;
        double r124819 = r124818 * r124785;
        double r124820 = r124767 / r124819;
        double r124821 = r124789 ? r124811 : r124820;
        double r124822 = r124766 ? r124787 : r124821;
        return r124822;
}

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 3 regimes

  2. if t < -7.321915440139023e-234

    1. Initial program 29.1

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

    3. Applied add-cube-cbrt29.3

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

    4. Applied unpow-prod-down29.3

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

    5. Applied times-frac22.1

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

    6. Applied associate-*l*20.1

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

    8. Applied unpow-prod-down20.1

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

    9. Applied associate-/l*14.8

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

    if -7.321915440139023e-234 < t < 9.314874640799083e-123

    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. Taylor expanded around -inf 41.3

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

    if 9.314874640799083e-123 < t

    1. Initial program 24.2

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

    3. Applied add-cube-cbrt24.3

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

    4. Applied unpow-prod-down24.3

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

    5. Applied times-frac17.8

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

    6. Applied associate-*l*15.8

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

    8. Applied *-un-lft-identity15.8

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

    9. Applied sqr-pow15.8

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

    10. Applied times-frac11.3

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

    11. Simplified11.3

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

    13. Applied associate-*l*10.6

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

    15. Applied associate-*l*10.6

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.3219154401390226 \cdot 10^{-234}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t \le 9.31487464079908344 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))