Average Error: 32.9 → 12.7
Time: 53.7s
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 r129970 = 2.0;
        double r129971 = t;
        double r129972 = 3.0;
        double r129973 = pow(r129971, r129972);
        double r129974 = l;
        double r129975 = r129974 * r129974;
        double r129976 = r129973 / r129975;
        double r129977 = k;
        double r129978 = sin(r129977);
        double r129979 = r129976 * r129978;
        double r129980 = tan(r129977);
        double r129981 = r129979 * r129980;
        double r129982 = 1.0;
        double r129983 = r129977 / r129971;
        double r129984 = pow(r129983, r129970);
        double r129985 = r129982 + r129984;
        double r129986 = r129985 + r129982;
        double r129987 = r129981 * r129986;
        double r129988 = r129970 / r129987;
        return r129988;
}
↓
double f(double t, double l, double k) {
        double r129989 = t;
        double r129990 = -4.860005812158333e-239;
        bool r129991 = r129989 <= r129990;
        double r129992 = 4.402616231627377e-246;
        bool r129993 = r129989 <= r129992;
        double r129994 = !r129993;
        bool r129995 = r129991 || r129994;
        double r129996 = 2.0;
        double r129997 = k;
        double r129998 = tan(r129997);
        double r129999 = l;
        double r130000 = cbrt(r129989);
        double r130001 = 3.0;
        double r130002 = pow(r130000, r130001);
        double r130003 = r129999 / r130002;
        double r130004 = r129998 / r130003;
        double r130005 = 1.0;
        double r130006 = r129997 / r129989;
        double r130007 = pow(r130006, r129996);
        double r130008 = r130005 + r130007;
        double r130009 = r130008 + r130005;
        double r130010 = r130004 * r130009;
        double r130011 = r130002 / r129999;
        double r130012 = sin(r129997);
        double r130013 = r130011 * r130012;
        double r130014 = r130002 * r130013;
        double r130015 = r130010 * r130014;
        double r130016 = r129996 / r130015;
        double r130017 = r129997 * r129989;
        double r130018 = r130017 / r129999;
        double r130019 = 0.16666666666666666;
        double r130020 = 3.0;
        double r130021 = pow(r129997, r130020);
        double r130022 = r130021 * r129989;
        double r130023 = r130022 / r129999;
        double r130024 = r130019 * r130023;
        double r130025 = r130018 - r130024;
        double r130026 = r130002 * r130025;
        double r130027 = r130026 * r129998;
        double r130028 = r130027 / r130003;
        double r130029 = r130028 * r130009;
        double r130030 = r129996 / r130029;
        double r130031 = r129995 ? r130016 : r130030;
        return r130031;
}

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))))