Average Error: 47.8 → 17.8
Time: 1.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.21656482861789566 \cdot 10^{146} \lor \neg \left(\ell \le -4.56574011928872929 \cdot 10^{-161}\right) \land \ell \le -4.17346075418630572 \cdot 10^{-255}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{{\ell}^{2}}{\sin k} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k}{\sin k}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \end{array}\]
\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}\;\ell \le -3.21656482861789566 \cdot 10^{146} \lor \neg \left(\ell \le -4.56574011928872929 \cdot 10^{-161}\right) \land \ell \le -4.17346075418630572 \cdot 10^{-255}:\\
\;\;\;\;\frac{\frac{2}{\left(\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}}{{\left(\frac{k}{t}\right)}^{2}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r112066 = 2.0;
        double r112067 = t;
        double r112068 = 3.0;
        double r112069 = pow(r112067, r112068);
        double r112070 = l;
        double r112071 = r112070 * r112070;
        double r112072 = r112069 / r112071;
        double r112073 = k;
        double r112074 = sin(r112073);
        double r112075 = r112072 * r112074;
        double r112076 = tan(r112073);
        double r112077 = r112075 * r112076;
        double r112078 = 1.0;
        double r112079 = r112073 / r112067;
        double r112080 = pow(r112079, r112066);
        double r112081 = r112078 + r112080;
        double r112082 = r112081 - r112078;
        double r112083 = r112077 * r112082;
        double r112084 = r112066 / r112083;
        return r112084;
}
↓
double f(double t, double l, double k) {
        double r112085 = l;
        double r112086 = -3.2165648286178957e+146;
        bool r112087 = r112085 <= r112086;
        double r112088 = -4.565740119288729e-161;
        bool r112089 = r112085 <= r112088;
        double r112090 = !r112089;
        double r112091 = -4.1734607541863057e-255;
        bool r112092 = r112085 <= r112091;
        bool r112093 = r112090 && r112092;
        bool r112094 = r112087 || r112093;
        double r112095 = 2.0;
        double r112096 = t;
        double r112097 = cbrt(r112096);
        double r112098 = r112097 * r112097;
        double r112099 = 3.0;
        double r112100 = pow(r112098, r112099);
        double r112101 = r112100 / r112085;
        double r112102 = pow(r112097, r112099);
        double r112103 = r112102 / r112085;
        double r112104 = r112101 * r112103;
        double r112105 = k;
        double r112106 = sin(r112105);
        double r112107 = r112104 * r112106;
        double r112108 = tan(r112105);
        double r112109 = r112107 * r112108;
        double r112110 = r112095 / r112109;
        double r112111 = r112105 / r112096;
        double r112112 = pow(r112111, r112095);
        double r112113 = r112110 / r112112;
        double r112114 = 2.0;
        double r112115 = pow(r112085, r112114);
        double r112116 = r112115 / r112106;
        double r112117 = 1.0;
        double r112118 = r112095 / r112114;
        double r112119 = pow(r112105, r112118);
        double r112120 = 1.0;
        double r112121 = pow(r112096, r112120);
        double r112122 = r112119 * r112121;
        double r112123 = r112117 / r112122;
        double r112124 = pow(r112123, r112120);
        double r112125 = cos(r112105);
        double r112126 = r112124 * r112125;
        double r112127 = r112126 / r112106;
        double r112128 = r112116 * r112127;
        double r112129 = r112117 / r112119;
        double r112130 = pow(r112129, r112120);
        double r112131 = r112128 * r112130;
        double r112132 = r112095 * r112131;
        double r112133 = r112094 ? r112113 : r112132;
        return r112133;
}

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 l < -3.2165648286178957e+146 or -4.565740119288729e-161 < l < -4.1734607541863057e-255

    1. Initial program 52.7

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

    2. Simplified48.4

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

    4. Applied add-cube-cbrt48.4

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

    5. Applied unpow-prod-down48.4

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

    6. Applied times-frac37.7

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

    if -3.2165648286178957e+146 < l < -4.565740119288729e-161 or -4.1734607541863057e-255 < l

    1. Initial program 46.7

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

    2. Simplified38.8

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

    3. Taylor expanded around inf 18.3

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

    5. Applied sqr-pow18.3

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

    6. Applied associate-*l*15.8

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

    8. Applied add-cube-cbrt15.8

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

    9. Applied times-frac15.6

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

    10. Applied unpow-prod-down15.6

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

    11. Applied associate-*l*13.8

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

    12. Simplified13.8

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

    14. Applied sqr-pow13.8

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

    15. Applied times-frac13.4

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

    16. Applied associate-*r*13.4

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

    17. Simplified13.4

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.21656482861789566 \cdot 10^{146} \lor \neg \left(\ell \le -4.56574011928872929 \cdot 10^{-161}\right) \land \ell \le -4.17346075418630572 \cdot 10^{-255}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{{\ell}^{2}}{\sin k} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k}{\sin k}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(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))))