Average Error: 48.3 → 10.7
Time: 4.7m
Precision: 64
\[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) - 1.0\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -7.701141167434097 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}}\\ \mathbf{elif}\;\ell \le 2.3132826167896824 \cdot 10^{+141}:\\ \;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0} \cdot \left(\left(\left(\cos k \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right) \cdot {\left(\frac{1}{{t}^{1.0}}\right)}^{1.0}\right)\right) \cdot 2.0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}}\\ \end{array}\]
\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) - 1.0\right)}
↓
\begin{array}{l}
\mathbf{if}\;\ell \le -7.701141167434097 \cdot 10^{+153}:\\
\;\;\;\;\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}}\\

\mathbf{elif}\;\ell \le 2.3132826167896824 \cdot 10^{+141}:\\
\;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0} \cdot \left(\left(\left(\cos k \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right) \cdot {\left(\frac{1}{{t}^{1.0}}\right)}^{1.0}\right)\right) \cdot 2.0\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}}\\

\end{array}
double f(double t, double l, double k) {
        double r7528831 = 2.0;
        double r7528832 = t;
        double r7528833 = 3.0;
        double r7528834 = pow(r7528832, r7528833);
        double r7528835 = l;
        double r7528836 = r7528835 * r7528835;
        double r7528837 = r7528834 / r7528836;
        double r7528838 = k;
        double r7528839 = sin(r7528838);
        double r7528840 = r7528837 * r7528839;
        double r7528841 = tan(r7528838);
        double r7528842 = r7528840 * r7528841;
        double r7528843 = 1.0;
        double r7528844 = r7528838 / r7528832;
        double r7528845 = pow(r7528844, r7528831);
        double r7528846 = r7528843 + r7528845;
        double r7528847 = r7528846 - r7528843;
        double r7528848 = r7528842 * r7528847;
        double r7528849 = r7528831 / r7528848;
        return r7528849;
}
↓
double f(double t, double l, double k) {
        double r7528850 = l;
        double r7528851 = -7.701141167434097e+153;
        bool r7528852 = r7528850 <= r7528851;
        double r7528853 = 2.0;
        double r7528854 = sqrt(r7528853);
        double r7528855 = k;
        double r7528856 = t;
        double r7528857 = r7528855 / r7528856;
        double r7528858 = 2.0;
        double r7528859 = r7528853 / r7528858;
        double r7528860 = pow(r7528857, r7528859);
        double r7528861 = r7528854 / r7528860;
        double r7528862 = cbrt(r7528856);
        double r7528863 = 3.0;
        double r7528864 = pow(r7528862, r7528863);
        double r7528865 = tan(r7528855);
        double r7528866 = r7528850 / r7528865;
        double r7528867 = r7528864 / r7528866;
        double r7528868 = r7528861 / r7528867;
        double r7528869 = r7528862 * r7528862;
        double r7528870 = pow(r7528869, r7528863);
        double r7528871 = sin(r7528855);
        double r7528872 = r7528850 / r7528871;
        double r7528873 = r7528870 / r7528872;
        double r7528874 = r7528861 / r7528873;
        double r7528875 = r7528868 * r7528874;
        double r7528876 = 2.3132826167896824e+141;
        bool r7528877 = r7528850 <= r7528876;
        double r7528878 = 1.0;
        double r7528879 = pow(r7528855, r7528859);
        double r7528880 = r7528878 / r7528879;
        double r7528881 = 1.0;
        double r7528882 = pow(r7528880, r7528881);
        double r7528883 = cos(r7528855);
        double r7528884 = r7528872 * r7528872;
        double r7528885 = r7528883 * r7528884;
        double r7528886 = r7528885 * r7528882;
        double r7528887 = pow(r7528856, r7528881);
        double r7528888 = r7528878 / r7528887;
        double r7528889 = pow(r7528888, r7528881);
        double r7528890 = r7528886 * r7528889;
        double r7528891 = r7528882 * r7528890;
        double r7528892 = r7528891 * r7528853;
        double r7528893 = r7528877 ? r7528892 : r7528875;
        double r7528894 = r7528852 ? r7528875 : r7528893;
        return r7528894;
}

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 < -7.701141167434097e+153 or 2.3132826167896824e+141 < l

    1. Initial program 63.3

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) - 1.0\right)}\]

    2. Simplified63.0

      \[\leadsto \color{blue}{\frac{\frac{2.0}{{\left(\frac{k}{t}\right)}^{2.0}}}{\frac{{t}^{3.0}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}\]
    3. Using strategy rm

    4. Applied times-frac63.0

      \[\leadsto \frac{\frac{2.0}{{\left(\frac{k}{t}\right)}^{2.0}}}{\frac{{t}^{3.0}}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}\]

    5. Applied add-cube-cbrt63.0

      \[\leadsto \frac{\frac{2.0}{{\left(\frac{k}{t}\right)}^{2.0}}}{\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3.0}}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}\]

    6. Applied unpow-prod-down63.0

      \[\leadsto \frac{\frac{2.0}{{\left(\frac{k}{t}\right)}^{2.0}}}{\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}\]

    7. Applied times-frac48.6

      \[\leadsto \frac{\frac{2.0}{{\left(\frac{k}{t}\right)}^{2.0}}}{\color{blue}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}}}\]

    8. Applied sqr-pow48.6

      \[\leadsto \frac{\frac{2.0}{\color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)} \cdot {\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}}\]

    9. Applied add-sqr-sqrt48.6

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{2.0}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)} \cdot {\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}}\]

    10. Applied times-frac48.4

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}} \cdot \frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}}\]

    11. Applied times-frac37.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}}}\]

    if -7.701141167434097e+153 < l < 2.3132826167896824e+141

    1. Initial program 45.4

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) - 1.0\right)}\]

    2. Simplified36.1

      \[\leadsto \color{blue}{\frac{\frac{2.0}{{\left(\frac{k}{t}\right)}^{2.0}}}{\frac{{t}^{3.0}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}\]

    3. Taylor expanded around inf 14.7

      \[\leadsto \color{blue}{2.0 \cdot \left({\left(\frac{1}{{t}^{1.0} \cdot {k}^{2.0}}\right)}^{1.0} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity14.7

      \[\leadsto 2.0 \cdot \left({\left(\frac{\color{blue}{1 \cdot 1}}{{t}^{1.0} \cdot {k}^{2.0}}\right)}^{1.0} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

    6. Applied times-frac14.8

      \[\leadsto 2.0 \cdot \left({\color{blue}{\left(\frac{1}{{t}^{1.0}} \cdot \frac{1}{{k}^{2.0}}\right)}}^{1.0} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

    7. Applied unpow-prod-down14.8

      \[\leadsto 2.0 \cdot \left(\color{blue}{\left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot {\left(\frac{1}{{k}^{2.0}}\right)}^{1.0}\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

    8. Applied associate-*l*15.6

      \[\leadsto 2.0 \cdot \color{blue}{\left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \left({\left(\frac{1}{{k}^{2.0}}\right)}^{1.0} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)}\]

    9. Simplified12.5

      \[\leadsto 2.0 \cdot \left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \color{blue}{\left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot {\left(\frac{1}{{k}^{2.0}}\right)}^{1.0}\right)}\right)\]
    10. Using strategy rm

    11. Applied sqr-pow12.5

      \[\leadsto 2.0 \cdot \left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot {\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2.0}{2}\right)} \cdot {k}^{\left(\frac{2.0}{2}\right)}}}\right)}^{1.0}\right)\right)\]

    12. Applied *-un-lft-identity12.5

      \[\leadsto 2.0 \cdot \left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot {\left(\frac{\color{blue}{1 \cdot 1}}{{k}^{\left(\frac{2.0}{2}\right)} \cdot {k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right)\right)\]

    13. Applied times-frac12.4

      \[\leadsto 2.0 \cdot \left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot {\color{blue}{\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}} \cdot \frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}}^{1.0}\right)\right)\]

    14. Applied unpow-prod-down12.4

      \[\leadsto 2.0 \cdot \left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot \color{blue}{\left({\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0} \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right)}\right)\right)\]

    15. Applied associate-*r*8.6

      \[\leadsto 2.0 \cdot \left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \color{blue}{\left(\left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right)}\right)\]
    16. Using strategy rm

    17. Applied associate-*r*5.5

      \[\leadsto 2.0 \cdot \color{blue}{\left(\left({\left(\frac{1}{{t}^{1.0}}\right)}^{1.0} \cdot \left(\left(\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right) \cdot \cos k\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -7.701141167434097 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}}\\ \mathbf{elif}\;\ell \le 2.3132826167896824 \cdot 10^{+141}:\\ \;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0} \cdot \left(\left(\left(\cos k \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2.0}{2}\right)}}\right)}^{1.0}\right) \cdot {\left(\frac{1}{{t}^{1.0}}\right)}^{1.0}\right)\right) \cdot 2.0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\tan k}}} \cdot \frac{\frac{\sqrt{2.0}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2.0}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\frac{\ell}{\sin k}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))