Average Error: 48.8 → 10.6
Time: 4.5m
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 -3.2121767733629187 \cdot 10^{153}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -4.46760789925130493 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left({\left(\frac{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}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 3.6473195150792043 \cdot 10^{-200}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\\ \mathbf{elif}\;\ell \le 1.3529385316824283 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 9.37568809863626905 \cdot 10^{197}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(-\left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\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 -3.2121767733629187 \cdot 10^{153}:\\
\;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\

\mathbf{elif}\;\ell \le -4.46760789925130493 \cdot 10^{-137}:\\
\;\;\;\;2 \cdot \left({\left(\frac{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}}{{\left(\sin k\right)}^{2}}\right)\right)\\

\mathbf{elif}\;\ell \le 3.6473195150792043 \cdot 10^{-200}:\\
\;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\\

\mathbf{elif}\;\ell \le 1.3529385316824283 \cdot 10^{146}:\\
\;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\right)\\

\mathbf{elif}\;\ell \le 9.37568809863626905 \cdot 10^{197}:\\
\;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot e^{\left(-\left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}\\

\end{array}
double f(double t, double l, double k) {
        double r347050 = 2.0;
        double r347051 = t;
        double r347052 = 3.0;
        double r347053 = pow(r347051, r347052);
        double r347054 = l;
        double r347055 = r347054 * r347054;
        double r347056 = r347053 / r347055;
        double r347057 = k;
        double r347058 = sin(r347057);
        double r347059 = r347056 * r347058;
        double r347060 = tan(r347057);
        double r347061 = r347059 * r347060;
        double r347062 = 1.0;
        double r347063 = r347057 / r347051;
        double r347064 = pow(r347063, r347050);
        double r347065 = r347062 + r347064;
        double r347066 = r347065 - r347062;
        double r347067 = r347061 * r347066;
        double r347068 = r347050 / r347067;
        return r347068;
}
↓
double f(double t, double l, double k) {
        double r347069 = l;
        double r347070 = -3.2121767733629187e+153;
        bool r347071 = r347069 <= r347070;
        double r347072 = 2.0;
        double r347073 = 1.0;
        double r347074 = k;
        double r347075 = 2.0;
        double r347076 = r347072 / r347075;
        double r347077 = pow(r347074, r347076);
        double r347078 = t;
        double r347079 = 1.0;
        double r347080 = pow(r347078, r347079);
        double r347081 = r347077 * r347080;
        double r347082 = r347077 * r347081;
        double r347083 = r347073 / r347082;
        double r347084 = pow(r347083, r347079);
        double r347085 = sin(r347074);
        double r347086 = cbrt(r347085);
        double r347087 = 4.0;
        double r347088 = pow(r347086, r347087);
        double r347089 = r347088 / r347069;
        double r347090 = r347073 / r347089;
        double r347091 = r347086 * r347086;
        double r347092 = cbrt(r347091);
        double r347093 = pow(r347092, r347075);
        double r347094 = r347090 / r347093;
        double r347095 = r347084 * r347094;
        double r347096 = cos(r347074);
        double r347097 = r347073 / r347069;
        double r347098 = r347096 / r347097;
        double r347099 = cbrt(r347086);
        double r347100 = pow(r347099, r347075);
        double r347101 = r347098 / r347100;
        double r347102 = r347095 * r347101;
        double r347103 = r347072 * r347102;
        double r347104 = -4.467607899251305e-137;
        bool r347105 = r347069 <= r347104;
        double r347106 = r347073 / r347077;
        double r347107 = pow(r347106, r347079);
        double r347108 = r347073 / r347081;
        double r347109 = pow(r347108, r347079);
        double r347110 = pow(r347069, r347075);
        double r347111 = r347096 * r347110;
        double r347112 = pow(r347085, r347075);
        double r347113 = r347111 / r347112;
        double r347114 = r347109 * r347113;
        double r347115 = r347107 * r347114;
        double r347116 = r347072 * r347115;
        double r347117 = 3.647319515079204e-200;
        bool r347118 = r347069 <= r347117;
        double r347119 = cbrt(r347069);
        double r347120 = r347119 * r347119;
        double r347121 = r347073 / r347120;
        double r347122 = r347121 / r347073;
        double r347123 = r347073 / r347122;
        double r347124 = r347123 / r347086;
        double r347125 = r347084 * r347124;
        double r347126 = r347088 / r347119;
        double r347127 = r347126 / r347069;
        double r347128 = r347096 / r347127;
        double r347129 = r347128 / r347086;
        double r347130 = r347125 * r347129;
        double r347131 = r347072 * r347130;
        double r347132 = 1.3529385316824283e+146;
        bool r347133 = r347069 <= r347132;
        double r347134 = sqrt(r347073);
        double r347135 = r347134 / r347077;
        double r347136 = pow(r347135, r347079);
        double r347137 = r347134 / r347081;
        double r347138 = pow(r347137, r347079);
        double r347139 = r347089 / r347069;
        double r347140 = r347096 / r347139;
        double r347141 = pow(r347086, r347075);
        double r347142 = r347140 / r347141;
        double r347143 = r347138 * r347142;
        double r347144 = r347136 * r347143;
        double r347145 = r347072 * r347144;
        double r347146 = 9.375688098636269e+197;
        bool r347147 = r347069 <= r347146;
        double r347148 = cbrt(r347096);
        double r347149 = r347148 * r347148;
        double r347150 = pow(r347092, r347087);
        double r347151 = r347150 / r347120;
        double r347152 = r347151 / r347120;
        double r347153 = r347149 / r347152;
        double r347154 = r347099 * r347099;
        double r347155 = pow(r347154, r347075);
        double r347156 = r347153 / r347155;
        double r347157 = r347084 * r347156;
        double r347158 = pow(r347099, r347087);
        double r347159 = r347158 / r347119;
        double r347160 = r347159 / r347119;
        double r347161 = r347148 / r347160;
        double r347162 = r347161 / r347100;
        double r347163 = r347157 * r347162;
        double r347164 = r347072 * r347163;
        double r347165 = log(r347077);
        double r347166 = log(r347074);
        double r347167 = r347166 * r347076;
        double r347168 = log(r347078);
        double r347169 = r347168 * r347079;
        double r347170 = r347167 + r347169;
        double r347171 = r347165 + r347170;
        double r347172 = -r347171;
        double r347173 = r347172 * r347079;
        double r347174 = log(r347096);
        double r347175 = log(r347088);
        double r347176 = log(r347069);
        double r347177 = r347175 - r347176;
        double r347178 = r347177 - r347176;
        double r347179 = r347174 - r347178;
        double r347180 = log(r347141);
        double r347181 = r347179 - r347180;
        double r347182 = r347173 + r347181;
        double r347183 = exp(r347182);
        double r347184 = r347072 * r347183;
        double r347185 = r347147 ? r347164 : r347184;
        double r347186 = r347133 ? r347145 : r347185;
        double r347187 = r347118 ? r347131 : r347186;
        double r347188 = r347105 ? r347116 : r347187;
        double r347189 = r347071 ? r347103 : r347188;
        return r347189;
}

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 6 regimes

  2. if l < -3.2121767733629187e+153

    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. Simplified63.9

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

    3. Taylor expanded around inf 63.8

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

    5. Applied sqr-pow63.8

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

    6. Applied associate-*l*63.8

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

    8. Applied add-cube-cbrt63.8

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

    9. Applied unpow-prod-down63.8

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

    10. Applied associate-/r*63.8

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

    11. Simplified63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}\right)}^{2}}\right)\]

    14. Applied cbrt-prod63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}}^{2}}\right)\]

    15. Applied unpow-prod-down63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}}\right)\]

    16. Applied div-inv63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\color{blue}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell} \cdot \frac{1}{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    17. Applied *-un-lft-identity63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\color{blue}{1 \cdot \cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell} \cdot \frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    18. Applied times-frac63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \frac{\cos k}{\frac{1}{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    19. Applied times-frac63.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}} \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)}\right)\]

    20. Applied associate-*r*42.0

      \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)}\]

    if -3.2121767733629187e+153 < l < -4.467607899251305e-137

    1. Initial program 45.4

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

    2. Simplified36.0

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

    3. Taylor expanded around inf 12.1

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

    5. Applied sqr-pow12.1

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

    6. Applied associate-*l*7.8

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

    8. Applied *-un-lft-identity7.8

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

    9. Applied times-frac7.4

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

    10. Applied unpow-prod-down7.4

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

    11. Applied associate-*l*4.2

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

    if -4.467607899251305e-137 < l < 3.647319515079204e-200

    1. Initial program 47.0

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

    2. Simplified38.1

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

    3. Taylor expanded around inf 18.4

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

    5. Applied sqr-pow18.4

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

    6. Applied associate-*l*18.4

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

    8. Applied add-cube-cbrt18.4

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

    9. Applied unpow-prod-down18.4

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

    10. Applied associate-/r*18.2

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

    11. Simplified13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
    12. Using strategy rm

    13. Applied unpow213.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}\right)\]

    14. Applied *-un-lft-identity13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\color{blue}{1 \cdot \ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    15. Applied add-cube-cbrt13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{1 \cdot \ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    16. Applied *-un-lft-identity13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\color{blue}{\left(1 \cdot \sqrt[3]{\sin k}\right)}}^{4}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{1 \cdot \ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    17. Applied unpow-prod-down13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{\color{blue}{{1}^{4} \cdot {\left(\sqrt[3]{\sin k}\right)}^{4}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{1 \cdot \ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    18. Applied times-frac13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\color{blue}{\frac{{1}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}}{1 \cdot \ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    19. Applied times-frac13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\color{blue}{\frac{\frac{{1}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1} \cdot \frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    20. Applied *-un-lft-identity13.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\color{blue}{1 \cdot \cos k}}{\frac{\frac{{1}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1} \cdot \frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    21. Applied times-frac12.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{1}{\frac{\frac{{1}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}} \cdot \frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)\]

    22. Applied times-frac10.4

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{1}{\frac{\frac{{1}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)}\right)\]

    23. Applied associate-*r*7.3

      \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{{1}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)}\]

    24. Simplified7.3

      \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right)} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\]

    if 3.647319515079204e-200 < l < 1.3529385316824283e+146

    1. Initial program 45.5

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

    2. Simplified36.0

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

    3. Taylor expanded around inf 13.0

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

    5. Applied sqr-pow13.0

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

    6. Applied associate-*l*9.3

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

    8. Applied add-cube-cbrt9.9

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

    9. Applied unpow-prod-down9.9

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

    10. Applied associate-/r*9.5

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

    11. Simplified7.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt7.9

      \[\leadsto 2 \cdot \left({\left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

    14. Applied times-frac7.6

      \[\leadsto 2 \cdot \left({\color{blue}{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}} \cdot \frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

    15. Applied unpow-prod-down7.6

      \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot {\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

    16. Applied associate-*l*4.9

      \[\leadsto 2 \cdot \color{blue}{\left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\right)}\]

    if 1.3529385316824283e+146 < l < 9.375688098636269e+197

    1. Initial program 61.9

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

    2. Simplified61.8

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

    3. Taylor expanded around inf 60.1

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

    5. Applied sqr-pow60.1

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

    6. Applied associate-*l*58.8

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

    8. Applied add-cube-cbrt58.8

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

    9. Applied unpow-prod-down58.8

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

    10. Applied associate-/r*58.8

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

    11. Simplified58.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}}^{2}}\right)\]

    14. Applied unpow-prod-down58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}}\right)\]

    15. Applied add-cube-cbrt58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    16. Applied add-cube-cbrt58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    17. Applied add-cube-cbrt58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}\right)}^{4}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    18. Applied cbrt-prod58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}}^{4}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    19. Applied unpow-prod-down58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    20. Applied times-frac58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\color{blue}{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    21. Applied times-frac58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\color{blue}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    22. Applied add-cube-cbrt58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}\right) \cdot \sqrt[3]{\cos k}}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    23. Applied times-frac58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\]

    24. Applied times-frac58.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)}\right)\]

    25. Applied associate-*r*26.0

      \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)}\]

    if 9.375688098636269e+197 < l

    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. Simplified64.0

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

    3. Taylor expanded around inf 64.0

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

    5. Applied sqr-pow64.0

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

    6. Applied associate-*l*64.0

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

    8. Applied add-cube-cbrt64.0

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

    9. Applied unpow-prod-down64.0

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

    10. Applied associate-/r*64.0

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

    11. Simplified64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
    12. Using strategy rm

    13. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}}\right)\]

    14. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\color{blue}{e^{\log \ell}}}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    15. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\color{blue}{e^{\log \ell}}}}{e^{\log \ell}}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    16. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{\color{blue}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right)}}}{e^{\log \ell}}}{e^{\log \ell}}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    17. Applied div-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\color{blue}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell}}}{e^{\log \ell}}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    18. Applied div-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\color{blue}{e^{\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell}}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    19. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\color{blue}{e^{\log \left(\cos k\right)}}}{e^{\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    20. Applied div-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{e^{\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)}}}{e^{\log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    21. Applied div-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}\right)\]

    22. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(e^{\log t}\right)}}^{1}\right)}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    23. Applied pow-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{e^{\log t \cdot 1}}\right)}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    24. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({\color{blue}{\left(e^{\log k}\right)}}^{\left(\frac{2}{2}\right)} \cdot e^{\log t \cdot 1}\right)}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    25. Applied pow-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left(\color{blue}{e^{\log k \cdot \frac{2}{2}}} \cdot e^{\log t \cdot 1}\right)}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    26. Applied prod-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{e^{\log k \cdot \frac{2}{2} + \log t \cdot 1}}}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    27. Applied add-exp-log64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{e^{\log \left({k}^{\left(\frac{2}{2}\right)}\right)}} \cdot e^{\log k \cdot \frac{2}{2} + \log t \cdot 1}}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    28. Applied prod-exp64.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{e^{\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)}}}\right)}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    29. Applied rec-exp64.0

      \[\leadsto 2 \cdot \left({\color{blue}{\left(e^{-\left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)}\right)}}^{1} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    30. Applied pow-exp64.0

      \[\leadsto 2 \cdot \left(\color{blue}{e^{\left(-\left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1}} \cdot e^{\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\right)\]

    31. Applied prod-exp36.7

      \[\leadsto 2 \cdot \color{blue}{e^{\left(-\left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.2121767733629187 \cdot 10^{153}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -4.46760789925130493 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left({\left(\frac{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}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 3.6473195150792043 \cdot 10^{-200}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\\ \mathbf{elif}\;\ell \le 1.3529385316824283 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 9.37568809863626905 \cdot 10^{197}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(-\left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\left(\log \left({\left(\sqrt[3]{\sin k}\right)}^{4}\right) - \log \ell\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))