Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 12.0

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le 6.53962801490363253 \cdot 10^{-170}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le 2.1181975674542142 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r314066 = 2.0;
        double r314067 = t;
        double r314068 = 3.0;
        double r314069 = pow(r314067, r314068);
        double r314070 = l;
        double r314071 = r314070 * r314070;
        double r314072 = r314069 / r314071;
        double r314073 = k;
        double r314074 = sin(r314073);
        double r314075 = r314072 * r314074;
        double r314076 = tan(r314073);
        double r314077 = r314075 * r314076;
        double r314078 = 1.0;
        double r314079 = r314073 / r314067;
        double r314080 = pow(r314079, r314066);
        double r314081 = r314078 + r314080;
        double r314082 = r314081 - r314078;
        double r314083 = r314077 * r314082;
        double r314084 = r314066 / r314083;
        return r314084;
}

↓

double f(double t, double l, double k) {
        double r314085 = l;
        double r314086 = 6.5396280149036325e-170;
        bool r314087 = r314085 <= r314086;
        double r314088 = 2.0;
        double r314089 = 1.0;
        double r314090 = k;
        double r314091 = 2.0;
        double r314092 = r314088 / r314091;
        double r314093 = pow(r314090, r314092);
        double r314094 = t;
        double r314095 = 1.0;
        double r314096 = pow(r314094, r314095);
        double r314097 = r314093 * r314096;
        double r314098 = r314093 * r314097;
        double r314099 = r314089 / r314098;
        double r314100 = pow(r314099, r314095);
        double r314101 = sin(r314090);
        double r314102 = cbrt(r314101);
        double r314103 = 4.0;
        double r314104 = pow(r314102, r314103);
        double r314105 = r314104 / r314085;
        double r314106 = cbrt(r314105);
        double r314107 = r314106 * r314106;
        double r314108 = cbrt(r314085);
        double r314109 = r314108 * r314108;
        double r314110 = r314107 / r314109;
        double r314111 = r314089 / r314110;
        double r314112 = cbrt(r314089);
        double r314113 = pow(r314112, r314091);
        double r314114 = r314111 / r314113;
        double r314115 = r314100 * r314114;
        double r314116 = cos(r314090);
        double r314117 = r314106 / r314108;
        double r314118 = r314116 / r314117;
        double r314119 = pow(r314102, r314091);
        double r314120 = r314118 / r314119;
        double r314121 = r314115 * r314120;
        double r314122 = r314088 * r314121;
        double r314123 = 2.1181975674542142e+146;
        bool r314124 = r314085 <= r314123;
        double r314125 = r314089 / r314093;
        double r314126 = pow(r314125, r314095);
        double r314127 = r314089 / r314097;
        double r314128 = pow(r314127, r314095);
        double r314129 = pow(r314085, r314091);
        double r314130 = r314116 * r314129;
        double r314131 = pow(r314101, r314091);
        double r314132 = r314130 / r314131;
        double r314133 = r314128 * r314132;
        double r314134 = r314126 * r314133;
        double r314135 = r314088 * r314134;
        double r314136 = r314116 / r314105;
        double r314137 = cbrt(r314119);
        double r314138 = r314137 * r314137;
        double r314139 = r314136 / r314138;
        double r314140 = r314100 * r314139;
        double r314141 = r314085 / r314137;
        double r314142 = r314140 * r314141;
        double r314143 = r314088 * r314142;
        double r314144 = r314124 ? r314135 : r314143;
        double r314145 = r314087 ? r314122 : r314144;
        return r314145;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < 6.5396280149036325e-170

   1. Initial program 47.7

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

   2. Simplified 39.8

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

   3. Taylor expanded around inf 21.6

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

   5. Applied sqr-pow 21.6

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

   6. Applied associate-*l* 20.1

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

   8. Applied add-cube-cbrt 20.3

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

   9. Applied unpow-prod-down 20.3

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

   10. Applied associate-/r* 20.1

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

   11. Simplified 17.4

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

   13. Applied *-un-lft-identity 17.4

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

   14. Applied cbrt-prod 17.4

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

   15. Applied unpow-prod-down 17.4

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

   16. Applied add-cube-cbrt 17.4

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

   17. Applied add-cube-cbrt 17.4

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

   18. Applied times-frac 17.4

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

   19. Applied *-un-lft-identity 17.4

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

   20. Applied times-frac 17.2

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

   21. Applied times-frac 16.0

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

   22. Applied associate-*r* 12.0

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

   if 6.5396280149036325e-170 < l < 2.1181975674542142e+146

   1. Initial program 44.8

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

   2. Simplified 34.8

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

   3. Taylor expanded around inf 11.1

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

   5. Applied sqr-pow 11.1

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

   6. Applied associate-*l* 7.2

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

   8. Applied *-un-lft-identity 7.2

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

   9. Applied times-frac 6.8

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

   10. Applied unpow-prod-down 6.8

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

   11. Applied associate-*l* 3.8

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

   if 2.1181975674542142e+146 < l

   1. Initial program 63.6

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

   2. Simplified 63.2

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

   3. Taylor expanded around inf 62.6

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

   5. Applied sqr-pow 62.6

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

   6. Applied associate-*l* 62.2

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

   8. Applied add-cube-cbrt 62.2

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

   9. Applied unpow-prod-down 62.2

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

   10. Applied associate-/r* 62.2

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

   11. Simplified 62.2

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

   13. Applied add-cube-cbrt 62.2

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

   14. Applied associate-/r/ 62.2

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

   15. Applied times-frac 62.2

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

   16. Applied associate-*r* 38.5

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 6.53962801490363253 \cdot 10^{-170}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le 2.1181975674542142 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))