Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 10.7

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 1.6443394873045096 \cdot 10^{249}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{1}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r357012 = 2.0;
        double r357013 = t;
        double r357014 = 3.0;
        double r357015 = pow(r357013, r357014);
        double r357016 = l;
        double r357017 = r357016 * r357016;
        double r357018 = r357015 / r357017;
        double r357019 = k;
        double r357020 = sin(r357019);
        double r357021 = r357018 * r357020;
        double r357022 = tan(r357019);
        double r357023 = r357021 * r357022;
        double r357024 = 1.0;
        double r357025 = r357019 / r357013;
        double r357026 = pow(r357025, r357012);
        double r357027 = r357024 + r357026;
        double r357028 = r357027 - r357024;
        double r357029 = r357023 * r357028;
        double r357030 = r357012 / r357029;
        return r357030;
}

↓

double f(double t, double l, double k) {
        double r357031 = l;
        double r357032 = r357031 * r357031;
        double r357033 = 0.0;
        bool r357034 = r357032 <= r357033;
        double r357035 = 2.0;
        double r357036 = 1.0;
        double r357037 = k;
        double r357038 = 2.0;
        double r357039 = r357035 / r357038;
        double r357040 = pow(r357037, r357039);
        double r357041 = t;
        double r357042 = 1.0;
        double r357043 = pow(r357041, r357042);
        double r357044 = r357040 * r357043;
        double r357045 = r357040 * r357044;
        double r357046 = r357036 / r357045;
        double r357047 = pow(r357046, r357042);
        double r357048 = sin(r357037);
        double r357049 = cbrt(r357048);
        double r357050 = 4.0;
        double r357051 = pow(r357049, r357050);
        double r357052 = r357051 / r357031;
        double r357053 = cbrt(r357052);
        double r357054 = r357053 * r357053;
        double r357055 = cbrt(r357031);
        double r357056 = r357055 * r357055;
        double r357057 = r357054 / r357056;
        double r357058 = r357036 / r357057;
        double r357059 = r357058 / r357036;
        double r357060 = r357047 * r357059;
        double r357061 = cos(r357037);
        double r357062 = r357053 / r357055;
        double r357063 = r357061 / r357062;
        double r357064 = pow(r357049, r357038);
        double r357065 = r357063 / r357064;
        double r357066 = r357060 * r357065;
        double r357067 = r357035 * r357066;
        double r357068 = 1.6443394873045096e+249;
        bool r357069 = r357032 <= r357068;
        double r357070 = r357036 / r357040;
        double r357071 = pow(r357070, r357042);
        double r357072 = r357036 / r357044;
        double r357073 = pow(r357072, r357042);
        double r357074 = pow(r357031, r357038);
        double r357075 = r357061 * r357074;
        double r357076 = pow(r357048, r357038);
        double r357077 = r357075 / r357076;
        double r357078 = r357073 * r357077;
        double r357079 = r357071 * r357078;
        double r357080 = r357035 * r357079;
        double r357081 = cbrt(r357061);
        double r357082 = r357081 * r357081;
        double r357083 = r357082 / r357052;
        double r357084 = pow(r357036, r357038);
        double r357085 = r357083 / r357084;
        double r357086 = r357047 * r357085;
        double r357087 = r357036 / r357031;
        double r357088 = r357081 / r357087;
        double r357089 = r357088 / r357064;
        double r357090 = r357086 * r357089;
        double r357091 = r357035 * r357090;
        double r357092 = r357069 ? r357080 : r357091;
        double r357093 = r357034 ? r357067 : r357092;
        return r357093;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 0.0

   1. Initial program 46.7

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

   2. Simplified 37.8

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

   3. Taylor expanded around inf 20.4

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

   5. Applied sqr-pow 20.4

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

   6. Applied associate-*l* 20.4

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

   8. Applied add-cube-cbrt 20.4

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

   9. Applied unpow-prod-down 20.4

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

   10. Applied associate-/r* 20.3

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

   11. Simplified 13.9

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

   13. Applied *-un-lft-identity 13.9

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

   14. Applied add-cube-cbrt 13.9

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

   15. Applied add-cube-cbrt 13.9

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

   16. Applied times-frac 13.9

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

   17. Applied *-un-lft-identity 13.9

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

   18. Applied times-frac 13.5

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

   19. Applied times-frac 11.5

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

   20. Applied associate-*r* 7.6

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

   if 0.0 < (* l l) < 1.6443394873045096e+249

   1. Initial program 44.0

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

   2. Simplified 34.1

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

   3. Taylor expanded around inf 9.7

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

   5. Applied sqr-pow 9.7

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

   6. Applied associate-*l* 5.9

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

   8. Applied *-un-lft-identity 5.9

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

   9. Applied times-frac 5.5

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

   10. Applied unpow-prod-down 5.5

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

   11. Applied associate-*l* 3.1

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

   if 1.6443394873045096e+249 < (* l l)

   1. Initial program 61.2

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

   2. Simplified 60.2

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

   3. Taylor expanded around inf 58.0

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

   5. Applied sqr-pow 58.0

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

   6. Applied associate-*l* 56.2

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

   8. Applied add-cube-cbrt 56.3

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

   9. Applied unpow-prod-down 56.3

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

   10. Applied associate-/r* 56.3

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

   11. Simplified 56.3

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

   13. Applied *-un-lft-identity 56.3

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

   14. Applied unpow-prod-down 56.3

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

   15. Applied div-inv 56.3

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

   16. Applied add-cube-cbrt 56.3

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

   17. Applied times-frac 56.3

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

   18. Applied times-frac 56.3

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

   19. Applied associate-*r* 36.5

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 1.6443394873045096 \cdot 10^{249}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{1}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(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))))