Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 14.6

Time: 1.3m

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}\;t \le -8.225955376089109082384490045820019778998 \cdot 10^{83}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t \le -1.220017042562660892612086221071422485758 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell \cdot \cos k} + 2 \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t \cdot \left(t \cdot t\right)}{\cos k}\right)}\\ \mathbf{elif}\;t \le 3.713773245832630524294817139824313844179 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \frac{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\sqrt[3]{\ell}}\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \mathbf{elif}\;t \le 308603786621388752962972221964288:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell \cdot \cos k} + 2 \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t \cdot \left(t \cdot t\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r10809877 = 2.0;
        double r10809878 = t;
        double r10809879 = 3.0;
        double r10809880 = pow(r10809878, r10809879);
        double r10809881 = l;
        double r10809882 = r10809881 * r10809881;
        double r10809883 = r10809880 / r10809882;
        double r10809884 = k;
        double r10809885 = sin(r10809884);
        double r10809886 = r10809883 * r10809885;
        double r10809887 = tan(r10809884);
        double r10809888 = r10809886 * r10809887;
        double r10809889 = 1.0;
        double r10809890 = r10809884 / r10809878;
        double r10809891 = pow(r10809890, r10809877);
        double r10809892 = r10809889 + r10809891;
        double r10809893 = r10809892 + r10809889;
        double r10809894 = r10809888 * r10809893;
        double r10809895 = r10809877 / r10809894;
        return r10809895;
}

↓

double f(double t, double l, double k) {
        double r10809896 = t;
        double r10809897 = -8.225955376089109e+83;
        bool r10809898 = r10809896 <= r10809897;
        double r10809899 = 2.0;
        double r10809900 = cbrt(r10809896);
        double r10809901 = 3.0;
        double r10809902 = pow(r10809900, r10809901);
        double r10809903 = l;
        double r10809904 = cbrt(r10809903);
        double r10809905 = r10809904 * r10809904;
        double r10809906 = r10809902 / r10809905;
        double r10809907 = r10809902 / r10809904;
        double r10809908 = r10809902 / r10809903;
        double r10809909 = k;
        double r10809910 = sin(r10809909);
        double r10809911 = r10809908 * r10809910;
        double r10809912 = r10809907 * r10809911;
        double r10809913 = tan(r10809909);
        double r10809914 = r10809912 * r10809913;
        double r10809915 = r10809906 * r10809914;
        double r10809916 = 1.0;
        double r10809917 = r10809909 / r10809896;
        double r10809918 = pow(r10809917, r10809899);
        double r10809919 = r10809916 + r10809918;
        double r10809920 = r10809919 + r10809916;
        double r10809921 = r10809915 * r10809920;
        double r10809922 = r10809899 / r10809921;
        double r10809923 = -1.2200170425626609e-116;
        bool r10809924 = r10809896 <= r10809923;
        double r10809925 = r10809896 / r10809903;
        double r10809926 = r10809909 * r10809910;
        double r10809927 = r10809926 * r10809926;
        double r10809928 = cos(r10809909);
        double r10809929 = r10809903 * r10809928;
        double r10809930 = r10809927 / r10809929;
        double r10809931 = r10809925 * r10809930;
        double r10809932 = r10809910 / r10809903;
        double r10809933 = r10809932 * r10809932;
        double r10809934 = r10809896 * r10809896;
        double r10809935 = r10809896 * r10809934;
        double r10809936 = r10809935 / r10809928;
        double r10809937 = r10809933 * r10809936;
        double r10809938 = r10809899 * r10809937;
        double r10809939 = r10809931 + r10809938;
        double r10809940 = r10809899 / r10809939;
        double r10809941 = 3.7137732458326305e-280;
        bool r10809942 = r10809896 <= r10809941;
        double r10809943 = cbrt(r10809920);
        double r10809944 = r10809943 * r10809943;
        double r10809945 = r10809912 / r10809904;
        double r10809946 = r10809907 * r10809945;
        double r10809947 = r10809913 * r10809946;
        double r10809948 = r10809944 * r10809947;
        double r10809949 = r10809948 * r10809943;
        double r10809950 = r10809899 / r10809949;
        double r10809951 = 3.0860378662138875e+32;
        bool r10809952 = r10809896 <= r10809951;
        double r10809953 = r10809952 ? r10809940 : r10809922;
        double r10809954 = r10809942 ? r10809950 : r10809953;
        double r10809955 = r10809924 ? r10809940 : r10809954;
        double r10809956 = r10809898 ? r10809922 : r10809955;
        return r10809956;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -8.225955376089109e+83 or 3.0860378662138875e+32 < t

   1. Initial program 23.1

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

   3. Applied add-cube-cbrt 23.1

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

   4. Applied unpow-prod-down 23.1

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

   5. Applied times-frac 15.4

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

   6. Applied associate-*l* 14.2

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

   8. Applied add-cube-cbrt 14.2

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

   9. Applied unpow-prod-down 14.2

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

   10. Applied times-frac 7.3

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

   12. Applied associate-*l* 5.7

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

   14. Applied associate-*l* 3.2

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

   if -8.225955376089109e+83 < t < -1.2200170425626609e-116 or 3.7137732458326305e-280 < t < 3.0860378662138875e+32

   1. Initial program 35.8

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

   3. Applied add-cube-cbrt 36.0

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

   4. Applied unpow-prod-down 36.0

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

   5. Applied times-frac 29.6

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

   6. Applied associate-*l* 26.2

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

   7. Taylor expanded around inf 28.8

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

   8. Simplified 19.9

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

   if -1.2200170425626609e-116 < t < 3.7137732458326305e-280

   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. Using strategy rm

   3. Applied add-cube-cbrt 64.0

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

   4. Applied unpow-prod-down 64.0

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

   5. Applied times-frac 57.5

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

   6. Applied associate-*l* 57.5

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

   8. Applied add-cube-cbrt 57.5

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

   9. Applied unpow-prod-down 57.5

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

   10. Applied times-frac 48.0

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

   12. Applied associate-*l* 48.0

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

   14. Applied add-cube-cbrt 48.0

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

   15. Applied associate-*r* 48.0

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

   16. Simplified 48.0

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

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.225955376089109082384490045820019778998 \cdot 10^{83}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t \le -1.220017042562660892612086221071422485758 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell \cdot \cos k} + 2 \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t \cdot \left(t \cdot t\right)}{\cos k}\right)}\\ \mathbf{elif}\;t \le 3.713773245832630524294817139824313844179 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \frac{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\sqrt[3]{\ell}}\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \mathbf{elif}\;t \le 308603786621388752962972221964288:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell \cdot \cos k} + 2 \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t \cdot \left(t \cdot t\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))