Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 11.3

Time: 48.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.530327949244370572056415908655916911895 \cdot 10^{-149} \lor \neg \left(t \le 3.991020417903674316972710306867471371038 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{\frac{1}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \tan k\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\frac{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\left(\sin k\right)}^{2} \cdot \left(k \cdot k\right)}\right) \cdot 2 - \frac{4}{{\left(\sin k\right)}^{2}} \cdot \frac{t \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{{k}^{4}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r124047 = 2.0;
        double r124048 = t;
        double r124049 = 3.0;
        double r124050 = pow(r124048, r124049);
        double r124051 = l;
        double r124052 = r124051 * r124051;
        double r124053 = r124050 / r124052;
        double r124054 = k;
        double r124055 = sin(r124054);
        double r124056 = r124053 * r124055;
        double r124057 = tan(r124054);
        double r124058 = r124056 * r124057;
        double r124059 = 1.0;
        double r124060 = r124054 / r124048;
        double r124061 = pow(r124060, r124047);
        double r124062 = r124059 + r124061;
        double r124063 = r124062 + r124059;
        double r124064 = r124058 * r124063;
        double r124065 = r124047 / r124064;
        return r124065;
}

↓

double f(double t, double l, double k) {
        double r124066 = t;
        double r124067 = -9.53032794924437e-149;
        bool r124068 = r124066 <= r124067;
        double r124069 = 3.9910204179036743e-109;
        bool r124070 = r124066 <= r124069;
        double r124071 = !r124070;
        bool r124072 = r124068 || r124071;
        double r124073 = 1.0;
        double r124074 = cbrt(r124066);
        double r124075 = r124074 * r124074;
        double r124076 = 3.0;
        double r124077 = 2.0;
        double r124078 = r124076 / r124077;
        double r124079 = pow(r124075, r124078);
        double r124080 = l;
        double r124081 = cbrt(r124080);
        double r124082 = r124079 / r124081;
        double r124083 = k;
        double r124084 = tan(r124083);
        double r124085 = r124082 * r124084;
        double r124086 = r124085 * r124082;
        double r124087 = r124073 / r124086;
        double r124088 = 1.0;
        double r124089 = r124088 + r124088;
        double r124090 = r124083 / r124066;
        double r124091 = 2.0;
        double r124092 = pow(r124090, r124091);
        double r124093 = r124089 + r124092;
        double r124094 = sin(r124083);
        double r124095 = r124093 * r124094;
        double r124096 = pow(r124074, r124076);
        double r124097 = r124091 / r124096;
        double r124098 = r124097 * r124081;
        double r124099 = r124095 / r124098;
        double r124100 = r124099 / r124080;
        double r124101 = r124087 / r124100;
        double r124102 = pow(r124066, r124088);
        double r124103 = r124073 / r124102;
        double r124104 = pow(r124103, r124088);
        double r124105 = r124080 * r124080;
        double r124106 = cos(r124083);
        double r124107 = r124105 * r124106;
        double r124108 = pow(r124094, r124077);
        double r124109 = r124083 * r124083;
        double r124110 = r124108 * r124109;
        double r124111 = r124107 / r124110;
        double r124112 = r124104 * r124111;
        double r124113 = r124112 * r124091;
        double r124114 = 4.0;
        double r124115 = r124114 / r124108;
        double r124116 = r124066 * r124107;
        double r124117 = 4.0;
        double r124118 = pow(r124083, r124117);
        double r124119 = r124116 / r124118;
        double r124120 = r124115 * r124119;
        double r124121 = r124113 - r124120;
        double r124122 = r124072 ? r124101 : r124121;
        return r124122;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -9.53032794924437e-149 or 3.9910204179036743e-109 < t

   1. Initial program 24.5

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

   2. Simplified 19.4

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

   4. Applied add-cube-cbrt 19.6

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

   5. Applied add-cube-cbrt 19.7

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

   6. Applied unpow-prod-down 19.7

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

   7. Applied times-frac 16.8

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

   8. Applied associate-*r* 15.8

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

   10. Applied sqr-pow 15.8

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

   11. Applied times-frac 13.4

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

   12. Applied associate-*r* 12.5

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

   14. Applied *-un-lft-identity 12.5

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

   15. Applied times-frac 12.2

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

   16. Applied associate-/l* 9.5

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

   17. Simplified 9.5

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

   19. Applied associate-*l/ 7.0

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

   20. Applied associate-*l/ 7.6

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

   21. Applied associate-/r/ 7.5

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

   22. Applied associate-/l* 7.0

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

   23. Simplified 7.0

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

   if -9.53032794924437e-149 < t < 3.9910204179036743e-109

   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. Simplified 64.0

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

   4. Applied add-cube-cbrt 64.0

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

   5. Applied add-cube-cbrt 64.0

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

   6. Applied unpow-prod-down 64.0

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

   7. Applied times-frac 58.4

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

   8. Applied associate-*r* 57.5

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

   10. Applied sqr-pow 57.5

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

   11. Applied times-frac 49.0

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

   12. Applied associate-*r* 47.3

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

   13. Taylor expanded around inf 29.4

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

   14. Simplified 28.9

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.530327949244370572056415908655916911895 \cdot 10^{-149} \lor \neg \left(t \le 3.991020417903674316972710306867471371038 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{\frac{1}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \tan k\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\frac{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\left(\sin k\right)}^{2} \cdot \left(k \cdot k\right)}\right) \cdot 2 - \frac{4}{{\left(\sin k\right)}^{2}} \cdot \frac{t \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{{k}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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))))