Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 11.2

Time: 20.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.17724582674678948 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}} \cdot \sin k\right)\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \mathbf{elif}\;t \le 1.13364796751045195 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{t}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell}\right) + {\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r118991 = 2.0;
        double r118992 = t;
        double r118993 = 3.0;
        double r118994 = pow(r118992, r118993);
        double r118995 = l;
        double r118996 = r118995 * r118995;
        double r118997 = r118994 / r118996;
        double r118998 = k;
        double r118999 = sin(r118998);
        double r119000 = r118997 * r118999;
        double r119001 = tan(r118998);
        double r119002 = r119000 * r119001;
        double r119003 = 1.0;
        double r119004 = r118998 / r118992;
        double r119005 = pow(r119004, r118991);
        double r119006 = r119003 + r119005;
        double r119007 = r119006 + r119003;
        double r119008 = r119002 * r119007;
        double r119009 = r118991 / r119008;
        return r119009;
}

↓

double f(double t, double l, double k) {
        double r119010 = t;
        double r119011 = -6.1772458267467895e-105;
        bool r119012 = r119010 <= r119011;
        double r119013 = 2.0;
        double r119014 = cbrt(r119010);
        double r119015 = 3.0;
        double r119016 = pow(r119014, r119015);
        double r119017 = l;
        double r119018 = r119016 / r119017;
        double r119019 = cbrt(r119018);
        double r119020 = r119019 * r119019;
        double r119021 = k;
        double r119022 = sin(r119021);
        double r119023 = r119019 * r119022;
        double r119024 = r119020 * r119023;
        double r119025 = r119016 * r119024;
        double r119026 = r119025 * r119022;
        double r119027 = 1.0;
        double r119028 = r119021 / r119010;
        double r119029 = pow(r119028, r119013);
        double r119030 = r119027 + r119029;
        double r119031 = r119030 + r119027;
        double r119032 = r119026 * r119031;
        double r119033 = r119017 / r119016;
        double r119034 = cos(r119021);
        double r119035 = r119033 * r119034;
        double r119036 = r119032 / r119035;
        double r119037 = r119013 / r119036;
        double r119038 = 1.133647967510452e-100;
        bool r119039 = r119010 <= r119038;
        double r119040 = 1.0;
        double r119041 = -1.0;
        double r119042 = pow(r119041, r119013);
        double r119043 = r119040 / r119042;
        double r119044 = pow(r119043, r119027);
        double r119045 = 2.0;
        double r119046 = pow(r119010, r119045);
        double r119047 = pow(r119022, r119045);
        double r119048 = r119046 * r119047;
        double r119049 = r119048 / r119017;
        double r119050 = r119044 * r119049;
        double r119051 = r119013 * r119050;
        double r119052 = pow(r119021, r119045);
        double r119053 = r119052 * r119047;
        double r119054 = r119053 / r119017;
        double r119055 = r119044 * r119054;
        double r119056 = r119051 + r119055;
        double r119057 = r119056 / r119035;
        double r119058 = r119013 / r119057;
        double r119059 = sqrt(r119010);
        double r119060 = cbrt(r119059);
        double r119061 = pow(r119060, r119015);
        double r119062 = cbrt(r119017);
        double r119063 = r119062 * r119062;
        double r119064 = r119014 * r119014;
        double r119065 = cbrt(r119064);
        double r119066 = pow(r119065, r119015);
        double r119067 = r119063 / r119066;
        double r119068 = r119061 / r119067;
        double r119069 = cbrt(r119014);
        double r119070 = pow(r119069, r119015);
        double r119071 = r119062 / r119070;
        double r119072 = r119061 / r119071;
        double r119073 = r119018 * r119022;
        double r119074 = r119072 * r119073;
        double r119075 = r119068 * r119074;
        double r119076 = tan(r119021);
        double r119077 = r119075 * r119076;
        double r119078 = r119077 * r119031;
        double r119079 = r119013 / r119078;
        double r119080 = r119039 ? r119058 : r119079;
        double r119081 = r119012 ? r119037 : r119080;
        return r119081;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -6.1772458267467895e-105

   1. Initial program 22.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. 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 17.1

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

   6. Applied associate-*l* 15.1

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

   8. Applied unpow-prod-down 15.1

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

   9. Applied associate-/l* 10.4

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

   11. Applied tan-quot 10.4

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

   12. Applied associate-*l/ 9.5

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

   13. Applied frac-times 7.7

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

   14. Applied associate-*l/ 6.6

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

   16. Applied add-cube-cbrt 6.7

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

   17. Applied associate-*l* 6.7

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

   if -6.1772458267467895e-105 < t < 1.133647967510452e-100

   1. Initial program 63.4

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

   3. Applied add-cube-cbrt 63.4

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

   4. Applied unpow-prod-down 63.4

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

   5. Applied times-frac 53.6

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

   6. Applied associate-*l* 53.6

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

   8. Applied unpow-prod-down 53.6

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

   9. Applied associate-/l* 44.7

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

   11. Applied tan-quot 44.7

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

   12. Applied associate-*l/ 44.7

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

   13. Applied frac-times 45.6

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

   14. Applied associate-*l/ 41.9

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

   15. Taylor expanded around -inf 23.0

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

   if 1.133647967510452e-100 < t

   1. Initial program 23.5

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

   3. Applied add-cube-cbrt 23.7

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

   4. Applied unpow-prod-down 23.7

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

   5. Applied times-frac 17.4

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

   6. Applied associate-*l* 14.9

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

   8. Applied unpow-prod-down 14.9

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

   9. Applied associate-/l* 9.7

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

   11. Applied add-cube-cbrt 9.7

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

   12. Applied cbrt-prod 9.7

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

   13. Applied unpow-prod-down 9.7

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

   14. Applied add-cube-cbrt 9.7

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

   15. Applied times-frac 9.7

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

   16. Applied add-sqr-sqrt 9.7

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

   17. Applied cbrt-prod 9.8

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

   18. Applied unpow-prod-down 9.7

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

   19. Applied times-frac 9.7

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

   20. Applied associate-*l* 8.7

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.17724582674678948 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}} \cdot \sin k\right)\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \mathbf{elif}\;t \le 1.13364796751045195 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{t}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell}\right) + {\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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))))