Toniolo and Linder, Equation (10+)

Average Error: 31.6 → 8.5

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.980748262550128 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\tan k}}{t \cdot \frac{\sin k \cdot t}{\ell}}}{-\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\frac{\ell}{\sqrt[3]{t}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\\ \mathbf{elif}\;t \le 6.457003182851085 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sin k \cdot \frac{t \cdot t}{\ell}}{\cos k} \cdot -2 + \left(-\frac{\frac{k \cdot k}{\ell} \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{-2}{\sin k}}{t}}{-2 - \frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\ell \cdot \frac{1}{\tan k \cdot t}\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3642727 = 2.0;
        double r3642728 = t;
        double r3642729 = 3.0;
        double r3642730 = pow(r3642728, r3642729);
        double r3642731 = l;
        double r3642732 = r3642731 * r3642731;
        double r3642733 = r3642730 / r3642732;
        double r3642734 = k;
        double r3642735 = sin(r3642734);
        double r3642736 = r3642733 * r3642735;
        double r3642737 = tan(r3642734);
        double r3642738 = r3642736 * r3642737;
        double r3642739 = 1.0;
        double r3642740 = r3642734 / r3642728;
        double r3642741 = pow(r3642740, r3642727);
        double r3642742 = r3642739 + r3642741;
        double r3642743 = r3642742 + r3642739;
        double r3642744 = r3642738 * r3642743;
        double r3642745 = r3642727 / r3642744;
        return r3642745;
}

↓

double f(double t, double l, double k) {
        double r3642746 = t;
        double r3642747 = -4.980748262550128e+31;
        bool r3642748 = r3642746 <= r3642747;
        double r3642749 = -2.0;
        double r3642750 = k;
        double r3642751 = tan(r3642750);
        double r3642752 = r3642749 / r3642751;
        double r3642753 = sin(r3642750);
        double r3642754 = r3642753 * r3642746;
        double r3642755 = l;
        double r3642756 = r3642754 / r3642755;
        double r3642757 = r3642746 * r3642756;
        double r3642758 = r3642752 / r3642757;
        double r3642759 = cbrt(r3642746);
        double r3642760 = r3642759 * r3642759;
        double r3642761 = -r3642760;
        double r3642762 = r3642758 / r3642761;
        double r3642763 = r3642755 / r3642759;
        double r3642764 = r3642750 / r3642746;
        double r3642765 = r3642764 * r3642764;
        double r3642766 = 2.0;
        double r3642767 = r3642765 + r3642766;
        double r3642768 = r3642763 / r3642767;
        double r3642769 = r3642762 * r3642768;
        double r3642770 = 6.457003182851085e-47;
        bool r3642771 = r3642746 <= r3642770;
        double r3642772 = r3642746 / r3642755;
        double r3642773 = r3642753 * r3642772;
        double r3642774 = r3642749 / r3642773;
        double r3642775 = r3642746 * r3642746;
        double r3642776 = r3642775 / r3642755;
        double r3642777 = r3642753 * r3642776;
        double r3642778 = cos(r3642750);
        double r3642779 = r3642777 / r3642778;
        double r3642780 = r3642779 * r3642749;
        double r3642781 = r3642750 * r3642750;
        double r3642782 = r3642781 / r3642755;
        double r3642783 = r3642782 * r3642753;
        double r3642784 = r3642783 / r3642778;
        double r3642785 = -r3642784;
        double r3642786 = r3642780 + r3642785;
        double r3642787 = r3642774 / r3642786;
        double r3642788 = r3642755 / r3642746;
        double r3642789 = r3642749 / r3642753;
        double r3642790 = r3642789 / r3642746;
        double r3642791 = r3642749 - r3642765;
        double r3642792 = r3642790 / r3642791;
        double r3642793 = 1.0;
        double r3642794 = r3642751 * r3642746;
        double r3642795 = r3642793 / r3642794;
        double r3642796 = r3642755 * r3642795;
        double r3642797 = r3642792 * r3642796;
        double r3642798 = r3642788 * r3642797;
        double r3642799 = r3642771 ? r3642787 : r3642798;
        double r3642800 = r3642748 ? r3642769 : r3642799;
        return r3642800;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -4.980748262550128e+31

   1. Initial program 23.0

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

   2. Simplified 11.2

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

   4. Applied frac-2neg 11.2

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

   5. Simplified 6.1

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

   7. Applied *-un-lft-identity 6.1

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

   8. Applied distribute-lft-neg-in 6.1

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

   9. Applied add-cube-cbrt 6.3

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

   10. Applied associate-*r/ 6.3

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

   11. Applied associate-/r/ 5.7

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

   12. Applied times-frac 4.9

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

   13. Applied times-frac 4.5

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

   14. Simplified 3.9

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

   if -4.980748262550128e+31 < t < 6.457003182851085e-47

   1. Initial program 47.9

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

   2. Simplified 37.1

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

   4. Applied frac-2neg 37.1

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

   5. Simplified 34.7

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

   7. Applied *-un-lft-identity 34.7

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

   8. Applied div-inv 34.7

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

   9. Applied times-frac 34.1

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

   10. Applied times-frac 34.1

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

   11. Applied associate-/l* 30.0

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

   12. Simplified 26.4

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

   13. Taylor expanded around inf 18.5

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

   14. Simplified 16.3

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

   if 6.457003182851085e-47 < t

   1. Initial program 21.9

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

   2. Simplified 12.0

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

   4. Applied frac-2neg 12.0

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

   5. Simplified 7.7

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

   7. Applied *-un-lft-identity 7.7

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

   8. Applied div-inv 7.7

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

   9. Applied times-frac 6.9

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

   10. Applied times-frac 3.8

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

   11. Applied associate-/l* 3.2

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

   12. Simplified 3.2

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

   14. Applied add-cube-cbrt 3.4

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

   15. Applied associate-/r* 3.4

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

   17. Applied add-cube-cbrt 3.4

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

   18. Applied associate-*r/ 3.5

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

   19. Applied associate-/r/ 3.5

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

   20. Applied times-frac 3.5

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

   21. Applied times-frac 4.2

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

   22. Simplified 4.3

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

   23. Simplified 4.3

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.980748262550128 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\tan k}}{t \cdot \frac{\sin k \cdot t}{\ell}}}{-\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\frac{\ell}{\sqrt[3]{t}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\\ \mathbf{elif}\;t \le 6.457003182851085 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sin k \cdot \frac{t \cdot t}{\ell}}{\cos k} \cdot -2 + \left(-\frac{\frac{k \cdot k}{\ell} \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{-2}{\sin k}}{t}}{-2 - \frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\ell \cdot \frac{1}{\tan k \cdot t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))