Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 3.2

Time: 24.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 9.695058772041076 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\sin k} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\sin k} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{1}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{\left(\frac{1}{2}\right)}}\right)}^{1} \cdot \cos k\right) \cdot \ell}}\\ \end{array}\]

C

double code(double t, double l, double k) {
	return (2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0)));
}

↓

double code(double t, double l, double k) {
	double VAR;
	if ((t <= 9.695058772041076e-72)) {
		VAR = (2.0 * (((l / sin(k)) * pow((1.0 / pow(k, (2.0 / 2.0))), 1.0)) / (sin(k) / ((pow(((1.0 / pow(k, (2.0 / 2.0))) / pow(t, 1.0)), 1.0) * cos(k)) * l))));
	} else {
		VAR = (2.0 * (((l / sin(k)) * pow((1.0 / (pow(k, (2.0 / 2.0)) * pow(t, (1.0 / 2.0)))), 1.0)) / (sin(k) / ((pow(((1.0 / pow(k, (2.0 / 2.0))) / pow(t, (1.0 / 2.0))), 1.0) * cos(k)) * l))));
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < 9.695058772041076e-72

   1. Initial program 50.7

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

   2. Simplified 44.5

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

   3. Taylor expanded around inf 22.5

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

   5. Applied sqr-pow 22.5

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

   6. Applied sqr-pow 22.5

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

   7. Applied associate-*r* 22.5

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

   8. Applied times-frac 20.9

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

   9. Applied associate-*r* 16.5

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

   10. Simplified 16.5

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

   12. Applied sqr-pow 16.5

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

   13. Applied associate-*l* 10.7

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

   14. Applied associate-/r* 10.4

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

   16. Applied div-inv 10.4

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

   17. Applied times-frac 10.5

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

   18. Applied unpow-prod-down 10.5

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

   19. Applied associate-*l* 10.5

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

   20. Applied associate-*l* 5.1

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

   21. Applied associate-/l* 5.0

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

   22. Applied associate-*l/ 3.9

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

   23. Simplified 3.9

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

   if 9.695058772041076e-72 < t

   1. Initial program 42.4

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

   2. Simplified 31.5

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

   3. Taylor expanded around inf 19.9

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

   5. Applied sqr-pow 19.9

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

   6. Applied sqr-pow 19.9

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

   7. Applied associate-*r* 19.9

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

   8. Applied times-frac 17.7

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

   9. Applied associate-*r* 13.4

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

   10. Simplified 13.3

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

   12. Applied sqr-pow 13.3

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

   13. Applied associate-*l* 11.1

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

   14. Applied associate-/r* 10.8

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

   16. Applied sqr-pow 10.9

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

   17. Applied associate-*r* 10.9

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

   18. Applied div-inv 10.9

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

   19. Applied times-frac 10.9

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

   20. Applied unpow-prod-down 10.9

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

   21. Applied associate-*l* 10.9

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

   22. Applied associate-*l* 4.9

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

   23. Applied associate-/l* 4.9

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

   24. Applied associate-*l/ 2.0

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

   25. Simplified 2.0

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 9.695058772041076 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\sin k} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\sin k} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{1}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{\left(\frac{1}{2}\right)}}\right)}^{1} \cdot \cos k\right) \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(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))))