Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 4.0

Time: 1.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2.8947889996786224 \cdot 10^{106}:\\ \;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{{1}^{1}}{\sin k} \cdot \frac{{\ell}^{1}}{\sin k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {1}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{1}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \end{array}\]

C

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

↓

double code(double t, double l, double k) {
	double VAR;
	if ((t <= 2.8947889996786224e+106)) {
		VAR = ((double) (2.0 * ((double) (((double) (((double) (((double) cos(k)) * l)) * ((double) pow(((double) (1.0 / ((double) pow(k, ((double) (2.0 / 2.0)))))), 1.0)))) * ((double) (((double) pow(((double) (1.0 / ((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) pow(t, 1.0)))))), 1.0)) * ((double) (((double) (((double) pow(1.0, 1.0)) / ((double) sin(k)))) * ((double) (((double) pow(l, 1.0)) / ((double) sin(k))))))))))));
	} else {
		VAR = ((double) (2.0 * ((double) (((double) (((double) (((double) (((double) cos(k)) * l)) * ((double) pow(((double) (1.0 / ((double) pow(k, ((double) (2.0 / 2.0)))))), 1.0)))) * ((double) pow(((double) (((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))) / ((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) pow(1.0, 1.0)))))), 1.0)))) * ((double) (((double) pow(((double) (((double) cbrt(1.0)) / ((double) pow(t, 1.0)))), 1.0)) * ((double) (((double) pow(l, 1.0)) / ((double) pow(((double) sin(k)), 2.0))))))))));
	}
	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 < 2.8947889996786224e+106

   1. Initial program 46.9

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

   2. Simplified 40.4

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

   3. Taylor expanded around inf 22.1

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

   5. Applied *-un-lft-identity 22.1

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

   6. Applied unpow-prod-down 22.1

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

   7. Applied sqr-pow 22.1

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

   8. Applied associate-*r* 22.1

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

   9. Applied times-frac 20.2

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

   10. Applied associate-*r* 15.9

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

   11. Simplified 15.9

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

   13. Applied sqr-pow 15.9

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

   14. Applied associate-*l* 12.2

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

   15. Applied *-un-lft-identity 12.2

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

   16. Applied times-frac 11.8

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

   17. Applied unpow-prod-down 11.8

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

   18. Applied associate-*r* 6.2

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

   19. Applied associate-*l* 5.3

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

   20. Simplified 5.3

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

   22. Applied unpow2 5.3

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

   23. Applied *-un-lft-identity 5.3

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

   24. Applied unpow-prod-down 5.3

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

   25. Applied times-frac 4.0

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

   if 2.8947889996786224e+106 < t

   1. Initial program 52.0

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

   2. Simplified 38.5

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

   3. Taylor expanded around inf 19.6

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

   5. Applied *-un-lft-identity 19.6

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

   6. Applied unpow-prod-down 19.6

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

   7. Applied sqr-pow 19.6

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

   8. Applied associate-*r* 19.6

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

   9. Applied times-frac 17.3

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

   10. Applied associate-*r* 13.3

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

   11. Simplified 13.3

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

   13. Applied sqr-pow 13.3

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

   14. Applied associate-*l* 13.3

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

   15. Applied *-un-lft-identity 13.3

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

   16. Applied times-frac 13.0

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

   17. Applied unpow-prod-down 13.0

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

   18. Applied associate-*r* 10.7

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

   19. Applied associate-*l* 10.5

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

   20. Simplified 10.5

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

   22. Applied *-un-lft-identity 10.5

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

   23. Applied unpow-prod-down 10.5

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

   24. Applied associate-*r* 10.5

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

   25. Applied add-cube-cbrt 10.5

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

   26. Applied times-frac 10.0

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

   27. Applied unpow-prod-down 10.0

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

   28. Applied associate-*l* 3.9

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

   29. Applied associate-*r* 4.2

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.8947889996786224 \cdot 10^{106}:\\ \;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{{1}^{1}}{\sin k} \cdot \frac{{\ell}^{1}}{\sin k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {1}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{1}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(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))))