Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 6.0

Time: 6.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 1.80151547661768517 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{\sqrt{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}}\right)\right) \cdot \frac{{\ell}^{1}}{\sqrt{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\\ \mathbf{elif}\;k \le 1.82445258730066349 \cdot 10^{113}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{\sqrt{1}}{{k}^{2} \cdot 1}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{\sqrt{{\left(\sin k\right)}^{2}}}\right)\right) \cdot \frac{{\ell}^{1}}{\sqrt{{\left(\sin k\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{\sqrt{{\left(\sin k\right)}^{2}}}\right)\right) \cdot \frac{{\ell}^{1}}{\sqrt{\sqrt{{\left(\sin k\right)}^{2}}}}}{\sqrt{\sqrt{{\left(\sin k\right)}^{2}}}}\\ \end{array}\]

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 ((k <= 1.8015154766176852e-93)) {
		VAR = ((double) (2.0 * ((double) (((double) (((double) (((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) cos(k)) * l)) / ((double) sqrt(((double) pow(((double) (((double) cbrt(((double) sin(k)))) * ((double) cbrt(((double) sin(k)))))), 2.0)))))))))) * ((double) (((double) pow(l, 1.0)) / ((double) sqrt(((double) pow(((double) (((double) cbrt(((double) sin(k)))) * ((double) cbrt(((double) sin(k)))))), 2.0)))))))) / ((double) (((double) sqrt(((double) pow(((double) cbrt(((double) sin(k)))), 2.0)))) * ((double) sqrt(((double) pow(((double) cbrt(((double) sin(k)))), 2.0))))))))));
	} else {
		double VAR_1;
		if ((k <= 1.8244525873006635e+113)) {
			VAR_1 = ((double) (2.0 * ((double) (((double) (((double) pow(((double) (((double) sqrt(1.0)) / ((double) (((double) pow(k, 2.0)) * 1.0)))), 1.0)) * ((double) (((double) pow(((double) (((double) sqrt(1.0)) / ((double) pow(t, 1.0)))), 1.0)) * ((double) (((double) (((double) cos(k)) * l)) / ((double) sqrt(((double) pow(((double) sin(k)), 2.0)))))))))) * ((double) (((double) pow(l, 1.0)) / ((double) sqrt(((double) pow(((double) sin(k)), 2.0))))))))));
		} else {
			VAR_1 = ((double) (2.0 * ((double) (((double) (((double) (((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) cos(k)) * l)) / ((double) sqrt(((double) pow(((double) sin(k)), 2.0)))))))))) * ((double) (((double) pow(l, 1.0)) / ((double) sqrt(((double) sqrt(((double) pow(((double) sin(k)), 2.0)))))))))) / ((double) sqrt(((double) sqrt(((double) pow(((double) sin(k)), 2.0))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < 1.8015154766176852e-93

   1. Initial program 50.0

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

   2. Simplified 42.6

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

   3. Taylor expanded around inf 24.2

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

   5. Applied add-sqr-sqrt 24.2

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

   6. Applied sqr-pow 24.2

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

   7. Applied associate-*r* 24.2

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

   8. Applied times-frac 21.0

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

   9. Applied associate-*r* 17.1

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

   10. Simplified 17.1

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

   12. Applied sqr-pow 17.1

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

   13. Applied associate-*l* 14.4

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

   14. Applied *-un-lft-identity 14.4

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

   15. Applied times-frac 14.0

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

   16. Applied unpow-prod-down 14.0

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

   17. Applied associate-*l* 9.0

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

   19. Applied add-cube-cbrt 9.3

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

   20. Applied unpow-prod-down 9.3

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

   21. Applied sqrt-prod 9.3

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

   22. Applied associate-/r* 9.3

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

   23. Applied add-cube-cbrt 9.5

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

   24. Applied unpow-prod-down 9.5

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

   25. Applied sqrt-prod 7.0

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

   26. Applied associate-/r* 7.0

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

   27. Applied associate-*r/ 7.1

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

   28. Applied associate-*r/ 7.1

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

   29. Applied frac-times 6.9

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

   30. Simplified 6.9

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

   if 1.8015154766176852e-93 < k < 1.8244525873006635e+113

   1. Initial program 53.3

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

   2. Simplified 41.9

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

   3. Taylor expanded around inf 15.4

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

   5. Applied add-sqr-sqrt 15.4

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

   6. Applied sqr-pow 15.4

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

   7. Applied associate-*r* 15.4

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

   8. Applied times-frac 13.7

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

   9. Applied associate-*r* 6.5

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

   10. Simplified 6.5

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

   12. Applied *-un-lft-identity 6.5

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

   13. Applied associate-*r* 6.5

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

   14. Applied add-sqr-sqrt 6.5

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

   15. Applied times-frac 6.1

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

   16. Applied unpow-prod-down 6.1

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

   17. Applied associate-*l* 1.8

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

   if 1.8244525873006635e+113 < k

   1. Initial program 40.7

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

   2. Simplified 35.2

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

   3. Taylor expanded around inf 23.1

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

   5. Applied add-sqr-sqrt 23.1

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

   6. Applied sqr-pow 23.1

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

   7. Applied associate-*r* 23.1

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

   8. Applied times-frac 23.1

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

   9. Applied associate-*r* 20.8

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

   10. Simplified 20.8

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

   12. Applied sqr-pow 20.8

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

   13. Applied associate-*l* 13.9

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

   14. Applied *-un-lft-identity 13.9

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

   15. Applied times-frac 13.5

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

   16. Applied unpow-prod-down 13.5

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

   17. Applied associate-*l* 7.6

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

   19. Applied add-sqr-sqrt 7.6

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

   20. Applied sqrt-prod 7.7

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

   21. Applied associate-/r* 7.6

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

   22. Applied associate-*r/ 7.6

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

   23. Simplified 7.6

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

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.80151547661768517 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{\sqrt{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}}\right)\right) \cdot \frac{{\ell}^{1}}{\sqrt{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\\ \mathbf{elif}\;k \le 1.82445258730066349 \cdot 10^{113}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{\sqrt{1}}{{k}^{2} \cdot 1}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{\sqrt{{\left(\sin k\right)}^{2}}}\right)\right) \cdot \frac{{\ell}^{1}}{\sqrt{{\left(\sin k\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{\sqrt{{\left(\sin k\right)}^{2}}}\right)\right) \cdot \frac{{\ell}^{1}}{\sqrt{\sqrt{{\left(\sin k\right)}^{2}}}}}{\sqrt{\sqrt{{\left(\sin k\right)}^{2}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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))))