Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 18.3

Time: 18.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -8.131248210282814 \cdot 10^{-222}:\\ \;\;\;\;\ell \cdot \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\left(\ell \cdot \frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{\sqrt[3]{2}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}\right)\right)\\ \mathbf{elif}\;t \leq 2.669625230460847 \cdot 10^{+132}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left({\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt[3]{2} \cdot \left(\ell \cdot \left(\sqrt[3]{2} \cdot \frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)\right)\right)}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\\ \end{array}\]

C

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

↓

double code(double t, double l, double k) {
	double VAR;
	if ((t <= -8.131248210282814e-222)) {
		VAR = ((double) (l * ((double) ((((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))) / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))) * ((double) (((double) (l * (((double) cbrt(1.0)) / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))))) * (((double) cbrt(2.0)) / ((double) (((double) sin(k)) * ((double) (((double) tan(k)) * ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow((k / t), 2.0)))))) * ((double) pow(((double) cbrt(t)), 3.0)))))))))))))));
	} else {
		double VAR_1;
		if ((t <= 2.669625230460847e+132)) {
			VAR_1 = ((double) (l * ((double) (l * (2.0 / ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))) * ((double) (((double) sin(k)) * ((double) (((double) (2.0 * ((double) (((double) pow((1.0 / ((double) pow((1.0 / ((double) pow(t, 2.0))), 1.0))), 1.0)) * (((double) sin(k)) / ((double) cos(k))))))) + ((double) (((double) sin(k)) * (((double) (k * k)) / ((double) cos(k))))))))))))))));
		} else {
			VAR_1 = (((double) (l * ((double) (((double) cbrt(2.0)) * ((double) (l * ((double) (((double) cbrt(2.0)) * (((double) cbrt(2.0)) / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))))))))))) / ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))) * ((double) (((double) sin(k)) * ((double) (((double) tan(k)) * ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow((k / t), 2.0)))))) * ((double) pow(((double) cbrt(t)), 3.0)))))))))));
		}
		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 t < -8.1312482102828139e-222

   1. Initial program 29.3

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

   2. Simplified 30.3

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

   4. Applied add-cube-cbrt 30.4

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

   5. Applied unpow-prod-down 30.4

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

   6. Applied associate-*l* 27.7

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

   7. Simplified 24.4

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

   9. Applied sqr-pow 24.4

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

   10. Applied associate-*l* 21.1

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

   11. Simplified 19.7

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

   13. Applied add-cube-cbrt 19.7

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

   14. Applied times-frac 19.6

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

   15. Applied associate-*l* 17.5

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

   16. Simplified 17.9

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

   18. Applied *-un-lft-identity 17.9

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

   19. Applied cbrt-prod 17.9

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

   20. Applied times-frac 18.2

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

   21. Applied associate-*r* 17.5

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

   if -8.1312482102828139e-222 < t < 2.66962523046084687e132

   1. Initial program 42.2

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

   2. Simplified 41.3

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

   4. Applied add-cube-cbrt 41.5

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

   5. Applied unpow-prod-down 41.5

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

   6. Applied associate-*l* 38.4

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

   7. Simplified 37.5

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

   9. Applied sqr-pow 37.5

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

   10. Applied associate-*l* 29.1

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

   11. Simplified 28.2

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

   12. Taylor expanded around inf 26.4

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

   13. Simplified 23.2

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

   if 2.66962523046084687e132 < 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. Simplified 26.2

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

   4. Applied add-cube-cbrt 26.2

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

   5. Applied unpow-prod-down 26.2

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

   6. Applied associate-*l* 25.6

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

   7. Simplified 19.7

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

   9. Applied sqr-pow 19.7

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

   10. Applied associate-*l* 15.7

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

   11. Simplified 12.7

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

   13. Applied add-cube-cbrt 12.7

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

   14. Applied times-frac 12.5

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

   15. Applied associate-*l* 10.8

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

   16. Simplified 11.7

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

   18. Applied associate-*r/ 11.7

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

   19. Applied associate-*r/ 11.7

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

   20. Applied associate-*r/ 11.8

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

   21. Simplified 11.8

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.131248210282814 \cdot 10^{-222}:\\ \;\;\;\;\ell \cdot \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\left(\ell \cdot \frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{\sqrt[3]{2}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}\right)\right)\\ \mathbf{elif}\;t \leq 2.669625230460847 \cdot 10^{+132}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left({\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt[3]{2} \cdot \left(\ell \cdot \left(\sqrt[3]{2} \cdot \frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)\right)\right)}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))