Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 17.0

Time: 16.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.9470297795445965 \cdot 10^{-264}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left({\left({t}^{1}\right)}^{1} \cdot \frac{\sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 5.3780304422650812 \cdot 10^{-244}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}, \frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 9.816394660137228 \cdot 10^{-177}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left({t}^{2}\right)}^{1} \cdot \frac{1}{\ell}\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 2.9646084215984668 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}, \frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(\left({\left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k\right)}^{1}}\\ \end{array}\]

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 (((l * l) <= 9.947029779544597e-264)) {
		VAR = (2.0 / ((((pow(cbrt(t), 3.0) * (pow(cbrt(t), 3.0) / l)) * (pow(pow(t, 1.0), 1.0) * (sin(k) / l))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)));
	} else {
		double VAR_1;
		if (((l * l) <= 5.378030442265081e-244)) {
			VAR_1 = (2.0 / fma(2.0, ((pow(t, 3.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))), ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)))));
		} else {
			double VAR_2;
			if (((l * l) <= 9.816394660137228e-177)) {
				VAR_2 = (2.0 / ((((pow(pow(t, 2.0), 1.0) * (1.0 / l)) * ((pow(cbrt(t), 3.0) / l) * sin(k))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)));
			} else {
				double VAR_3;
				if (((l * l) <= 2.964608421598467e-73)) {
					VAR_3 = (2.0 / fma(2.0, ((pow(t, 3.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))), ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)))));
				} else {
					VAR_3 = (2.0 / pow(((((pow(cbrt(((pow(cbrt(t), 3.0) / l) * sin(k))), 3.0) * pow(cbrt(t), 3.0)) * (pow(cbrt(t), 3.0) / l)) * ((1.0 + pow((k / t), 2.0)) + 1.0)) * tan(k)), 1.0));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if (* l l) < 9.947029779544597e-264

   1. Initial program 24.9

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

   3. Applied add-cube-cbrt 25.0

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

   4. Applied unpow-prod-down 25.0

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

   5. Applied times-frac 18.2

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

   6. Applied associate-*l* 15.6

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

   8. Applied *-un-lft-identity 15.6

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

   9. Applied unpow-prod-down 15.6

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

   10. Applied times-frac 10.3

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

   11. Simplified 10.3

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

   12. Taylor expanded around inf 10.2

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

   if 9.947029779544597e-264 < (* l l) < 5.378030442265081e-244 or 9.816394660137228e-177 < (* l l) < 2.964608421598467e-73

   1. Initial program 22.6

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

   2. Taylor expanded around inf 14.6

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

   3. Simplified 14.6

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

   if 5.378030442265081e-244 < (* l l) < 9.816394660137228e-177

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.8

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

   4. Applied unpow-prod-down 21.8

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

   5. Applied times-frac 18.7

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

   6. Applied associate-*l* 16.3

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

   7. Taylor expanded around 0 16.1

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

   if 2.964608421598467e-73 < (* l l)

   1. Initial program 41.9

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

   3. Applied add-cube-cbrt 42.1

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

   4. Applied unpow-prod-down 42.1

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

   5. Applied times-frac 33.1

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

   6. Applied associate-*l* 32.2

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

   8. Applied *-un-lft-identity 32.2

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

   9. Applied unpow-prod-down 32.2

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

   10. Applied times-frac 24.6

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

   11. Simplified 24.6

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

   13. Applied add-cube-cbrt 24.6

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

   15. Applied pow1 24.6

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

   16. Applied pow1 24.6

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

   17. Applied pow1 24.6

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

   18. Applied pow1 24.6

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

   19. Applied pow1 24.6

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

   20. Applied pow-prod-down 24.6

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

   21. Applied pow-prod-down 24.6

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

   22. Applied pow1 24.6

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

   23. Applied pow-prod-down 24.6

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

   24. Applied pow-prod-down 24.6

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

   25. Applied pow-prod-down 24.6

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

   26. Simplified 22.8

       \[\leadsto \frac{2}{{\color{blue}{\left(\left(\left(\left({\left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k\right)}}^{1}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.9470297795445965 \cdot 10^{-264}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left({\left({t}^{1}\right)}^{1} \cdot \frac{\sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 5.3780304422650812 \cdot 10^{-244}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}, \frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 9.816394660137228 \cdot 10^{-177}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left({t}^{2}\right)}^{1} \cdot \frac{1}{\ell}\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 2.9646084215984668 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}, \frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(\left({\left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k\right)}^{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +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))))