Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 29.9

Time: 23.8s

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 \le -3.2347895977646959 \cdot 10^{82}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{elif}\;t \le -1.21814902124711208 \cdot 10^{-79}:\\ \;\;\;\;\left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \le -1.6429965258483605 \cdot 10^{-127}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)} \cdot \frac{\ell \cdot \ell}{\sin k}\\ \mathbf{elif}\;t \le 1.63591626494623807 \cdot 10^{-222}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{elif}\;t \le 3.2617513714426391 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot \tan k\right)} \cdot \frac{\ell \cdot \ell}{\sin k}\\ \mathbf{elif}\;t \le 6.54064412827118068 \cdot 10^{78}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sqrt[3]{\sin k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{3}\right) \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\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 <= -3.234789597764696e+82)) {
		VAR = ((double) log(((double) pow(((double) exp(((double) (((double) (2.0 * 1.0)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))), ((double) (((double) (l * l)) / ((double) sin(k))))))));
	} else {
		double VAR_1;
		if ((t <= -1.2181490212471121e-79)) {
			VAR_1 = ((double) (((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(t, 3.0)))))) * ((double) (l / ((double) tan(k)))))) * ((double) (l / ((double) sin(k))))));
		} else {
			double VAR_2;
			if ((t <= -1.6429965258483605e-127)) {
				VAR_2 = ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), 3.0)))) * ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) tan(k)))))))) * ((double) (((double) (l * l)) / ((double) sin(k))))));
			} else {
				double VAR_3;
				if ((t <= 1.635916264946238e-222)) {
					VAR_3 = ((double) log(((double) pow(((double) exp(((double) (((double) (2.0 * 1.0)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))), ((double) (((double) (l * l)) / ((double) sin(k))))))));
				} else {
					double VAR_4;
					if ((t <= 3.261751371442639e-26)) {
						VAR_4 = ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(t, ((double) (3.0 / 2.0)))))) * ((double) pow(t, ((double) (3.0 / 2.0)))))) * ((double) tan(k)))))))) * ((double) (((double) (l * l)) / ((double) sin(k))))));
					} else {
						double VAR_5;
						if ((t <= 6.540644128271181e+78)) {
							VAR_5 = ((double) (((double) (((double) (2.0 * ((double) (l / ((double) (((double) cbrt(((double) sin(k)))) * ((double) cbrt(((double) sin(k)))))))))) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((double) (l / ((double) cbrt(((double) sin(k))))))));
						} else {
							VAR_5 = ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(t, 3.0)))) * ((double) tan(k)))))))) * ((double) (l * ((double) (l / ((double) sin(k))))))));
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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 6 regimes

2. if t < -3.2347895977646959e82 or -1.6429965258483605e-127 < t < 1.63591626494623807e-222

   1. Initial program 56.7

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

   2. Simplified 50.2

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

   4. Applied sqr-pow 50.2

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

   5. Applied associate-*l* 46.2

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

   7. Applied times-frac 46.1

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

   9. Applied add-log-exp 46.6

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

   10. Simplified 40.6

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

   if -3.2347895977646959e82 < t < -1.21814902124711208e-79

   1. Initial program 29.0

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

   2. Simplified 21.1

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

   4. Applied sqr-pow 21.1

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

   5. Applied associate-*l* 20.1

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

   7. Applied times-frac 19.5

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

   9. Applied associate-*r* 19.5

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

   11. Applied *-un-lft-identity 19.5

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

   12. Applied times-frac 19.0

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

   13. Applied associate-*r* 13.7

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

   14. Simplified 14.8

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

   if -1.21814902124711208e-79 < t < -1.6429965258483605e-127

   1. Initial program 45.7

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

   2. Simplified 44.8

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

   4. Applied sqr-pow 44.8

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

   5. Applied associate-*l* 39.8

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

   7. Applied times-frac 39.7

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

   9. Applied associate-*r* 38.4

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

   11. Applied add-cube-cbrt 38.7

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

   12. Applied unpow-prod-down 38.6

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

   13. Applied associate-*r* 20.8

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

   if 1.63591626494623807e-222 < t < 3.2617513714426391e-26

   1. Initial program 48.3

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

   2. Simplified 47.3

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

   4. Applied sqr-pow 47.3

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

   5. Applied associate-*l* 44.9

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

   7. Applied times-frac 44.8

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

   9. Applied associate-*r* 44.4

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

   11. Applied sqr-pow 44.4

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

   12. Applied associate-*r* 26.3

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

   if 3.2617513714426391e-26 < t < 6.54064412827118068e78

   1. Initial program 30.7

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

   2. Simplified 21.4

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

   4. Applied sqr-pow 21.4

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

   5. Applied associate-*l* 21.3

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

   7. Applied times-frac 20.4

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

   9. Applied add-cube-cbrt 20.6

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

   10. Applied times-frac 19.5

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

   11. Applied associate-*r* 14.1

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

   12. Simplified 14.7

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

   if 6.54064412827118068e78 < t

   1. Initial program 50.4

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

   2. Simplified 37.8

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

   4. Applied sqr-pow 37.8

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

   5. Applied associate-*l* 30.9

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

   7. Applied times-frac 30.7

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

   9. Applied associate-*r* 30.7

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

   11. Applied *-un-lft-identity 30.7

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

   12. Applied times-frac 30.6

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

   13. Simplified 30.6

       \[\leadsto \frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{3}\right) \cdot \tan k\right)} \cdot \left(\color{blue}{\ell} \cdot \frac{\ell}{\sin k}\right)\]
3. Recombined 6 regimes into one program.

4. Final simplification 29.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.2347895977646959 \cdot 10^{82}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{elif}\;t \le -1.21814902124711208 \cdot 10^{-79}:\\ \;\;\;\;\left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \le -1.6429965258483605 \cdot 10^{-127}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)} \cdot \frac{\ell \cdot \ell}{\sin k}\\ \mathbf{elif}\;t \le 1.63591626494623807 \cdot 10^{-222}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{elif}\;t \le 3.2617513714426391 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot \tan k\right)} \cdot \frac{\ell \cdot \ell}{\sin k}\\ \mathbf{elif}\;t \le 6.54064412827118068 \cdot 10^{78}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sqrt[3]{\sin k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{3}\right) \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))))