Average Error: 32.7 → 17.7
Time: 21.5s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.3013132694615146 \cdot 10^{-252}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}\right)\right)\\ \mathbf{elif}\;t \le 2.2844510146699535 \cdot 10^{-95}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left({\left(\frac{1}{{\left({1}^{2} \cdot \frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\ell \cdot \frac{2}{\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}\right)\right)\\ \end{array}\]
\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 -3.3013132694615146 \cdot 10^{-252}:\\
\;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{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 \le 2.2844510146699535 \cdot 10^{-95}:\\
\;\;\;\;\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({1}^{2} \cdot \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}:\\
\;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\ell \cdot \frac{2}{\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({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}\right)\right)\\

\end{array}
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 <= -3.3013132694615146e-252)) {
		VAR = ((double) (l * ((double) (((double) (1.0 / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (((double) (l / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (2.0 / ((double) (((double) sin(k)) * ((double) (((double) tan(k)) * ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))) * ((double) pow(((double) cbrt(t)), 3.0))))))))))))))));
	} else {
		double VAR_1;
		if ((t <= 2.2844510146699535e-95)) {
			VAR_1 = ((double) (l * ((double) (l * ((double) (2.0 / ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))) * ((double) (((double) sin(k)) * ((double) (((double) (2.0 * ((double) (((double) pow(((double) (1.0 / ((double) pow(((double) (((double) pow(1.0, 2.0)) * ((double) (1.0 / ((double) pow(t, 2.0)))))), 1.0)))), 1.0)) * ((double) (((double) sin(k)) / ((double) cos(k)))))))) + ((double) (((double) sin(k)) * ((double) (((double) (k * k)) / ((double) cos(k))))))))))))))))));
		} else {
			VAR_1 = ((double) (l * ((double) (((double) (1.0 / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (l * ((double) (2.0 / ((double) (((double) (((double) tan(k)) * ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))) * ((double) pow(((double) cbrt(t)), 3.0)))))) * ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))) * ((double) sin(k))))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -3.3013132694615146e-252

    1. Initial program 30.8

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

    2. Simplified31.5

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

    4. Applied add-cube-cbrt31.6

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

    5. Applied unpow-prod-down31.6

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

    6. Applied associate-*l*29.0

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

    7. Simplified26.3

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

    9. Applied sqr-pow26.3

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

    10. Applied associate-*l*22.2

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

    11. Simplified20.6

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

    13. Applied *-un-lft-identity20.6

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

    14. Applied times-frac20.4

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

    15. Applied associate-*l*18.3

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

    16. Simplified18.8

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

    18. Applied *-un-lft-identity18.8

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

    19. Applied times-frac19.0

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

    20. Applied associate-*r*17.9

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

    21. Simplified17.9

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

    if -3.3013132694615146e-252 < t < 2.2844510146699535e-95

    1. Initial program 62.7

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

    2. Simplified62.6

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

    4. Applied add-cube-cbrt62.6

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

    5. Applied unpow-prod-down62.6

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

    6. Applied associate-*l*56.4

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

    7. Simplified56.4

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

    9. Applied sqr-pow56.4

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

    10. Applied associate-*l*40.1

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

    11. Simplified40.1

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

    12. Taylor expanded around inf 32.4

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

    13. Simplified28.3

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

    if 2.2844510146699535e-95 < 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. Simplified24.7

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

    4. Applied add-cube-cbrt24.9

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

    5. Applied unpow-prod-down24.9

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

    6. Applied associate-*l*24.1

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

    7. Simplified20.3

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

    9. Applied sqr-pow20.3

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

    10. Applied associate-*l*18.8

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

    11. Simplified16.8

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

    13. Applied *-un-lft-identity16.8

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

    14. Applied times-frac16.7

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

    15. Applied associate-*l*14.4

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

    16. Simplified15.3

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

    18. Applied associate-*r*13.4

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.3013132694615146 \cdot 10^{-252}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}\right)\right)\\ \mathbf{elif}\;t \le 2.2844510146699535 \cdot 10^{-95}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left({\left(\frac{1}{{\left({1}^{2} \cdot \frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\ell \cdot \frac{2}{\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))