Average Error: 47.9 → 30.8
Time: 1.3min
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.43511274737075789 \cdot 10^{-108}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{\sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)\right)\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le -5.07998190631281 \cdot 10^{-309}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{{-1}^{5}}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot \left({\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}\right)}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 5.993295287220876 \cdot 10^{-135}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{\sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)\right)\right)\right) \cdot \sin k}\\ \end{array}\]
\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 -2.43511274737075789 \cdot 10^{-108}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{\sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)\right)\right)\right) \cdot \sin k}\\

\mathbf{elif}\;t \le -5.07998190631281 \cdot 10^{-309}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{{-1}^{5}}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot \left({\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}\right)}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \sin k}\\

\mathbf{elif}\;t \le 5.993295287220876 \cdot 10^{-135}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) \cdot \sin k}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{\sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)\right)\right)\right) \cdot \sin k}\\

\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 <= -2.435112747370758e-108)) {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (1.0 / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (k / ((double) cbrt(t)))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))))) * ((double) sin(k))))));
	} else {
		double VAR_1;
		if ((t <= -5.07998190631281e-309)) {
			VAR_1 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (((double) pow(-1.0, 5.0)) / ((double) (((double) pow(((double) cbrt(-1.0)), 3.0)) * ((double) (((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / k)))))))))), 1.0)) * ((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / t)))))))))), 1.0)))))))), 1.0)) * ((double) (((double) sin(k)) / ((double) cos(k)))))))) * ((double) sin(k))))));
		} else {
			double VAR_2;
			if ((t <= 5.993295287220876e-135)) {
				VAR_2 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) sin(k)) / ((double) cos(k)))) * ((double) pow(((double) (1.0 / ((double) (((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(((double) (1.0 / t)))) + ((double) log(1.0)))))))), 1.0)) * ((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (1.0 / k)))))))))), 1.0)))))), 1.0)))))) * ((double) sin(k))))));
			} else {
				VAR_2 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (1.0 / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (k / ((double) cbrt(t)))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))))) * ((double) sin(k))))));
			}
			VAR_1 = VAR_2;
		}
		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 < -2.43511274737075789e-108 or 5.993295287220876e-135 < t

    1. Initial program 42.9

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

    2. Simplified32.2

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

    4. Applied sqr-pow32.2

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

    5. Applied associate-*l*27.6

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

    7. Applied add-cube-cbrt27.6

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

    8. Applied *-un-lft-identity27.6

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

    9. Applied times-frac27.6

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

    10. Applied unpow-prod-down27.6

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

    11. Applied associate-*l*27.7

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

    13. Applied div-inv27.7

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

    14. Applied unpow-prod-down27.7

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

    15. Applied associate-*l*27.2

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

    if -2.43511274737075789e-108 < t < -5.07998190631281e-309

    1. Initial program 63.9

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

    2. Simplified63.9

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

    4. Applied sqr-pow63.9

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

    5. Applied associate-*l*63.9

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

    7. Applied add-cube-cbrt63.9

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

    8. Applied *-un-lft-identity63.9

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

    9. Applied times-frac63.9

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

    10. Applied unpow-prod-down63.9

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

    11. Applied associate-*l*63.9

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

    13. Applied div-inv63.9

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

    14. Applied unpow-prod-down63.9

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

    15. Applied associate-*l*63.9

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

    16. Taylor expanded around -inf 42.4

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left({\left(\frac{{-1}^{5}}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot \left({\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}\right)}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)}\right) \cdot \sin k}\]

    if -5.07998190631281e-309 < t < 5.993295287220876e-135

    1. Initial program 64.0

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

    2. Simplified64.0

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

    4. Applied sqr-pow64.0

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

    5. Applied associate-*l*64.0

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

    7. Applied add-cube-cbrt64.0

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

    8. Applied *-un-lft-identity64.0

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

    9. Applied times-frac64.0

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

    10. Applied unpow-prod-down64.0

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

    11. Applied associate-*l*64.0

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

    13. Applied div-inv64.0

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

    14. Applied unpow-prod-down64.0

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

    15. Applied associate-*l*64.0

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

    16. Taylor expanded around inf 42.4

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

  4. Final simplification30.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.43511274737075789 \cdot 10^{-108}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{\sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)\right)\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le -5.07998190631281 \cdot 10^{-309}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{{-1}^{5}}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot \left({\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}\right)}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 5.993295287220876 \cdot 10^{-135}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{\sqrt[3]{t}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)\right)\right)\right) \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(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))))