Average Error: 32.4 → 16.5
Time: 14.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.3078668923076751 \cdot 10^{-190}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}\right)\right)\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \le 1.3337804818741451 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \le 2.67098617363212593 \cdot 10^{169}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.45585033432551279 \cdot 10^{227}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\tan k \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)\right)\right)\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\\ \end{array}\]
\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 -1.3078668923076751 \cdot 10^{-190}:\\
\;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \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)\right)\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\

\mathbf{elif}\;t \le 1.3337804818741451 \cdot 10^{-151}:\\
\;\;\;\;0\\

\mathbf{elif}\;t \le 2.67098617363212593 \cdot 10^{169}:\\
\;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)\right)\right)}\\

\mathbf{elif}\;t \le 1.45585033432551279 \cdot 10^{227}:\\
\;\;\;\;\frac{2}{\frac{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\tan k \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)\right)\right)\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell}}\right)\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 <= -1.3078668923076751e-190)) {
		VAR = ((double) (2.0 / ((double) (((double) (((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / ((double) (l / ((double) pow(((double) cbrt(t)), 3.0)))))) * ((double) (((double) cbrt(((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)) * ((double) sin(k)))))) * ((double) (((double) cbrt(((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)) * ((double) sin(k)))))) * ((double) cbrt(((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)) * ((double) sin(k)))))))))))) * ((double) tan(k)))) * ((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0))))))))));
	} else {
		double VAR_1;
		if ((t <= 1.3337804818741451e-151)) {
			VAR_1 = 0.0;
		} else {
			double VAR_2;
			if ((t <= 2.670986173632126e+169)) {
				VAR_2 = ((double) (2.0 / ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))) * ((double) (((double) tan(k)) * ((double) (((double) (((double) pow(t, ((double) (3.0 / 2.0)))) / l)) * ((double) (((double) sin(k)) * ((double) (((double) pow(t, ((double) (3.0 / 2.0)))) / l))))))))))));
			} else {
				double VAR_3;
				if ((t <= 1.4558503343255128e+227)) {
					VAR_3 = ((double) (2.0 / ((double) (((double) (((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) (((double) tan(k)) * ((double) (((double) cbrt(((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)) * ((double) sin(k)))))) * ((double) (((double) cbrt(((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)) * ((double) sin(k)))))) * ((double) cbrt(((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) sin(k)))))))))))))))) / ((double) (((double) (l / ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) cbrt(l))))))));
				} else {
					VAR_3 = ((double) (2.0 / ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))) * ((double) (((double) tan(k)) * ((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) / ((double) (l / ((double) pow(((double) cbrt(t)), 3.0)))))) * ((double) (((double) (((double) pow(((double) cbrt(((double) sqrt(t)))), 3.0)) / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))) * ((double) (((double) sin(k)) * ((double) (((double) pow(((double) cbrt(((double) sqrt(t)))), 3.0)) / ((double) cbrt(l))))))))))))))));
				}
				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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if t < -1.3078668923076751e-190

    1. Initial program 27.7

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

    3. Applied add-cube-cbrt27.8

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

    4. Applied unpow-prod-down27.8

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

    5. Applied times-frac19.7

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

    6. Applied associate-*l*17.9

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

    8. Applied unpow-prod-down17.9

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

    9. Applied associate-/l*13.0

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

    11. Applied add-cube-cbrt13.1

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

    if -1.3078668923076751e-190 < t < 1.3337804818741451e-151

    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. Using strategy rm

    3. Applied add-cube-cbrt64.0

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

    4. Applied unpow-prod-down64.0

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

    5. Applied times-frac63.0

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

    6. Applied associate-*l*63.0

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

    8. Applied unpow-prod-down63.0

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

    9. Applied associate-/l*51.8

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

    10. Taylor expanded around inf 41.6

      \[\leadsto \color{blue}{0}\]

    if 1.3337804818741451e-151 < t < 2.67098617363212593e169

    1. Initial program 29.6

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

    3. Applied sqr-pow29.6

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

    4. Applied times-frac17.5

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

    5. Applied associate-*l*13.7

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

    if 2.67098617363212593e169 < t < 1.45585033432551279e227

    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. Using strategy rm

    3. Applied add-cube-cbrt22.6

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

    4. Applied unpow-prod-down22.6

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

    5. Applied times-frac21.6

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

    6. Applied associate-*l*21.6

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

    8. Applied unpow-prod-down21.6

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

    9. Applied associate-/l*7.0

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

    11. Applied add-cube-cbrt7.1

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

    13. Applied associate-*l/7.1

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

    14. Applied cbrt-div7.1

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

    15. Applied associate-*r/7.0

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

    16. Applied frac-times5.3

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

    17. Applied associate-*l/2.0

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

    18. Applied associate-*l/2.7

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

    19. Simplified6.2

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

    if 1.45585033432551279e227 < t

    1. Initial program 18.6

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

    3. Applied add-cube-cbrt18.6

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

    4. Applied unpow-prod-down18.6

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

    5. Applied times-frac13.6

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

    6. Applied associate-*l*13.6

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

    8. Applied unpow-prod-down13.6

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

    9. Applied associate-/l*7.1

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

    11. Applied add-cube-cbrt7.1

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

    12. Applied add-sqr-sqrt7.1

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

    13. Applied cbrt-prod7.1

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

    14. Applied unpow-prod-down7.1

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

    15. Applied times-frac7.1

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

    16. Applied associate-*l*7.1

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

    17. Simplified7.1

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \color{blue}{\left(\sin k \cdot \frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell}}\right)}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3078668923076751 \cdot 10^{-190}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}\right)\right)\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \le 1.3337804818741451 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \le 2.67098617363212593 \cdot 10^{169}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.45585033432551279 \cdot 10^{227}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\tan k \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)\right)\right)\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{\sqrt{t}}\right)}^{3}}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))))