Average Error: 48.3 → 2.5
Time: 1.8m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.505324626283498 \cdot 10^{-283} \lor \neg \left(\ell \le 3.19149458410540197 \cdot 10^{25}\right):\\ \;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\ell}^{1}}{{\left(\sin k\right)}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{1}}{\sin k}}{\sin k}\right)\\ \end{array}\]
\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}\;\ell \le -3.505324626283498 \cdot 10^{-283} \lor \neg \left(\ell \le 3.19149458410540197 \cdot 10^{25}\right):\\
\;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\ell}^{1}}{{\left(\sin k\right)}^{2}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{1}}{\sin k}}{\sin k}\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 (((l <= -3.505324626283498e-283) || !(l <= 3.191494584105402e+25))) {
		VAR = ((double) (2.0 * ((double) (((double) (((double) (((double) cos(k)) * l)) * ((double) pow(((double) (1.0 / ((double) pow(k, ((double) (2.0 / 2.0)))))), 1.0)))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))) / ((double) pow(t, 1.0)))), 1.0)) * ((double) (((double) pow(((double) (((double) cbrt(1.0)) / ((double) pow(k, ((double) (2.0 / 2.0)))))), 1.0)) * ((double) (((double) pow(l, 1.0)) / ((double) pow(((double) sin(k)), 2.0))))))))))));
	} else {
		VAR = ((double) (2.0 * ((double) (((double) (((double) (((double) cos(k)) * l)) * ((double) pow(((double) (1.0 / ((double) pow(k, ((double) (2.0 / 2.0)))))), 1.0)))) * ((double) (((double) (((double) pow(((double) (1.0 / ((double) (((double) pow(k, ((double) (2.0 / 2.0)))) * ((double) pow(t, 1.0)))))), 1.0)) * ((double) (((double) pow(l, 1.0)) / ((double) sin(k)))))) / ((double) sin(k))))))));
	}
	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 2 regimes

  2. if l < -3.505324626283498e-283 or 3.191494584105402e+25 < l

    1. Initial program 49.9

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

    2. Simplified43.0

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

    3. Taylor expanded around inf 27.1

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

    5. Applied *-un-lft-identity27.1

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

    6. Applied unpow-prod-down27.1

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

    7. Applied sqr-pow27.1

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

    8. Applied associate-*r*27.1

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

    9. Applied times-frac25.7

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

    10. Applied associate-*r*20.4

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

    11. Simplified20.4

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

    13. Applied sqr-pow20.4

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

    14. Applied associate-*l*16.0

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

    15. Applied *-un-lft-identity16.0

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

    16. Applied times-frac15.4

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

    17. Applied unpow-prod-down15.4

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

    18. Applied associate-*r*8.8

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

    19. Applied associate-*l*7.4

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

    20. Simplified7.4

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

    22. Applied *-commutative7.4

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

    23. Applied add-cube-cbrt7.4

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

    24. Applied times-frac6.9

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

    25. Applied unpow-prod-down6.9

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

    26. Applied associate-*l*2.7

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

    if -3.505324626283498e-283 < l < 3.191494584105402e+25

    1. Initial program 45.6

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

    2. Simplified36.4

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

    3. Taylor expanded around inf 13.4

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

    5. Applied *-un-lft-identity13.4

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

    6. Applied unpow-prod-down13.4

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

    7. Applied sqr-pow13.4

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

    8. Applied associate-*r*13.3

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

    9. Applied times-frac10.1

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

    10. Applied associate-*r*7.9

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

    11. Simplified7.9

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

    13. Applied sqr-pow7.9

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

    14. Applied associate-*l*6.6

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

    15. Applied *-un-lft-identity6.6

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

    16. Applied times-frac6.6

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

    17. Applied unpow-prod-down6.6

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

    18. Applied associate-*r*4.7

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

    19. Applied associate-*l*4.9

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

    20. Simplified4.9

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

    22. Applied unpow24.9

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

    23. Applied associate-/r*2.0

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

    24. Applied associate-*r/2.0

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.505324626283498 \cdot 10^{-283} \lor \neg \left(\ell \le 3.19149458410540197 \cdot 10^{25}\right):\\ \;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\ell}^{1}}{{\left(\sin k\right)}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(\cos k \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{1}}{\sin k}}{\sin k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))