Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 27.8

Time: 47.5s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -7.4653916594980344 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(2 \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{{\left(e^{2 \cdot \left(\log \left(\frac{-1}{k}\right) + \log 1\right)}\right)}^{1}}{{\left(\sqrt[3]{-1}\right)}^{6}}\right)}^{1}\right)\right)\\ \mathbf{elif}\;k \le 21.185942955599955:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1}\right) - 0.333333333333333315 \cdot \left({\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1} \cdot {\ell}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\\ \end{array}\]

C

double code(double t, double l, double k) {
	return (2.0 / ((double) (((double) (((double) ((((double) pow(t, 3.0)) / ((double) (l * l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow((k / t), 2.0)))) - 1.0)))));
}

↓

double code(double t, double l, double k) {
	double VAR;
	if ((k <= -7.465391659498034e-155)) {
		VAR = ((double) ((1.0 / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))) * ((double) (2.0 * ((double) ((((double) (((double) cos(k)) * ((double) pow(l, 2.0)))) / ((double) pow(((double) sin(k)), 2.0))) * ((double) pow((((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log((-1.0 / k))) + ((double) log(1.0)))))))), 1.0)) / ((double) pow(((double) cbrt(-1.0)), 6.0))), 1.0))))))));
	} else {
		double VAR_1;
		if ((k <= 21.185942955599955)) {
			VAR_1 = ((double) ((1.0 / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))) * ((double) (((double) (2.0 * ((double) ((((double) pow(l, 2.0)) / ((double) pow(k, 2.0))) * ((double) pow((1.0 / ((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log(1.0)) + ((double) log(k)))))))), 1.0))), 1.0)))))) - ((double) (0.3333333333333333 * ((double) (((double) pow((1.0 / ((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log(1.0)) + ((double) log(k)))))))), 1.0))), 1.0)) * ((double) pow(l, 2.0))))))))));
		} else {
			VAR_1 = (((double) ((2.0 / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))) * l)) / ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))) * (((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) (((double) pow((k / t), 2.0)) * ((double) (((double) sin(k)) * ((double) tan(k)))))))) / l))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -7.4653916594980344e-155

   1. Initial program 47.6

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

   2. Simplified 40.1

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

   4. Applied add-cube-cbrt 40.2

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

   5. Applied unpow-prod-down 40.2

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

   6. Applied div-inv 40.2

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

   7. Applied times-frac 37.7

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

   8. Applied associate-*l* 37.1

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

   9. Simplified 36.9

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

   11. Applied sqr-pow 36.9

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

   12. Applied *-un-lft-identity 36.9

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

   13. Applied times-frac 36.8

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

   14. Applied associate-*l* 31.5

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

   15. Taylor expanded around -inf 27.9

       \[\leadsto \frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \color{blue}{\left(2 \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{{\left(e^{2 \cdot \left(\log \left(\frac{-1}{k}\right) + \log 1\right)}\right)}^{1}}{{\left(\sqrt[3]{-1}\right)}^{6}}\right)}^{1}\right)\right)}\]

   if -7.4653916594980344e-155 < k < 21.185942955599955

   1. Initial program 61.5

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

   2. Simplified 57.5

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

   4. Applied add-cube-cbrt 57.6

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

   5. Applied unpow-prod-down 57.6

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

   6. Applied div-inv 57.6

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

   7. Applied times-frac 56.5

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

   8. Applied associate-*l* 56.3

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

   9. Simplified 55.9

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

   11. Applied sqr-pow 55.9

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

   12. Applied *-un-lft-identity 55.9

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

   13. Applied times-frac 55.9

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

   14. Applied associate-*l* 55.1

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

   15. Taylor expanded around 0 47.0

       \[\leadsto \frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \color{blue}{\left(2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1}\right) - 0.333333333333333315 \cdot \left({\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1} \cdot {\ell}^{2}\right)\right)}\]

   if 21.185942955599955 < k

   1. Initial program 44.0

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

   2. Simplified 35.4

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

   4. Applied add-cube-cbrt 35.5

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

   5. Applied unpow-prod-down 35.5

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

   6. Applied div-inv 35.5

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

   7. Applied times-frac 33.0

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

   8. Applied associate-*l* 32.9

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

   9. Simplified 32.7

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

   11. Applied sqr-pow 32.7

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

   12. Applied *-un-lft-identity 32.7

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

   13. Applied times-frac 32.7

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

   14. Applied associate-*l* 26.5

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

   16. Applied associate-/l* 23.7

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

   18. Applied associate-*r/ 21.7

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

   19. Applied frac-times 20.9

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

   20. Simplified 20.9

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

4. Final simplification 27.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -7.4653916594980344 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(2 \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{{\left(e^{2 \cdot \left(\log \left(\frac{-1}{k}\right) + \log 1\right)}\right)}^{1}}{{\left(\sqrt[3]{-1}\right)}^{6}}\right)}^{1}\right)\right)\\ \mathbf{elif}\;k \le 21.185942955599955:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1}\right) - 0.333333333333333315 \cdot \left({\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1} \cdot {\ell}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))