Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 17.1

Time: 16.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -8.11827642384914891 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\frac{1}{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \le 7.2711535224133574 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \left(\left(\frac{k}{\frac{\ell}{{t}^{1}}} + \frac{{k}^{3}}{\frac{\ell}{t}} \cdot \frac{-1}{6}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\\ \end{array}\]

C

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 <= -8.118276423849149e-147)) {
		VAR = ((double) (2.0 / ((double) (((double) (((double) (((double) (((double) sqrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) / ((double) (1.0 / ((double) sqrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))))))) * ((double) (((double) (((double) sin(k)) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)))) * ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))) / l)))))) * ((double) tan(k)))) * ((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0))))))))));
	} else {
		double VAR_1;
		if ((t <= 7.271153522413357e-72)) {
			VAR_1 = ((double) (2.0 / ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))) * ((double) (((double) tan(k)) * ((double) (((double) (1.0 / ((double) (((double) (((double) cbrt(l)) * ((double) cbrt(l)))) / ((double) (((double) cbrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) cbrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))))))))) * ((double) (((double) (((double) (k / ((double) (l / ((double) pow(t, 1.0)))))) + ((double) (((double) (((double) pow(k, 3.0)) / ((double) (l / t)))) * -0.16666666666666666)))) * ((double) (((double) cbrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))) / ((double) cbrt(l))))))))))))))));
		} else {
			VAR_1 = ((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)), ((double) (3.0 / 2.0)))) / ((double) (((double) (((double) cbrt(l)) * ((double) cbrt(l)))) / ((double) sqrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))))))) * ((double) (((double) (((double) sin(k)) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) / l)))) * ((double) (((double) sqrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (((double) pow(((double) cbrt(t)), ((double) (3.0 / 2.0)))) / ((double) cbrt(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 t < -8.11827642384914891e-147

   1. Initial program 26.1

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

   3. Applied add-cube-cbrt 26.2

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

   4. Applied unpow-prod-down 26.2

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

   5. Applied times-frac 18.3

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

   6. Applied associate-*l* 16.2

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

   7. Simplified 16.2

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

   9. Applied sqr-pow 16.2

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

   10. Applied associate-/l* 12.0

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

   12. Applied add-sqr-sqrt 12.0

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

   13. Applied *-un-lft-identity 12.0

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

   14. Applied times-frac 12.0

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

   15. Applied add-sqr-sqrt 12.0

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

   16. Applied times-frac 12.0

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

   17. Applied associate-*l* 11.6

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

   18. Simplified 11.6

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

   if -8.11827642384914891e-147 < t < 7.2711535224133574e-72

   1. Initial program 60.8

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

   3. Applied add-cube-cbrt 60.8

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

   4. Applied unpow-prod-down 60.8

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

   5. Applied times-frac 55.2

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

   6. Applied associate-*l* 55.0

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

   7. Simplified 55.0

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

   9. Applied sqr-pow 55.0

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

   10. Applied associate-/l* 45.8

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

   12. Applied add-cube-cbrt 45.8

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

   13. Applied add-cube-cbrt 45.8

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

   14. Applied times-frac 45.8

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

   15. Applied *-un-lft-identity 45.8

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

   16. Applied times-frac 45.8

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

   17. Applied associate-*l* 45.8

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

   18. Simplified 45.8

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

   19. Taylor expanded around 0 50.5

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

   20. Simplified 42.2

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

   if 7.2711535224133574e-72 < t

   1. Initial program 22.7

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

   3. Applied add-cube-cbrt 22.9

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

   4. Applied unpow-prod-down 22.9

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

   5. Applied times-frac 17.0

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

   6. Applied associate-*l* 14.5

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

   7. Simplified 14.5

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

   9. Applied sqr-pow 14.5

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

   10. Applied associate-/l* 9.4

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

   12. Applied add-sqr-sqrt 9.4

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

   13. Applied add-cube-cbrt 9.4

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

   14. Applied times-frac 9.4

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

   15. Applied unpow-prod-down 9.4

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

   16. Applied times-frac 9.4

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

   17. Applied associate-*l* 8.2

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

   18. Simplified 8.2

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.11827642384914891 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\frac{1}{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \le 7.2711535224133574 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \left(\left(\frac{k}{\frac{\ell}{{t}^{1}}} + \frac{{k}^{3}}{\frac{\ell}{t}} \cdot \frac{-1}{6}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))