Toniolo and Linder, Equation (10+)

Average Error: 31.9 → 21.1

Time: 18.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -4.4345504407829408 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;\ell \le -3.0650235703512473 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{1} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{elif}\;\ell \le 2.5281842998290106 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;\ell \le 2.43691444302226172 \cdot 10^{106}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

C

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

↓

double code(double t, double l, double k) {
	double VAR;
	if ((l <= -4.434550440782941e+123)) {
		VAR = ((((1.0 / pow((cbrt(t) * cbrt(t)), (3.0 / 2.0))) * (((2.0 / (pow(cbrt(t), 3.0) * sin(k))) * l) / pow((cbrt(t) * cbrt(t)), (3.0 / 2.0)))) / tan(k)) * (l / fma(2.0, 1.0, pow((k / t), 2.0))));
	} else {
		double VAR_1;
		if ((l <= -3.065023570351247e-188)) {
			VAR_1 = (((1.0 / pow((cbrt(t) * cbrt(t)), 3.0)) / 1.0) * ((((2.0 / (pow(cbrt(t), 3.0) * sin(k))) * l) / tan(k)) * (l / fma(2.0, 1.0, pow((k / t), 2.0)))));
		} else {
			double VAR_2;
			if ((l <= 2.5281842998290106e-199)) {
				VAR_2 = ((((1.0 / pow(cbrt(t), 3.0)) * (((2.0 / (pow(cbrt(t), 3.0) * sin(k))) * l) / pow(cbrt(t), 3.0))) / tan(k)) * (l / fma(2.0, 1.0, pow((k / t), 2.0))));
			} else {
				double VAR_3;
				if ((l <= 2.4369144430222617e+106)) {
					VAR_3 = (((1.0 / pow((cbrt(t) * cbrt(t)), 3.0)) / (cbrt(tan(k)) * cbrt(tan(k)))) * ((((2.0 / (pow(cbrt(t), 3.0) * sin(k))) * l) / cbrt(tan(k))) * (l / fma(2.0, 1.0, pow((k / t), 2.0)))));
				} else {
					VAR_3 = ((((1.0 / pow(cbrt(t), 3.0)) * (((2.0 / (pow(cbrt(t), 3.0) * sin(k))) * l) / pow(cbrt(t), 3.0))) / tan(k)) * (l / fma(2.0, 1.0, pow((k / t), 2.0))));
				}
				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

Derivation

1. Split input into 4 regimes

2. if l < -4.434550440782941e+123

   1. Initial program 59.0

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

   2. Simplified 58.6

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

   4. Applied *-un-lft-identity 58.6

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

   5. Applied times-frac 54.3

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

   6. Applied associate-*r* 47.1

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

   7. Simplified 47.1

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

   9. Applied add-cube-cbrt 47.3

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

   10. Applied unpow-prod-down 47.3

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

   11. Applied associate-*l* 45.9

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

   13. Applied *-un-lft-identity 45.9

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

   14. Applied times-frac 45.6

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

   15. Applied associate-*l* 39.5

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

   17. Applied sqr-pow 39.5

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

   18. Applied *-un-lft-identity 39.5

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

   19. Applied times-frac 38.8

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

   20. Applied associate-*l* 32.1

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

   21. Simplified 32.1

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

   if -4.434550440782941e+123 < l < -3.065023570351247e-188

   1. Initial program 24.9

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

   2. Simplified 25.9

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

   4. Applied *-un-lft-identity 25.9

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

   5. Applied times-frac 25.9

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

   6. Applied associate-*r* 25.4

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

   7. Simplified 24.4

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

   9. Applied add-cube-cbrt 24.7

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

   10. Applied unpow-prod-down 24.7

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

   11. Applied associate-*l* 23.1

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

   13. Applied *-un-lft-identity 23.1

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

   14. Applied times-frac 23.1

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

   15. Applied associate-*l* 21.3

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

   17. Applied *-un-lft-identity 21.3

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

   18. Applied times-frac 22.5

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

   19. Applied associate-*l* 20.9

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

   if -3.065023570351247e-188 < l < 2.5281842998290106e-199 or 2.4369144430222617e+106 < l

   1. Initial program 35.6

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

   2. Simplified 35.8

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

   4. Applied *-un-lft-identity 35.8

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

   5. Applied times-frac 34.6

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

   6. Applied associate-*r* 30.7

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

   7. Simplified 29.0

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

   9. Applied add-cube-cbrt 29.2

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

   10. Applied unpow-prod-down 29.2

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

   11. Applied associate-*l* 28.7

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

   13. Applied *-un-lft-identity 28.7

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

   14. Applied times-frac 28.5

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

   15. Applied associate-*l* 24.1

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

   17. Applied unpow-prod-down 24.1

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

   18. Applied *-un-lft-identity 24.1

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

   19. Applied times-frac 23.9

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

   20. Applied associate-*l* 21.0

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

   21. Simplified 21.0

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

   if 2.5281842998290106e-199 < l < 2.4369144430222617e+106

   1. Initial program 23.5

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

   2. Simplified 23.6

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

   4. Applied *-un-lft-identity 23.6

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

   5. Applied times-frac 23.6

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

   6. Applied associate-*r* 23.2

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

   7. Simplified 21.7

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

   9. Applied add-cube-cbrt 22.0

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

   10. Applied unpow-prod-down 22.0

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

   11. Applied associate-*l* 20.5

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

   13. Applied *-un-lft-identity 20.5

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

   14. Applied times-frac 20.5

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

   15. Applied associate-*l* 19.3

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

   17. Applied add-cube-cbrt 19.4

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

   18. Applied times-frac 19.1

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

   19. Applied associate-*l* 17.1

       \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.4345504407829408 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;\ell \le -3.0650235703512473 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{1} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{elif}\;\ell \le 2.5281842998290106 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;\ell \le 2.43691444302226172 \cdot 10^{106}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(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))))