Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 1.2

Time: 5.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.19264518939479538 \cdot 10^{-95} \lor \neg \left(k \le 3.0397255266168896 \cdot 10^{-158}\right):\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(-{\ell}^{1}\right)}{\frac{1}{{\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1}}}}{\frac{\sin k}{\left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \cos k\right) \cdot \ell} \cdot \left(-{\left(\sin k\right)}^{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(-\sqrt[3]{{\ell}^{1}} \cdot \sqrt[3]{{\ell}^{1}}\right)}{\frac{1}{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k}} \cdot \frac{\sqrt[3]{{\ell}^{1}}}{\frac{\sin k}{\ell} \cdot \left(-{\left(\sin k\right)}^{1}\right)}\right)\\ \end{array}\]

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 temp;
	if (((k <= -1.1926451893947954e-95) || !(k <= 3.0397255266168896e-158))) {
		temp = (2.0 * (((pow((1.0 / pow(k, (2.0 / 2.0))), 1.0) * -pow(l, 1.0)) / (1.0 / pow((sqrt(1.0) / pow(t, 1.0)), 1.0))) / ((sin(k) / ((pow((sqrt(1.0) / pow(k, (2.0 / 2.0))), 1.0) * cos(k)) * l)) * -pow(sin(k), 1.0))));
	} else {
		temp = (2.0 * (((pow((1.0 / pow(k, (2.0 / 2.0))), 1.0) * -(cbrt(pow(l, 1.0)) * cbrt(pow(l, 1.0)))) / (1.0 / (pow((1.0 / (pow(k, (2.0 / 2.0)) * pow(t, 1.0))), 1.0) * cos(k)))) * (cbrt(pow(l, 1.0)) / ((sin(k) / l) * -pow(sin(k), 1.0)))));
	}
	return temp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -1.1926451893947954e-95 or 3.0397255266168896e-158 < k

   1. Initial program 47.4

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

   2. Simplified 38.8

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

   3. Taylor expanded around inf 19.7

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

   5. Applied sqr-pow 19.7

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

   6. Applied sqr-pow 19.7

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

   7. Applied associate-*r* 19.7

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

   8. Applied times-frac 18.2

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

   9. Applied associate-*r* 14.0

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

   10. Simplified 14.0

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

   12. Applied frac-2neg 14.0

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

   13. Applied sqr-pow 14.0

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

   14. Applied associate-*l* 10.6

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

   15. Applied *-un-lft-identity 10.6

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

   16. Applied times-frac 10.2

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

   17. Applied unpow-prod-down 10.2

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

   18. Applied associate-*l* 10.2

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

   19. Applied associate-*l* 5.8

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

   20. Applied associate-/l* 5.8

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

   21. Applied frac-times 5.0

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

   22. Simplified 5.0

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

   23. Simplified 5.0

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

   25. Applied *-commutative 5.0

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

   26. Applied add-sqr-sqrt 5.0

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

   27. Applied times-frac 4.7

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

   28. Applied unpow-prod-down 4.7

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

   29. Applied associate-*l* 4.7

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

   30. Applied associate-*l* 1.1

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

   31. Applied *-un-lft-identity 1.1

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

   32. Applied times-frac 0.8

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

   33. Applied associate-*l* 1.0

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

   34. Applied associate-/r* 1.0

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

   if -1.1926451893947954e-95 < k < 3.0397255266168896e-158

   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. Simplified 64.0

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

   3. Taylor expanded around inf 52.5

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

   5. Applied sqr-pow 52.5

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

   6. Applied sqr-pow 52.5

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

   7. Applied associate-*r* 52.5

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

   8. Applied times-frac 42.1

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

   9. Applied associate-*r* 40.3

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

   10. Simplified 40.2

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

   12. Applied frac-2neg 40.2

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

   13. Applied sqr-pow 40.2

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

   14. Applied associate-*l* 15.0

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

   15. Applied *-un-lft-identity 15.0

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

   16. Applied times-frac 15.0

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

   17. Applied unpow-prod-down 15.0

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

   18. Applied associate-*l* 15.0

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

   19. Applied associate-*l* 14.9

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

   20. Applied associate-/l* 14.9

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

   21. Applied frac-times 14.1

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

   22. Simplified 14.1

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

   23. Simplified 14.1

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

   25. Applied *-un-lft-identity 14.1

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

   26. Applied times-frac 7.2

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

   27. Applied associate-*l* 7.1

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

   28. Applied add-cube-cbrt 7.9

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

   29. Applied distribute-lft-neg-in 7.9

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

   30. Applied associate-*r* 7.9

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

   31. Applied times-frac 4.2

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.19264518939479538 \cdot 10^{-95} \lor \neg \left(k \le 3.0397255266168896 \cdot 10^{-158}\right):\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(-{\ell}^{1}\right)}{\frac{1}{{\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1}}}}{\frac{\sin k}{\left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \cos k\right) \cdot \ell} \cdot \left(-{\left(\sin k\right)}^{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(-\sqrt[3]{{\ell}^{1}} \cdot \sqrt[3]{{\ell}^{1}}\right)}{\frac{1}{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k}} \cdot \frac{\sqrt[3]{{\ell}^{1}}}{\frac{\sin k}{\ell} \cdot \left(-{\left(\sin k\right)}^{1}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(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))))