Toniolo and Linder, Equation (10+)

Average Error: 31.9 → 11.3

Time: 1.9m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r4528608 = 2.0;
        double r4528609 = t;
        double r4528610 = 3.0;
        double r4528611 = pow(r4528609, r4528610);
        double r4528612 = l;
        double r4528613 = r4528612 * r4528612;
        double r4528614 = r4528611 / r4528613;
        double r4528615 = k;
        double r4528616 = sin(r4528615);
        double r4528617 = r4528614 * r4528616;
        double r4528618 = tan(r4528615);
        double r4528619 = r4528617 * r4528618;
        double r4528620 = 1.0;
        double r4528621 = r4528615 / r4528609;
        double r4528622 = pow(r4528621, r4528608);
        double r4528623 = r4528620 + r4528622;
        double r4528624 = r4528623 + r4528620;
        double r4528625 = r4528619 * r4528624;
        double r4528626 = r4528608 / r4528625;
        return r4528626;
}

↓

double f(double t, double l, double k) {
        double r4528627 = 1.0;
        double r4528628 = t;
        double r4528629 = cbrt(r4528628);
        double r4528630 = r4528627 / r4528629;
        double r4528631 = k;
        double r4528632 = cos(r4528631);
        double r4528633 = r4528632 * r4528632;
        double r4528634 = sin(r4528631);
        double r4528635 = r4528633 * r4528634;
        double r4528636 = r4528631 / r4528628;
        double r4528637 = r4528636 * r4528634;
        double r4528638 = r4528636 * r4528637;
        double r4528639 = r4528638 * r4528632;
        double r4528640 = r4528632 * r4528634;
        double r4528641 = r4528639 + r4528640;
        double r4528642 = r4528632 * r4528641;
        double r4528643 = r4528635 + r4528642;
        double r4528644 = cbrt(r4528643);
        double r4528645 = r4528644 * r4528644;
        double r4528646 = r4528633 * r4528632;
        double r4528647 = cbrt(r4528646);
        double r4528648 = r4528647 * r4528647;
        double r4528649 = l;
        double r4528650 = r4528649 / r4528628;
        double r4528651 = cbrt(r4528634);
        double r4528652 = r4528651 * r4528651;
        double r4528653 = r4528650 / r4528652;
        double r4528654 = r4528629 / r4528653;
        double r4528655 = r4528627 / r4528654;
        double r4528656 = r4528648 * r4528655;
        double r4528657 = r4528645 / r4528656;
        double r4528658 = r4528630 / r4528657;
        double r4528659 = 2.0;
        double r4528660 = r4528650 / r4528651;
        double r4528661 = r4528629 / r4528660;
        double r4528662 = r4528659 / r4528661;
        double r4528663 = tan(r4528631);
        double r4528664 = r4528663 * r4528636;
        double r4528665 = r4528636 * r4528664;
        double r4528666 = r4528663 + r4528665;
        double r4528667 = r4528666 + r4528663;
        double r4528668 = cbrt(r4528667);
        double r4528669 = r4528662 / r4528668;
        double r4528670 = r4528658 * r4528669;
        return r4528670;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 31.9

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

2. Simplified 18.4

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

4. Applied add-cube-cbrt 18.5

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

5. Applied add-cube-cbrt 18.6

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

6. Applied times-frac 17.1

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

7. Applied add-cube-cbrt 17.1

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

8. Applied times-frac 16.5

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

9. Applied *-un-lft-identity 16.5

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

10. Applied times-frac 16.3

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

11. Applied times-frac 11.8

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

13. Applied *-un-lft-identity 11.8

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

14. Applied times-frac 11.8

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

15. Applied *-un-lft-identity 11.8

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

16. Applied times-frac 11.8

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

17. Applied associate-/l* 11.3

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

19. Applied tan-quot 11.3

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

20. Applied tan-quot 11.3

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

21. Applied associate-*l/ 11.3

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

22. Applied associate-*l/ 11.3

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

23. Applied tan-quot 11.3

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

24. Applied frac-add 11.3

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

25. Applied frac-add 11.3

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

26. Applied cbrt-div 11.3

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

27. Applied tan-quot 11.3

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

28. Applied tan-quot 11.3

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

29. Applied associate-*l/ 11.3

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

30. Applied associate-*l/ 11.3

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

31. Applied tan-quot 11.3

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

32. Applied frac-add 11.3

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

33. Applied frac-add 11.3

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

34. Applied cbrt-div 11.3

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

35. Applied frac-times 11.3

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

36. Applied associate-/l/ 11.3

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

37. Final simplification 11.3

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

Reproduce

herbie shell --seed 2019134 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))