Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 12.2

Time: 3.0m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r7395564 = 2.0;
        double r7395565 = t;
        double r7395566 = 3.0;
        double r7395567 = pow(r7395565, r7395566);
        double r7395568 = l;
        double r7395569 = r7395568 * r7395568;
        double r7395570 = r7395567 / r7395569;
        double r7395571 = k;
        double r7395572 = sin(r7395571);
        double r7395573 = r7395570 * r7395572;
        double r7395574 = tan(r7395571);
        double r7395575 = r7395573 * r7395574;
        double r7395576 = 1.0;
        double r7395577 = r7395571 / r7395565;
        double r7395578 = pow(r7395577, r7395564);
        double r7395579 = r7395576 + r7395578;
        double r7395580 = r7395579 + r7395576;
        double r7395581 = r7395575 * r7395580;
        double r7395582 = r7395564 / r7395581;
        return r7395582;
}

↓

double f(double t, double l, double k) {
        double r7395583 = l;
        double r7395584 = 2.0;
        double r7395585 = 1.0;
        double r7395586 = k;
        double r7395587 = t;
        double r7395588 = r7395586 / r7395587;
        double r7395589 = 2.0;
        double r7395590 = pow(r7395588, r7395589);
        double r7395591 = fma(r7395584, r7395585, r7395590);
        double r7395592 = cbrt(r7395591);
        double r7395593 = r7395583 / r7395592;
        double r7395594 = cbrt(r7395589);
        double r7395595 = cbrt(r7395587);
        double r7395596 = 3.0;
        double r7395597 = pow(r7395595, r7395596);
        double r7395598 = cbrt(r7395597);
        double r7395599 = cbrt(r7395583);
        double r7395600 = r7395598 / r7395599;
        double r7395601 = r7395594 / r7395600;
        double r7395602 = tan(r7395586);
        double r7395603 = r7395601 / r7395602;
        double r7395604 = 1.0;
        double r7395605 = r7395592 * r7395592;
        double r7395606 = r7395604 / r7395605;
        double r7395607 = r7395603 * r7395606;
        double r7395608 = r7395598 * r7395598;
        double r7395609 = r7395604 / r7395608;
        double r7395610 = r7395607 * r7395609;
        double r7395611 = r7395593 * r7395610;
        double r7395612 = r7395595 * r7395595;
        double r7395613 = r7395596 / r7395584;
        double r7395614 = pow(r7395612, r7395613);
        double r7395615 = r7395614 / r7395599;
        double r7395616 = r7395594 / r7395615;
        double r7395617 = sin(r7395586);
        double r7395618 = cbrt(r7395617);
        double r7395619 = r7395616 / r7395618;
        double r7395620 = r7395611 * r7395619;
        double r7395621 = r7395618 * r7395618;
        double r7395622 = r7395616 / r7395621;
        double r7395623 = r7395620 * r7395622;
        return r7395623;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.8

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

2. Simplified 32.6

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

4. Applied add-cube-cbrt 32.7

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

5. Applied add-cube-cbrt 32.8

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

6. Applied unpow-prod-down 32.8

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

7. Applied times-frac 30.0

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

8. Applied add-cube-cbrt 30.0

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

9. Applied times-frac 30.0

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

10. Applied times-frac 24.6

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

11. Applied associate-*l* 22.5

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

13. Applied add-cube-cbrt 22.5

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

14. Applied sqr-pow 22.5

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

15. Applied times-frac 18.6

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

16. Applied times-frac 18.4

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

17. Applied times-frac 15.7

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

18. Applied associate-*l* 12.9

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

20. Applied add-cube-cbrt 12.9

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

21. Applied *-un-lft-identity 12.9

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

22. Applied times-frac 12.9

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

23. Applied associate-*r* 12.2

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

25. Applied *-un-lft-identity 12.2

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

26. Applied *-un-lft-identity 12.2

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

27. Applied add-cube-cbrt 12.2

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

28. Applied times-frac 12.2

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

29. Applied *-un-lft-identity 12.2

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

30. Applied times-frac 12.2

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

31. Applied times-frac 12.2

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

32. Applied associate-*l* 12.2

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

33. Final simplification 12.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))