Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 12.4

Time: 2.9m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r355913 = 2.0;
        double r355914 = t;
        double r355915 = 3.0;
        double r355916 = pow(r355914, r355915);
        double r355917 = l;
        double r355918 = r355917 * r355917;
        double r355919 = r355916 / r355918;
        double r355920 = k;
        double r355921 = sin(r355920);
        double r355922 = r355919 * r355921;
        double r355923 = tan(r355920);
        double r355924 = r355922 * r355923;
        double r355925 = 1.0;
        double r355926 = r355920 / r355914;
        double r355927 = pow(r355926, r355913);
        double r355928 = r355925 + r355927;
        double r355929 = r355928 + r355925;
        double r355930 = r355924 * r355929;
        double r355931 = r355913 / r355930;
        return r355931;
}

↓

double f(double t, double l, double k) {
        double r355932 = 2.0;
        double r355933 = cbrt(r355932);
        double r355934 = t;
        double r355935 = cbrt(r355934);
        double r355936 = 3.0;
        double r355937 = pow(r355935, r355936);
        double r355938 = l;
        double r355939 = cbrt(r355938);
        double r355940 = r355937 / r355939;
        double r355941 = r355933 / r355940;
        double r355942 = k;
        double r355943 = sin(r355942);
        double r355944 = cbrt(r355943);
        double r355945 = r355944 * r355944;
        double r355946 = r355941 / r355945;
        double r355947 = r355941 / r355944;
        double r355948 = tan(r355942);
        double r355949 = r355940 * r355948;
        double r355950 = r355933 / r355949;
        double r355951 = cbrt(r355950);
        double r355952 = 2.0;
        double r355953 = 1.0;
        double r355954 = r355942 / r355934;
        double r355955 = pow(r355954, r355932);
        double r355956 = fma(r355952, r355953, r355955);
        double r355957 = cbrt(r355956);
        double r355958 = r355957 / r355939;
        double r355959 = r355951 / r355958;
        double r355960 = r355951 * r355951;
        double r355961 = r355958 * r355958;
        double r355962 = r355960 / r355961;
        double r355963 = r355959 * r355962;
        double r355964 = r355947 * r355963;
        double r355965 = r355946 * r355964;
        return r355965;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 33.1

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

2. Simplified 32.7

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

4. Applied add-cube-cbrt 32.8

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

5. Applied add-cube-cbrt 32.9

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

6. Applied unpow-prod-down 32.9

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

7. Applied times-frac 30.2

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

8. Applied add-cube-cbrt 30.2

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

9. Applied times-frac 30.1

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

10. Applied times-frac 24.8

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

11. Applied associate-*l* 22.8

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

12. Simplified 22.7

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

14. Applied add-cube-cbrt 22.7

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

15. Applied unpow-prod-down 22.7

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

16. Applied times-frac 18.5

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

17. Applied times-frac 18.3

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

18. Applied times-frac 15.6

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

19. Applied associate-*l* 12.5

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

20. Simplified 13.1

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

22. Applied add-cube-cbrt 13.1

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

23. Applied add-cube-cbrt 13.1

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

24. Applied times-frac 13.1

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

25. Applied add-cube-cbrt 13.1

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

26. Applied times-frac 12.4

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

27. Simplified 12.4

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

28. Final simplification 12.4

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

Reproduce

herbie shell --seed 2019174 +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))))