Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 12.5

Time: 1.2m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r163802 = 2.0;
        double r163803 = t;
        double r163804 = 3.0;
        double r163805 = pow(r163803, r163804);
        double r163806 = l;
        double r163807 = r163806 * r163806;
        double r163808 = r163805 / r163807;
        double r163809 = k;
        double r163810 = sin(r163809);
        double r163811 = r163808 * r163810;
        double r163812 = tan(r163809);
        double r163813 = r163811 * r163812;
        double r163814 = 1.0;
        double r163815 = r163809 / r163803;
        double r163816 = pow(r163815, r163802);
        double r163817 = r163814 + r163816;
        double r163818 = r163817 + r163814;
        double r163819 = r163813 * r163818;
        double r163820 = r163802 / r163819;
        return r163820;
}

↓

double f(double t, double l, double k) {
        double r163821 = 2.0;
        double r163822 = sqrt(r163821);
        double r163823 = sqrt(r163822);
        double r163824 = t;
        double r163825 = cbrt(r163824);
        double r163826 = r163825 * r163825;
        double r163827 = 3.0;
        double r163828 = 2.0;
        double r163829 = r163827 / r163828;
        double r163830 = pow(r163826, r163829);
        double r163831 = l;
        double r163832 = cbrt(r163831);
        double r163833 = r163830 / r163832;
        double r163834 = k;
        double r163835 = sin(r163834);
        double r163836 = cbrt(r163835);
        double r163837 = r163833 * r163836;
        double r163838 = r163823 / r163837;
        double r163839 = r163832 * r163822;
        double r163840 = pow(r163825, r163827);
        double r163841 = r163839 / r163840;
        double r163842 = tan(r163834);
        double r163843 = r163841 / r163842;
        double r163844 = 1.0;
        double r163845 = r163834 / r163824;
        double r163846 = pow(r163845, r163821);
        double r163847 = fma(r163828, r163844, r163846);
        double r163848 = sqrt(r163847);
        double r163849 = r163843 / r163848;
        double r163850 = r163831 / r163848;
        double r163851 = r163849 * r163850;
        double r163852 = r163838 * r163851;
        double r163853 = r163823 / r163833;
        double r163854 = r163836 * r163836;
        double r163855 = r163853 / r163854;
        double r163856 = r163852 * r163855;
        return r163856;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.3

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

2. Simplified 32.5

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

4. Applied add-cube-cbrt 32.5

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

5. Applied add-cube-cbrt 32.6

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

6. Applied unpow-prod-down 32.6

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

7. Applied times-frac 29.9

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

8. Applied add-sqr-sqrt 29.9

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

9. Applied times-frac 29.8

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

10. Applied times-frac 24.6

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

11. Applied associate-*l* 22.4

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

12. Simplified 22.4

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

14. Applied add-cube-cbrt 22.4

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

15. Applied sqr-pow 22.4

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

16. Applied times-frac 18.5

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

17. Applied add-sqr-sqrt 18.5

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

18. Applied sqrt-prod 18.5

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

19. Applied times-frac 18.3

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

20. Applied times-frac 15.6

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

21. Applied associate-*l* 12.9

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

22. Simplified 12.4

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

24. Applied add-sqr-sqrt 12.4

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

25. Applied times-frac 12.5

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

26. Final simplification 12.5

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

Reproduce

herbie shell --seed 2019196 +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))))