Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 12.3

Time: 1.1m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r173830 = 2.0;
        double r173831 = t;
        double r173832 = 3.0;
        double r173833 = pow(r173831, r173832);
        double r173834 = l;
        double r173835 = r173834 * r173834;
        double r173836 = r173833 / r173835;
        double r173837 = k;
        double r173838 = sin(r173837);
        double r173839 = r173836 * r173838;
        double r173840 = tan(r173837);
        double r173841 = r173839 * r173840;
        double r173842 = 1.0;
        double r173843 = r173837 / r173831;
        double r173844 = pow(r173843, r173830);
        double r173845 = r173842 + r173844;
        double r173846 = r173845 + r173842;
        double r173847 = r173841 * r173846;
        double r173848 = r173830 / r173847;
        return r173848;
}

↓

double f(double t, double l, double k) {
        double r173849 = 2.0;
        double r173850 = cbrt(r173849);
        double r173851 = t;
        double r173852 = cbrt(r173851);
        double r173853 = 3.0;
        double r173854 = pow(r173852, r173853);
        double r173855 = r173850 / r173854;
        double r173856 = 2.0;
        double r173857 = 1.0;
        double r173858 = k;
        double r173859 = r173858 / r173851;
        double r173860 = pow(r173859, r173849);
        double r173861 = fma(r173856, r173857, r173860);
        double r173862 = cbrt(r173861);
        double r173863 = r173862 * r173862;
        double r173864 = l;
        double r173865 = r173854 / r173864;
        double r173866 = sin(r173858);
        double r173867 = r173865 * r173866;
        double r173868 = r173850 / r173867;
        double r173869 = cbrt(r173868);
        double r173870 = r173869 * r173869;
        double r173871 = r173863 / r173870;
        double r173872 = r173855 / r173871;
        double r173873 = tan(r173858);
        double r173874 = r173850 / r173865;
        double r173875 = r173862 / r173869;
        double r173876 = r173874 / r173875;
        double r173877 = r173873 / r173876;
        double r173878 = r173872 / r173877;
        return r173878;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.5

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

2. Simplified 32.5

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

4. Applied add-cube-cbrt 32.6

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

5. Applied unpow-prod-down 32.6

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

6. Applied times-frac 25.5

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

7. Applied associate-*l* 23.7

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

9. Applied *-un-lft-identity 23.7

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

10. Applied unpow-prod-down 23.7

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

11. Applied times-frac 18.0

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

12. Simplified 18.0

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

14. Applied add-cube-cbrt 18.0

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

15. Applied times-frac 17.8

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

16. Applied associate-/l* 16.2

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

18. Applied add-cube-cbrt 16.3

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

19. Applied add-cube-cbrt 16.2

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

20. Applied times-frac 16.2

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

21. Applied times-frac 16.1

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

22. Applied times-frac 14.0

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

23. Applied associate-/l* 12.3

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

24. Final simplification 12.3

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(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))))