Toniolo and Linder, Equation (10-)

Average Error: 47.2 → 1.1

Time: 5.7m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r11587841 = 2.0;
        double r11587842 = t;
        double r11587843 = 3.0;
        double r11587844 = pow(r11587842, r11587843);
        double r11587845 = l;
        double r11587846 = r11587845 * r11587845;
        double r11587847 = r11587844 / r11587846;
        double r11587848 = k;
        double r11587849 = sin(r11587848);
        double r11587850 = r11587847 * r11587849;
        double r11587851 = tan(r11587848);
        double r11587852 = r11587850 * r11587851;
        double r11587853 = 1.0;
        double r11587854 = r11587848 / r11587842;
        double r11587855 = pow(r11587854, r11587841);
        double r11587856 = r11587853 + r11587855;
        double r11587857 = r11587856 - r11587853;
        double r11587858 = r11587852 * r11587857;
        double r11587859 = r11587841 / r11587858;
        return r11587859;
}

↓

double f(double t, double l, double k) {
        double r11587860 = 2.0;
        double r11587861 = t;
        double r11587862 = r11587860 / r11587861;
        double r11587863 = cbrt(r11587862);
        double r11587864 = r11587863 * r11587863;
        double r11587865 = cbrt(r11587864);
        double r11587866 = 1.0;
        double r11587867 = l;
        double r11587868 = cbrt(r11587867);
        double r11587869 = r11587868 * r11587868;
        double r11587870 = r11587866 / r11587869;
        double r11587871 = r11587865 / r11587870;
        double r11587872 = cbrt(r11587863);
        double r11587873 = k;
        double r11587874 = tan(r11587873);
        double r11587875 = cbrt(r11587874);
        double r11587876 = r11587872 / r11587875;
        double r11587877 = r11587873 / r11587868;
        double r11587878 = r11587876 / r11587877;
        double r11587879 = r11587871 * r11587878;
        double r11587880 = r11587863 / r11587875;
        double r11587881 = sin(r11587873);
        double r11587882 = r11587880 / r11587881;
        double r11587883 = r11587879 * r11587882;
        double r11587884 = r11587873 / r11587867;
        double r11587885 = r11587880 / r11587884;
        double r11587886 = r11587883 * r11587885;
        return r11587886;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 47.2

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

2. Simplified 30.8

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

4. Applied associate-/r/ 30.7

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

5. Applied times-frac 30.6

   \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\tan k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

6. Applied associate-/l* 28.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\tan k}}{\frac{\frac{k}{t} \cdot \frac{k}{t}}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}}}\]

7. Simplified 15.0

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

9. Applied add-cube-cbrt 15.2

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

10. Applied add-cube-cbrt 15.3

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

11. Applied times-frac 15.3

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

12. Applied times-frac 14.7

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{\frac{\frac{k}{t}}{\frac{\ell}{t}} \cdot \frac{\frac{k}{t}}{\frac{\ell}{t}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}}\]

13. Simplified 2.7

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

15. Applied associate-*l* 1.0

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

17. Applied add-cube-cbrt 1.1

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

18. Applied *-un-lft-identity 1.1

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

19. Applied times-frac 1.1

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

20. Applied *-un-lft-identity 1.1

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

21. Applied add-cube-cbrt 1.1

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

22. Applied cbrt-prod 1.1

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

23. Applied times-frac 1.1

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

24. Applied times-frac 1.1

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

25. Final simplification 1.1

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

Reproduce

herbie shell --seed 2019164 
(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))))