Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 22.8

Time: 20.2s

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r139863 = 2.0;
        double r139864 = t;
        double r139865 = 3.0;
        double r139866 = pow(r139864, r139865);
        double r139867 = l;
        double r139868 = r139867 * r139867;
        double r139869 = r139866 / r139868;
        double r139870 = k;
        double r139871 = sin(r139870);
        double r139872 = r139869 * r139871;
        double r139873 = tan(r139870);
        double r139874 = r139872 * r139873;
        double r139875 = 1.0;
        double r139876 = r139870 / r139864;
        double r139877 = pow(r139876, r139863);
        double r139878 = r139875 + r139877;
        double r139879 = r139878 + r139875;
        double r139880 = r139874 * r139879;
        double r139881 = r139863 / r139880;
        return r139881;
}

↓

double f(double t, double l, double k) {
        double r139882 = 2.0;
        double r139883 = 1.0;
        double r139884 = cbrt(r139883);
        double r139885 = t;
        double r139886 = cbrt(r139885);
        double r139887 = r139886 * r139886;
        double r139888 = 3.0;
        double r139889 = 2.0;
        double r139890 = r139888 / r139889;
        double r139891 = pow(r139887, r139890);
        double r139892 = r139884 / r139891;
        double r139893 = 1.0;
        double r139894 = pow(r139892, r139893);
        double r139895 = pow(r139886, r139888);
        double r139896 = r139884 / r139895;
        double r139897 = pow(r139896, r139893);
        double r139898 = l;
        double r139899 = k;
        double r139900 = sin(r139899);
        double r139901 = r139898 / r139900;
        double r139902 = r139897 * r139901;
        double r139903 = r139894 * r139902;
        double r139904 = r139894 * r139903;
        double r139905 = r139882 * r139904;
        double r139906 = tan(r139899);
        double r139907 = r139905 / r139906;
        double r139908 = r139899 / r139885;
        double r139909 = pow(r139908, r139882);
        double r139910 = fma(r139889, r139893, r139909);
        double r139911 = r139898 / r139910;
        double r139912 = r139907 * r139911;
        return r139912;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.4

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

2. Simplified 32.6

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

4. Applied *-un-lft-identity 32.6

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

5. Applied times-frac 31.8

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

6. Applied associate-*r* 29.3

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

7. Simplified 27.9

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

8. Taylor expanded around inf 27.7

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

10. Applied add-cube-cbrt 27.9

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

11. Applied unpow-prod-down 27.9

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

12. Applied add-cube-cbrt 27.9

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

13. Applied times-frac 27.7

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

14. Applied unpow-prod-down 27.7

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

15. Applied associate-*l* 24.1

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

17. Applied sqr-pow 24.1

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

18. Applied times-frac 24.0

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

19. Applied unpow-prod-down 24.0

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

20. Applied associate-*l* 22.8

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

21. Final simplification 22.8

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

Reproduce

herbie shell --seed 2020021 +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))))