Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 23.1

Time: 16.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.8718558274139864 \cdot 10^{287}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\left(\sqrt[3]{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell} \cdot \sqrt[3]{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}\right) \cdot \sqrt[3]{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r107877 = 2.0;
        double r107878 = t;
        double r107879 = 3.0;
        double r107880 = pow(r107878, r107879);
        double r107881 = l;
        double r107882 = r107881 * r107881;
        double r107883 = r107880 / r107882;
        double r107884 = k;
        double r107885 = sin(r107884);
        double r107886 = r107883 * r107885;
        double r107887 = tan(r107884);
        double r107888 = r107886 * r107887;
        double r107889 = 1.0;
        double r107890 = r107884 / r107878;
        double r107891 = pow(r107890, r107877);
        double r107892 = r107889 + r107891;
        double r107893 = r107892 + r107889;
        double r107894 = r107888 * r107893;
        double r107895 = r107877 / r107894;
        return r107895;
}

↓

double f(double t, double l, double k) {
        double r107896 = l;
        double r107897 = r107896 * r107896;
        double r107898 = 1.8718558274139864e+287;
        bool r107899 = r107897 <= r107898;
        double r107900 = 1.0;
        double r107901 = t;
        double r107902 = cbrt(r107901);
        double r107903 = r107902 * r107902;
        double r107904 = 3.0;
        double r107905 = pow(r107903, r107904);
        double r107906 = r107900 / r107905;
        double r107907 = k;
        double r107908 = tan(r107907);
        double r107909 = cbrt(r107908);
        double r107910 = r107909 * r107909;
        double r107911 = r107906 / r107910;
        double r107912 = 2.0;
        double r107913 = pow(r107902, r107904);
        double r107914 = sin(r107907);
        double r107915 = r107913 * r107914;
        double r107916 = r107912 / r107915;
        double r107917 = r107916 * r107896;
        double r107918 = r107917 / r107909;
        double r107919 = 2.0;
        double r107920 = 1.0;
        double r107921 = r107907 / r107901;
        double r107922 = pow(r107921, r107912);
        double r107923 = fma(r107919, r107920, r107922);
        double r107924 = r107896 / r107923;
        double r107925 = r107918 * r107924;
        double r107926 = r107911 * r107925;
        double r107927 = cbrt(r107917);
        double r107928 = r107927 * r107927;
        double r107929 = r107928 * r107927;
        double r107930 = r107906 * r107929;
        double r107931 = r107930 / r107908;
        double r107932 = r107931 * r107924;
        double r107933 = r107899 ? r107926 : r107932;
        return r107933;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (* l l) < 1.8718558274139864e+287

   1. Initial program 25.9

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

   2. Simplified 26.0

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

   4. Applied *-un-lft-identity 26.0

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

   5. Applied times-frac 26.0

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

   6. Applied associate-*r* 25.3

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

   7. Simplified 23.9

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

   9. Applied add-cube-cbrt 24.1

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

   10. Applied unpow-prod-down 24.1

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

   11. Applied associate-*l* 22.9

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

   13. Applied *-un-lft-identity 22.9

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

   14. Applied times-frac 22.8

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

   15. Applied associate-*l* 20.4

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

   17. Applied add-cube-cbrt 20.4

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

   18. Applied times-frac 20.2

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

   19. Applied associate-*l* 19.3

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

   if 1.8718558274139864e+287 < (* l l)

   1. Initial program 62.0

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

   2. Simplified 61.9

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

   4. Applied *-un-lft-identity 61.9

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

   5. Applied times-frac 57.4

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

   6. Applied associate-*r* 46.7

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

   7. Simplified 46.7

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

   9. Applied add-cube-cbrt 46.9

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

   10. Applied unpow-prod-down 46.9

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

   11. Applied associate-*l* 46.2

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

   13. Applied *-un-lft-identity 46.2

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

   14. Applied times-frac 45.9

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

   15. Applied associate-*l* 40.0

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

   17. Applied add-cube-cbrt 40.0

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

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.8718558274139864 \cdot 10^{287}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\left(\sqrt[3]{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell} \cdot \sqrt[3]{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}\right) \cdot \sqrt[3]{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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))))