Toniolo and Linder, Equation (10+)

Average Error: 31.5 → 8.8

Time: 54.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.5086094187532317 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \left(\sin k \cdot \frac{\sqrt[3]{\sqrt[3]{t}}}{\sqrt[3]{\frac{\ell}{t}}}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.5813717718336364 \cdot 10^{-60}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{k \cdot k}{\ell}, \frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \sin k\right)}{\cos k \cdot \ell} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \left(\sin k \cdot \frac{\sqrt[3]{\sqrt[3]{t}}}{\sqrt[3]{\frac{\ell}{t}}}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1377881 = 2.0;
        double r1377882 = t;
        double r1377883 = 3.0;
        double r1377884 = pow(r1377882, r1377883);
        double r1377885 = l;
        double r1377886 = r1377885 * r1377885;
        double r1377887 = r1377884 / r1377886;
        double r1377888 = k;
        double r1377889 = sin(r1377888);
        double r1377890 = r1377887 * r1377889;
        double r1377891 = tan(r1377888);
        double r1377892 = r1377890 * r1377891;
        double r1377893 = 1.0;
        double r1377894 = r1377888 / r1377882;
        double r1377895 = pow(r1377894, r1377881);
        double r1377896 = r1377893 + r1377895;
        double r1377897 = r1377896 + r1377893;
        double r1377898 = r1377892 * r1377897;
        double r1377899 = r1377881 / r1377898;
        return r1377899;
}

↓

double f(double t, double l, double k) {
        double r1377900 = t;
        double r1377901 = -4.5086094187532317e+30;
        bool r1377902 = r1377900 <= r1377901;
        double r1377903 = l;
        double r1377904 = r1377903 / r1377900;
        double r1377905 = 2.0;
        double r1377906 = k;
        double r1377907 = r1377906 / r1377900;
        double r1377908 = fma(r1377907, r1377907, r1377905);
        double r1377909 = tan(r1377906);
        double r1377910 = cbrt(r1377900);
        double r1377911 = r1377910 * r1377910;
        double r1377912 = cbrt(r1377911);
        double r1377913 = cbrt(r1377903);
        double r1377914 = 1.0;
        double r1377915 = r1377914 / r1377900;
        double r1377916 = cbrt(r1377915);
        double r1377917 = r1377913 * r1377916;
        double r1377918 = cbrt(r1377904);
        double r1377919 = r1377917 * r1377918;
        double r1377920 = r1377912 / r1377919;
        double r1377921 = sin(r1377906);
        double r1377922 = cbrt(r1377910);
        double r1377923 = r1377922 / r1377918;
        double r1377924 = r1377921 * r1377923;
        double r1377925 = r1377920 * r1377924;
        double r1377926 = r1377925 * r1377911;
        double r1377927 = r1377909 * r1377926;
        double r1377928 = r1377908 * r1377927;
        double r1377929 = r1377905 / r1377928;
        double r1377930 = r1377904 * r1377929;
        double r1377931 = 1.5813717718336364e-60;
        bool r1377932 = r1377900 <= r1377931;
        double r1377933 = r1377921 * r1377921;
        double r1377934 = cos(r1377906);
        double r1377935 = r1377933 / r1377934;
        double r1377936 = r1377906 * r1377906;
        double r1377937 = r1377936 / r1377903;
        double r1377938 = r1377900 * r1377900;
        double r1377939 = r1377938 * r1377933;
        double r1377940 = r1377934 * r1377903;
        double r1377941 = r1377939 / r1377940;
        double r1377942 = r1377941 * r1377905;
        double r1377943 = fma(r1377935, r1377937, r1377942);
        double r1377944 = r1377905 / r1377943;
        double r1377945 = r1377904 * r1377944;
        double r1377946 = r1377932 ? r1377945 : r1377930;
        double r1377947 = r1377902 ? r1377930 : r1377946;
        return r1377947;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -4.5086094187532317e+30 or 1.5813717718336364e-60 < t

   1. Initial program 22.3

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

   2. Simplified 11.4

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

   4. Applied add-cube-cbrt 11.5

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

   5. Applied times-frac 10.9

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

   6. Applied associate-*l* 6.8

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

   8. Applied associate-*l/ 6.7

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

   9. Applied associate-*l/ 4.8

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

   10. Applied associate-*l/ 4.3

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

   11. Applied associate-/r/ 4.2

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

   13. Applied add-cube-cbrt 4.3

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

   14. Applied add-cube-cbrt 4.3

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

   15. Applied cbrt-prod 4.3

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

   16. Applied times-frac 4.3

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

   17. Applied associate-*l* 3.5

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

   19. Applied div-inv 3.5

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

   20. Applied cbrt-prod 3.5

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

   if -4.5086094187532317e+30 < t < 1.5813717718336364e-60

   1. Initial program 48.2

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

   2. Simplified 35.5

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

   4. Applied add-cube-cbrt 35.6

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

   5. Applied times-frac 35.1

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

   6. Applied associate-*l* 33.7

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

   8. Applied associate-*l/ 33.7

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

   9. Applied associate-*l/ 34.5

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

   10. Applied associate-*l/ 31.7

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

   11. Applied associate-/r/ 31.7

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

   12. Taylor expanded around inf 20.3

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

   13. Simplified 18.4

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.5086094187532317 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \left(\sin k \cdot \frac{\sqrt[3]{\sqrt[3]{t}}}{\sqrt[3]{\frac{\ell}{t}}}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.5813717718336364 \cdot 10^{-60}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{k \cdot k}{\ell}, \frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \sin k\right)}{\cos k \cdot \ell} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \left(\sin k \cdot \frac{\sqrt[3]{\sqrt[3]{t}}}{\sqrt[3]{\frac{\ell}{t}}}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(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))))