Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 10.4

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3179268367824978449241969949568663552:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{elif}\;t \le 8.675237942045073407386696168565799397963 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \sin k\right)}{\ell \cdot \cos k}, 2, \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r5161886 = 2.0;
        double r5161887 = t;
        double r5161888 = 3.0;
        double r5161889 = pow(r5161887, r5161888);
        double r5161890 = l;
        double r5161891 = r5161890 * r5161890;
        double r5161892 = r5161889 / r5161891;
        double r5161893 = k;
        double r5161894 = sin(r5161893);
        double r5161895 = r5161892 * r5161894;
        double r5161896 = tan(r5161893);
        double r5161897 = r5161895 * r5161896;
        double r5161898 = 1.0;
        double r5161899 = r5161893 / r5161887;
        double r5161900 = pow(r5161899, r5161886);
        double r5161901 = r5161898 + r5161900;
        double r5161902 = r5161901 + r5161898;
        double r5161903 = r5161897 * r5161902;
        double r5161904 = r5161886 / r5161903;
        return r5161904;
}

↓

double f(double t, double l, double k) {
        double r5161905 = t;
        double r5161906 = -3.1792683678249784e+36;
        bool r5161907 = r5161905 <= r5161906;
        double r5161908 = 2.0;
        double r5161909 = k;
        double r5161910 = tan(r5161909);
        double r5161911 = cbrt(r5161905);
        double r5161912 = 3.0;
        double r5161913 = pow(r5161911, r5161912);
        double r5161914 = sin(r5161909);
        double r5161915 = l;
        double r5161916 = r5161913 / r5161915;
        double r5161917 = r5161914 * r5161916;
        double r5161918 = r5161913 * r5161917;
        double r5161919 = r5161910 * r5161918;
        double r5161920 = r5161909 / r5161905;
        double r5161921 = pow(r5161920, r5161908);
        double r5161922 = 1.0;
        double r5161923 = r5161921 + r5161922;
        double r5161924 = r5161923 + r5161922;
        double r5161925 = r5161919 * r5161924;
        double r5161926 = r5161915 / r5161913;
        double r5161927 = r5161925 / r5161926;
        double r5161928 = r5161908 / r5161927;
        double r5161929 = 8.675237942045073e-59;
        bool r5161930 = r5161905 <= r5161929;
        double r5161931 = 1.0;
        double r5161932 = -1.0;
        double r5161933 = pow(r5161932, r5161908);
        double r5161934 = r5161931 / r5161933;
        double r5161935 = pow(r5161934, r5161922);
        double r5161936 = r5161905 * r5161905;
        double r5161937 = r5161914 * r5161914;
        double r5161938 = r5161936 * r5161937;
        double r5161939 = cos(r5161909);
        double r5161940 = r5161915 * r5161939;
        double r5161941 = r5161938 / r5161940;
        double r5161942 = r5161935 * r5161941;
        double r5161943 = r5161909 * r5161914;
        double r5161944 = r5161943 * r5161943;
        double r5161945 = r5161944 / r5161940;
        double r5161946 = r5161945 * r5161935;
        double r5161947 = fma(r5161942, r5161908, r5161946);
        double r5161948 = r5161947 / r5161926;
        double r5161949 = r5161908 / r5161948;
        double r5161950 = r5161930 ? r5161949 : r5161928;
        double r5161951 = r5161907 ? r5161928 : r5161950;
        return r5161951;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -3.1792683678249784e+36 or 8.675237942045073e-59 < t

   1. Initial program 23.1

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

   3. Applied add-cube-cbrt 23.2

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

   4. Applied unpow-prod-down 23.2

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

   5. Applied times-frac 16.4

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

   7. Applied unpow-prod-down 16.4

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

   8. Applied associate-/l* 11.4

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

   10. Applied associate-*l/ 11.4

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

   11. Applied associate-*l/ 8.3

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

   12. Applied associate-*l/ 8.3

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

   13. Applied associate-*l/ 7.9

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

   15. Applied associate-*l* 3.9

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

   if -3.1792683678249784e+36 < t < 8.675237942045073e-59

   1. Initial program 50.1

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

   3. Applied add-cube-cbrt 50.2

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

   4. Applied unpow-prod-down 50.2

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

   5. Applied times-frac 42.9

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

   7. Applied unpow-prod-down 42.9

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

   8. Applied associate-/l* 37.3

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

   10. Applied associate-*l/ 37.3

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

   11. Applied associate-*l/ 36.1

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

   12. Applied associate-*l/ 37.3

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

   13. Applied associate-*l/ 33.9

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

   14. Taylor expanded around -inf 21.8

       \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot \left(\frac{{t}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot {\left(\sin k\right)}^{2}\right)}{\ell \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right) + \frac{{\left(\sin k\right)}^{2} \cdot {k}^{2}}{\cos k \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\]

   15. Simplified 21.8

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3179268367824978449241969949568663552:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{elif}\;t \le 8.675237942045073407386696168565799397963 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \sin k\right)}{\ell \cdot \cos k}, 2, \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))