Toniolo and Linder, Equation (10-)

Average Error: 46.7 → 3.0

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -4.979647532007629 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{\frac{t}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt[3]{k}}}} \cdot \left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt{2}}{\sqrt[3]{k} \cdot \frac{k}{\sqrt[3]{\ell}}}\right)\\ \mathbf{elif}\;k \le 1.6846040995988706 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{\sqrt{\sqrt{2}}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\sqrt{\sqrt{2}}}{t}\right) \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt[3]{k}}}} \cdot \left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt{2}}{\sqrt[3]{k} \cdot \frac{k}{\sqrt[3]{\ell}}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r4266903 = 2.0;
        double r4266904 = t;
        double r4266905 = 3.0;
        double r4266906 = pow(r4266904, r4266905);
        double r4266907 = l;
        double r4266908 = r4266907 * r4266907;
        double r4266909 = r4266906 / r4266908;
        double r4266910 = k;
        double r4266911 = sin(r4266910);
        double r4266912 = r4266909 * r4266911;
        double r4266913 = tan(r4266910);
        double r4266914 = r4266912 * r4266913;
        double r4266915 = 1.0;
        double r4266916 = r4266910 / r4266904;
        double r4266917 = pow(r4266916, r4266903);
        double r4266918 = r4266915 + r4266917;
        double r4266919 = r4266918 - r4266915;
        double r4266920 = r4266914 * r4266919;
        double r4266921 = r4266903 / r4266920;
        return r4266921;
}

↓

double f(double t, double l, double k) {
        double r4266922 = k;
        double r4266923 = -4.979647532007629e+167;
        bool r4266924 = r4266922 <= r4266923;
        double r4266925 = 1.0;
        double r4266926 = t;
        double r4266927 = l;
        double r4266928 = cbrt(r4266927);
        double r4266929 = r4266928 * r4266928;
        double r4266930 = cbrt(r4266922);
        double r4266931 = r4266930 * r4266930;
        double r4266932 = r4266929 / r4266931;
        double r4266933 = r4266926 / r4266932;
        double r4266934 = r4266925 / r4266933;
        double r4266935 = 2.0;
        double r4266936 = sqrt(r4266935);
        double r4266937 = r4266936 * r4266927;
        double r4266938 = tan(r4266922);
        double r4266939 = r4266937 / r4266938;
        double r4266940 = sin(r4266922);
        double r4266941 = r4266939 / r4266940;
        double r4266942 = r4266922 / r4266928;
        double r4266943 = r4266930 * r4266942;
        double r4266944 = r4266936 / r4266943;
        double r4266945 = r4266941 * r4266944;
        double r4266946 = r4266934 * r4266945;
        double r4266947 = 1.6846040995988706e+107;
        bool r4266948 = r4266922 <= r4266947;
        double r4266949 = sqrt(r4266936);
        double r4266950 = r4266927 / r4266922;
        double r4266951 = r4266922 / r4266950;
        double r4266952 = r4266949 / r4266951;
        double r4266953 = r4266949 / r4266926;
        double r4266954 = r4266952 * r4266953;
        double r4266955 = r4266936 / r4266938;
        double r4266956 = r4266927 / r4266940;
        double r4266957 = r4266955 * r4266956;
        double r4266958 = r4266954 * r4266957;
        double r4266959 = r4266948 ? r4266958 : r4266946;
        double r4266960 = r4266924 ? r4266946 : r4266959;
        return r4266960;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -4.979647532007629e+167 or 1.6846040995988706e+107 < k

   1. Initial program 37.9

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

   2. Simplified 19.9

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

   4. Applied div-inv 19.9

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

   5. Applied add-sqr-sqrt 19.9

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

   6. Applied times-frac 19.9

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

   7. Applied times-frac 20.2

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

   8. Simplified 7.2

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{t \cdot \frac{k}{1}}{\frac{\ell}{\frac{k}{1}}}}} \cdot \frac{\frac{\sqrt{2}}{\frac{1}{\ell}}}{\sin k \cdot \tan k}\]

   9. Simplified 7.2

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

   11. Applied add-sqr-sqrt 7.2

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

   12. Applied add-cube-cbrt 7.3

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

   13. Applied times-frac 7.3

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

   14. Applied add-cube-cbrt 7.4

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

   15. Applied times-frac 7.4

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

   16. Applied times-frac 8.6

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

   17. Applied *-un-lft-identity 8.6

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

   18. Applied times-frac 8.2

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

   19. Applied associate-*l* 4.3

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

   20. Simplified 4.3

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

   if -4.979647532007629e+167 < k < 1.6846040995988706e+107

   1. Initial program 53.4

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

   2. Simplified 13.9

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

   4. Applied div-inv 13.9

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

   5. Applied add-sqr-sqrt 14.1

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

   6. Applied times-frac 13.9

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

   7. Applied times-frac 10.9

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

   8. Simplified 6.9

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{t \cdot \frac{k}{1}}{\frac{\ell}{\frac{k}{1}}}}} \cdot \frac{\frac{\sqrt{2}}{\frac{1}{\ell}}}{\sin k \cdot \tan k}\]

   9. Simplified 3.6

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

   11. Applied add-cube-cbrt 3.6

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

   12. Applied *-un-lft-identity 3.6

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

   13. Applied times-frac 3.6

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

   14. Applied *-un-lft-identity 3.6

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

   15. Applied times-frac 3.6

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

   16. Applied times-frac 2.1

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

   17. Applied add-sqr-sqrt 2.1

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

   18. Applied sqrt-prod 2.0

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

   19. Applied times-frac 2.0

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

   20. Simplified 2.0

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

   21. Simplified 2.0

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.979647532007629 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{\frac{t}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt[3]{k}}}} \cdot \left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt{2}}{\sqrt[3]{k} \cdot \frac{k}{\sqrt[3]{\ell}}}\right)\\ \mathbf{elif}\;k \le 1.6846040995988706 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{\sqrt{\sqrt{2}}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\sqrt{\sqrt{2}}}{t}\right) \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{k} \cdot \sqrt[3]{k}}}} \cdot \left(\frac{\frac{\sqrt{2} \cdot \ell}{\tan k}}{\sin k} \cdot \frac{\sqrt{2}}{\sqrt[3]{k} \cdot \frac{k}{\sqrt[3]{\ell}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))))