Average Error: 46.8 → 5.6
Time: 6.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt{\sqrt{2}}}{\sin k \cdot \tan k}}{\frac{1}{\sqrt[3]{\ell}}} \cdot \frac{\frac{\sqrt{\sqrt{2}}}{k}}{\frac{1}{\sqrt[3]{\ell}}}\right) \cdot \frac{\ell \cdot \frac{\sqrt{2}}{k}}{\frac{t}{\sqrt[3]{\ell}}}\]
\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)}
↓
\left(\frac{\frac{\sqrt{\sqrt{2}}}{\sin k \cdot \tan k}}{\frac{1}{\sqrt[3]{\ell}}} \cdot \frac{\frac{\sqrt{\sqrt{2}}}{k}}{\frac{1}{\sqrt[3]{\ell}}}\right) \cdot \frac{\ell \cdot \frac{\sqrt{2}}{k}}{\frac{t}{\sqrt[3]{\ell}}}
double f(double t, double l, double k) {
        double r3229962 = 2.0;
        double r3229963 = t;
        double r3229964 = 3.0;
        double r3229965 = pow(r3229963, r3229964);
        double r3229966 = l;
        double r3229967 = r3229966 * r3229966;
        double r3229968 = r3229965 / r3229967;
        double r3229969 = k;
        double r3229970 = sin(r3229969);
        double r3229971 = r3229968 * r3229970;
        double r3229972 = tan(r3229969);
        double r3229973 = r3229971 * r3229972;
        double r3229974 = 1.0;
        double r3229975 = r3229969 / r3229963;
        double r3229976 = pow(r3229975, r3229962);
        double r3229977 = r3229974 + r3229976;
        double r3229978 = r3229977 - r3229974;
        double r3229979 = r3229973 * r3229978;
        double r3229980 = r3229962 / r3229979;
        return r3229980;
}
↓
double f(double t, double l, double k) {
        double r3229981 = 2.0;
        double r3229982 = sqrt(r3229981);
        double r3229983 = sqrt(r3229982);
        double r3229984 = k;
        double r3229985 = sin(r3229984);
        double r3229986 = tan(r3229984);
        double r3229987 = r3229985 * r3229986;
        double r3229988 = r3229983 / r3229987;
        double r3229989 = 1.0;
        double r3229990 = l;
        double r3229991 = cbrt(r3229990);
        double r3229992 = r3229989 / r3229991;
        double r3229993 = r3229988 / r3229992;
        double r3229994 = r3229983 / r3229984;
        double r3229995 = r3229994 / r3229992;
        double r3229996 = r3229993 * r3229995;
        double r3229997 = r3229982 / r3229984;
        double r3229998 = r3229990 * r3229997;
        double r3229999 = t;
        double r3230000 = r3229999 / r3229991;
        double r3230001 = r3229998 / r3230000;
        double r3230002 = r3229996 * r3230001;
        return r3230002;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.8

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

  2. Simplified30.0

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

  4. Applied associate-*r/30.3

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

  5. Applied frac-times41.0

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

  6. Applied associate-/r/41.0

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

  7. Applied times-frac37.1

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

  8. Simplified17.7

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

  10. Applied add-cube-cbrt17.9

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

  11. Applied *-un-lft-identity17.9

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

  12. Applied times-frac18.0

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

  13. Applied add-sqr-sqrt18.0

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

  14. Applied times-frac17.8

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

  15. Applied times-frac10.4

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

  16. Applied associate-*l*8.7

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

  17. Simplified7.2

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

  19. Applied *-un-lft-identity7.2

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

  20. Applied times-frac7.2

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

  21. Applied add-sqr-sqrt7.2

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

  22. Applied sqrt-prod7.2

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

  23. Applied times-frac7.2

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

  24. Applied times-frac5.6

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

  25. Final simplification5.6

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

Reproduce

herbie shell --seed 2019138 +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))))