Average Error: 46.8 → 1.4
Time: 5.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{1}{\sqrt[3]{\tan k}} \cdot \left(\left(\frac{1}{\sqrt[3]{\tan k}} \cdot \frac{1}{\sqrt[3]{\tan k}}\right) \cdot \frac{\frac{2}{t}}{\frac{k}{\ell}}\right)}{\frac{k}{\ell} \cdot \sin k}\]
\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)}
↓
\frac{\frac{1}{\sqrt[3]{\tan k}} \cdot \left(\left(\frac{1}{\sqrt[3]{\tan k}} \cdot \frac{1}{\sqrt[3]{\tan k}}\right) \cdot \frac{\frac{2}{t}}{\frac{k}{\ell}}\right)}{\frac{k}{\ell} \cdot \sin k}
double f(double t, double l, double k) {
        double r13712765 = 2.0;
        double r13712766 = t;
        double r13712767 = 3.0;
        double r13712768 = pow(r13712766, r13712767);
        double r13712769 = l;
        double r13712770 = r13712769 * r13712769;
        double r13712771 = r13712768 / r13712770;
        double r13712772 = k;
        double r13712773 = sin(r13712772);
        double r13712774 = r13712771 * r13712773;
        double r13712775 = tan(r13712772);
        double r13712776 = r13712774 * r13712775;
        double r13712777 = 1.0;
        double r13712778 = r13712772 / r13712766;
        double r13712779 = pow(r13712778, r13712765);
        double r13712780 = r13712777 + r13712779;
        double r13712781 = r13712780 - r13712777;
        double r13712782 = r13712776 * r13712781;
        double r13712783 = r13712765 / r13712782;
        return r13712783;
}
↓
double f(double t, double l, double k) {
        double r13712784 = 1.0;
        double r13712785 = k;
        double r13712786 = tan(r13712785);
        double r13712787 = cbrt(r13712786);
        double r13712788 = r13712784 / r13712787;
        double r13712789 = r13712788 * r13712788;
        double r13712790 = 2.0;
        double r13712791 = t;
        double r13712792 = r13712790 / r13712791;
        double r13712793 = l;
        double r13712794 = r13712785 / r13712793;
        double r13712795 = r13712792 / r13712794;
        double r13712796 = r13712789 * r13712795;
        double r13712797 = r13712788 * r13712796;
        double r13712798 = sin(r13712785);
        double r13712799 = r13712794 * r13712798;
        double r13712800 = r13712797 / r13712799;
        return r13712800;
}

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. Simplified30.2

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

  4. Applied associate-/r/30.1

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

  5. Applied times-frac30.0

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

  6. Applied associate-/l*27.9

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

  7. Simplified15.2

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

  9. Applied div-inv15.2

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

  10. Applied times-frac15.3

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

  11. Simplified4.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{1}}}}{\frac{k}{\frac{\ell}{1}}}} \cdot \frac{\frac{1}{\tan k}}{\sin k}\]
  12. Using strategy rm

  13. Applied frac-times1.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{1}}} \cdot \frac{1}{\tan k}}{\frac{k}{\frac{\ell}{1}} \cdot \sin k}}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt1.4

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

  16. Applied add-cube-cbrt1.4

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

  17. Applied times-frac1.4

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

  18. Applied associate-*r*1.4

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

  19. Simplified1.4

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

  20. Final simplification1.4

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

Reproduce

herbie shell --seed 2019158 
(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))))