Average Error: 47.1 → 1.0
Time: 5.7m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}\]
\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[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}
double f(double t, double l, double k) {
        double r11213757 = 2.0;
        double r11213758 = t;
        double r11213759 = 3.0;
        double r11213760 = pow(r11213758, r11213759);
        double r11213761 = l;
        double r11213762 = r11213761 * r11213761;
        double r11213763 = r11213760 / r11213762;
        double r11213764 = k;
        double r11213765 = sin(r11213764);
        double r11213766 = r11213763 * r11213765;
        double r11213767 = tan(r11213764);
        double r11213768 = r11213766 * r11213767;
        double r11213769 = 1.0;
        double r11213770 = r11213764 / r11213758;
        double r11213771 = pow(r11213770, r11213757);
        double r11213772 = r11213769 + r11213771;
        double r11213773 = r11213772 - r11213769;
        double r11213774 = r11213768 * r11213773;
        double r11213775 = r11213757 / r11213774;
        return r11213775;
}
↓
double f(double t, double l, double k) {
        double r11213776 = 2.0;
        double r11213777 = t;
        double r11213778 = r11213776 / r11213777;
        double r11213779 = cbrt(r11213778);
        double r11213780 = k;
        double r11213781 = tan(r11213780);
        double r11213782 = cbrt(r11213781);
        double r11213783 = r11213779 / r11213782;
        double r11213784 = l;
        double r11213785 = r11213780 / r11213784;
        double r11213786 = r11213783 / r11213785;
        double r11213787 = sin(r11213780);
        double r11213788 = r11213783 / r11213787;
        double r11213789 = r11213786 * r11213788;
        double r11213790 = r11213789 * r11213786;
        return r11213790;
}

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 47.1

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

  2. Simplified30.5

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

  4. Applied associate-/l*28.3

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

  5. Simplified15.4

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

  7. Applied add-cube-cbrt15.6

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

  8. Applied add-cube-cbrt15.7

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

  9. Applied times-frac15.7

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

  10. Applied times-frac15.1

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

  11. Simplified2.8

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

  13. Applied associate-*l*1.0

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

  14. Final simplification1.0

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

Reproduce

herbie shell --seed 2019168 +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))))