Average Error: 47.3 → 0.9
Time: 4.8m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\sqrt[3]{\cos k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\frac{k}{\ell}}\right)\right)}{\frac{k}{\ell} \cdot \sin k}\]
\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{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\sqrt[3]{\cos k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\frac{k}{\ell}}\right)\right)}{\frac{k}{\ell} \cdot \sin k}
double f(double t, double l, double k) {
        double r10116096 = 2.0;
        double r10116097 = t;
        double r10116098 = 3.0;
        double r10116099 = pow(r10116097, r10116098);
        double r10116100 = l;
        double r10116101 = r10116100 * r10116100;
        double r10116102 = r10116099 / r10116101;
        double r10116103 = k;
        double r10116104 = sin(r10116103);
        double r10116105 = r10116102 * r10116104;
        double r10116106 = tan(r10116103);
        double r10116107 = r10116105 * r10116106;
        double r10116108 = 1.0;
        double r10116109 = r10116103 / r10116097;
        double r10116110 = pow(r10116109, r10116096);
        double r10116111 = r10116108 + r10116110;
        double r10116112 = r10116111 - r10116108;
        double r10116113 = r10116107 * r10116112;
        double r10116114 = r10116096 / r10116113;
        return r10116114;
}
↓
double f(double t, double l, double k) {
        double r10116115 = 2.0;
        double r10116116 = t;
        double r10116117 = r10116115 / r10116116;
        double r10116118 = cbrt(r10116117);
        double r10116119 = k;
        double r10116120 = tan(r10116119);
        double r10116121 = cbrt(r10116120);
        double r10116122 = r10116118 / r10116121;
        double r10116123 = cos(r10116119);
        double r10116124 = cbrt(r10116123);
        double r10116125 = sin(r10116119);
        double r10116126 = cbrt(r10116125);
        double r10116127 = r10116118 / r10116126;
        double r10116128 = l;
        double r10116129 = r10116119 / r10116128;
        double r10116130 = r10116127 / r10116129;
        double r10116131 = r10116124 * r10116130;
        double r10116132 = r10116122 * r10116131;
        double r10116133 = r10116122 * r10116132;
        double r10116134 = r10116129 * r10116125;
        double r10116135 = r10116133 / r10116134;
        return r10116135;
}

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 47.3

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

  2. Simplified30.3

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

  4. Applied associate-/l*28.0

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

  5. Simplified15.3

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

  7. Applied add-cube-cbrt15.5

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

  8. Applied add-cube-cbrt15.5

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

  9. Applied times-frac15.5

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

  10. Applied times-frac14.9

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

  11. Simplified2.7

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

  13. Applied associate-*r/2.7

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

  14. Applied frac-times1.0

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

  16. Applied associate-/r/1.0

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

  17. Applied tan-quot0.9

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

  18. Applied cbrt-div0.9

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

  19. Applied associate-/r/0.9

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

  20. Applied times-frac0.9

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

  21. Final simplification0.9

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

Reproduce

herbie shell --seed 2019162 +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))))