Average Error: 47.3 → 0.9
Time: 4.6m
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 r10130105 = 2.0;
        double r10130106 = t;
        double r10130107 = 3.0;
        double r10130108 = pow(r10130106, r10130107);
        double r10130109 = l;
        double r10130110 = r10130109 * r10130109;
        double r10130111 = r10130108 / r10130110;
        double r10130112 = k;
        double r10130113 = sin(r10130112);
        double r10130114 = r10130111 * r10130113;
        double r10130115 = tan(r10130112);
        double r10130116 = r10130114 * r10130115;
        double r10130117 = 1.0;
        double r10130118 = r10130112 / r10130106;
        double r10130119 = pow(r10130118, r10130105);
        double r10130120 = r10130117 + r10130119;
        double r10130121 = r10130120 - r10130117;
        double r10130122 = r10130116 * r10130121;
        double r10130123 = r10130105 / r10130122;
        return r10130123;
}
↓
double f(double t, double l, double k) {
        double r10130124 = 2.0;
        double r10130125 = t;
        double r10130126 = r10130124 / r10130125;
        double r10130127 = cbrt(r10130126);
        double r10130128 = k;
        double r10130129 = tan(r10130128);
        double r10130130 = cbrt(r10130129);
        double r10130131 = r10130127 / r10130130;
        double r10130132 = cos(r10130128);
        double r10130133 = cbrt(r10130132);
        double r10130134 = sin(r10130128);
        double r10130135 = cbrt(r10130134);
        double r10130136 = r10130127 / r10130135;
        double r10130137 = l;
        double r10130138 = r10130128 / r10130137;
        double r10130139 = r10130136 / r10130138;
        double r10130140 = r10130133 * r10130139;
        double r10130141 = r10130131 * r10130140;
        double r10130142 = r10130131 * r10130141;
        double r10130143 = r10130138 * r10130134;
        double r10130144 = r10130142 / r10130143;
        return r10130144;
}

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))))