Average Error: 46.8 → 1.0
Time: 5.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}} \cdot \left(\frac{1}{\frac{\frac{k}{\ell}}{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}\right)\]
\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}}}{\frac{k}{\ell}} \cdot \left(\frac{1}{\frac{\frac{k}{\ell}}{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}\right)
double f(double t, double l, double k) {
        double r11580159 = 2.0;
        double r11580160 = t;
        double r11580161 = 3.0;
        double r11580162 = pow(r11580160, r11580161);
        double r11580163 = l;
        double r11580164 = r11580163 * r11580163;
        double r11580165 = r11580162 / r11580164;
        double r11580166 = k;
        double r11580167 = sin(r11580166);
        double r11580168 = r11580165 * r11580167;
        double r11580169 = tan(r11580166);
        double r11580170 = r11580168 * r11580169;
        double r11580171 = 1.0;
        double r11580172 = r11580166 / r11580160;
        double r11580173 = pow(r11580172, r11580159);
        double r11580174 = r11580171 + r11580173;
        double r11580175 = r11580174 - r11580171;
        double r11580176 = r11580170 * r11580175;
        double r11580177 = r11580159 / r11580176;
        return r11580177;
}
↓
double f(double t, double l, double k) {
        double r11580178 = 2.0;
        double r11580179 = t;
        double r11580180 = r11580178 / r11580179;
        double r11580181 = cbrt(r11580180);
        double r11580182 = k;
        double r11580183 = tan(r11580182);
        double r11580184 = cbrt(r11580183);
        double r11580185 = r11580181 / r11580184;
        double r11580186 = l;
        double r11580187 = r11580182 / r11580186;
        double r11580188 = r11580185 / r11580187;
        double r11580189 = 1.0;
        double r11580190 = r11580187 / r11580185;
        double r11580191 = r11580189 / r11580190;
        double r11580192 = sin(r11580182);
        double r11580193 = r11580185 / r11580192;
        double r11580194 = r11580191 * r11580193;
        double r11580195 = r11580188 * r11580194;
        return r11580195;
}

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

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

  4. Applied associate-/l*27.9

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

  5. Simplified15.2

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

  7. Applied add-cube-cbrt15.3

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

  8. Applied add-cube-cbrt15.4

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

  9. Applied times-frac15.4

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

  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-*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. Using strategy rm

  15. Applied clear-num1.0

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

  16. Final simplification1.0

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

Reproduce

herbie shell --seed 2019158 +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))))