Average Error: 46.8 → 4.7
Time: 9.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{2}{\frac{\sqrt[3]{t}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\frac{\frac{1}{\tan k}}{\frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}}{\frac{k}{\sqrt[3]{t}}}\right) \cdot \left(\frac{\frac{1}{t}}{\sqrt[3]{\sin k} \cdot \frac{1}{t}} \cdot \left(\frac{\frac{\ell}{\sqrt[3]{\sin k}}}{k} \cdot \frac{\frac{1}{t}}{\sqrt[3]{\sin k}}\right)\right)\]
\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)}
↓
\left(\frac{\frac{2}{\frac{\sqrt[3]{t}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\frac{\frac{1}{\tan k}}{\frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}}{\frac{k}{\sqrt[3]{t}}}\right) \cdot \left(\frac{\frac{1}{t}}{\sqrt[3]{\sin k} \cdot \frac{1}{t}} \cdot \left(\frac{\frac{\ell}{\sqrt[3]{\sin k}}}{k} \cdot \frac{\frac{1}{t}}{\sqrt[3]{\sin k}}\right)\right)
double f(double t, double l, double k) {
        double r14697204 = 2.0;
        double r14697205 = t;
        double r14697206 = 3.0;
        double r14697207 = pow(r14697205, r14697206);
        double r14697208 = l;
        double r14697209 = r14697208 * r14697208;
        double r14697210 = r14697207 / r14697209;
        double r14697211 = k;
        double r14697212 = sin(r14697211);
        double r14697213 = r14697210 * r14697212;
        double r14697214 = tan(r14697211);
        double r14697215 = r14697213 * r14697214;
        double r14697216 = 1.0;
        double r14697217 = r14697211 / r14697205;
        double r14697218 = pow(r14697217, r14697204);
        double r14697219 = r14697216 + r14697218;
        double r14697220 = r14697219 - r14697216;
        double r14697221 = r14697215 * r14697220;
        double r14697222 = r14697204 / r14697221;
        return r14697222;
}
↓
double f(double t, double l, double k) {
        double r14697223 = 2.0;
        double r14697224 = t;
        double r14697225 = cbrt(r14697224);
        double r14697226 = l;
        double r14697227 = cbrt(r14697226);
        double r14697228 = r14697225 / r14697227;
        double r14697229 = r14697228 * r14697228;
        double r14697230 = r14697223 / r14697229;
        double r14697231 = 1.0;
        double r14697232 = r14697225 * r14697225;
        double r14697233 = r14697231 / r14697232;
        double r14697234 = r14697230 / r14697233;
        double r14697235 = k;
        double r14697236 = tan(r14697235);
        double r14697237 = r14697231 / r14697236;
        double r14697238 = r14697237 / r14697228;
        double r14697239 = r14697235 / r14697225;
        double r14697240 = r14697238 / r14697239;
        double r14697241 = r14697234 * r14697240;
        double r14697242 = r14697231 / r14697224;
        double r14697243 = sin(r14697235);
        double r14697244 = cbrt(r14697243);
        double r14697245 = r14697244 * r14697242;
        double r14697246 = r14697242 / r14697245;
        double r14697247 = r14697226 / r14697244;
        double r14697248 = r14697247 / r14697235;
        double r14697249 = r14697242 / r14697244;
        double r14697250 = r14697248 * r14697249;
        double r14697251 = r14697246 * r14697250;
        double r14697252 = r14697241 * r14697251;
        return r14697252;
}

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. Simplified28.8

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

  4. Applied div-inv28.8

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

  5. Applied times-frac28.8

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

  6. Applied times-frac18.5

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

  8. Applied div-inv18.5

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

  9. Applied add-cube-cbrt18.7

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

  10. Applied add-sqr-sqrt18.7

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

  11. Applied times-frac18.4

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

  12. Applied times-frac17.5

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

  13. Applied times-frac11.3

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

  14. Simplified11.1

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

  16. Applied add-cube-cbrt11.2

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

  17. Applied *-un-lft-identity11.2

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

  18. Applied times-frac11.2

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

  19. Applied add-cube-cbrt11.3

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

  20. Applied add-cube-cbrt11.2

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

  21. Applied times-frac11.2

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

  22. Applied div-inv11.2

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

  23. Applied times-frac11.2

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

  24. Applied times-frac8.1

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

  25. Simplified8.1

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

  27. Applied *-un-lft-identity8.1

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

  28. Applied associate-/r/7.9

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

  29. Applied times-frac6.9

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

  30. Applied times-frac4.7

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

  31. Simplified4.7

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

  32. Final simplification4.7

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

Reproduce

herbie shell --seed 2019141 
(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))))