Average Error: 48.5 → 1.5
Time: 5.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{{\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right) \cdot \left(\frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)\right)\]
\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)}
↓
2 \cdot \left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{{\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right) \cdot \left(\frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{{\left(\frac{\frac{1}{\sqrt[3]{{k}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{{t}^{1}}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)\right)
double f(double t, double l, double k) {
        double r16029252 = 2.0;
        double r16029253 = t;
        double r16029254 = 3.0;
        double r16029255 = pow(r16029253, r16029254);
        double r16029256 = l;
        double r16029257 = r16029256 * r16029256;
        double r16029258 = r16029255 / r16029257;
        double r16029259 = k;
        double r16029260 = sin(r16029259);
        double r16029261 = r16029258 * r16029260;
        double r16029262 = tan(r16029259);
        double r16029263 = r16029261 * r16029262;
        double r16029264 = 1.0;
        double r16029265 = r16029259 / r16029253;
        double r16029266 = pow(r16029265, r16029252);
        double r16029267 = r16029264 + r16029266;
        double r16029268 = r16029267 - r16029264;
        double r16029269 = r16029263 * r16029268;
        double r16029270 = r16029252 / r16029269;
        return r16029270;
}
↓
double f(double t, double l, double k) {
        double r16029271 = 2.0;
        double r16029272 = k;
        double r16029273 = cos(r16029272);
        double r16029274 = sin(r16029272);
        double r16029275 = l;
        double r16029276 = r16029274 / r16029275;
        double r16029277 = r16029273 / r16029276;
        double r16029278 = 1.0;
        double r16029279 = 2.0;
        double r16029280 = r16029271 / r16029279;
        double r16029281 = pow(r16029272, r16029280);
        double r16029282 = r16029278 / r16029281;
        double r16029283 = t;
        double r16029284 = 1.0;
        double r16029285 = pow(r16029283, r16029284);
        double r16029286 = cbrt(r16029285);
        double r16029287 = r16029282 / r16029286;
        double r16029288 = pow(r16029287, r16029284);
        double r16029289 = cbrt(r16029276);
        double r16029290 = r16029288 / r16029289;
        double r16029291 = r16029277 * r16029290;
        double r16029292 = cbrt(r16029281);
        double r16029293 = r16029292 * r16029292;
        double r16029294 = r16029278 / r16029293;
        double r16029295 = r16029294 / r16029286;
        double r16029296 = pow(r16029295, r16029284);
        double r16029297 = r16029296 / r16029289;
        double r16029298 = r16029278 / r16029292;
        double r16029299 = r16029298 / r16029286;
        double r16029300 = pow(r16029299, r16029284);
        double r16029301 = r16029300 / r16029289;
        double r16029302 = r16029297 * r16029301;
        double r16029303 = r16029291 * r16029302;
        double r16029304 = r16029271 * r16029303;
        return r16029304;
}

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 48.5

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

  2. Simplified41.1

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

  3. Taylor expanded around inf 22.3

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

  4. Simplified20.3

    \[\leadsto \color{blue}{\frac{{\left(\frac{\frac{1}{{k}^{2}}}{{t}^{1}}\right)}^{1} \cdot \cos k}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}} \cdot 2}\]
  5. Using strategy rm

  6. Applied times-frac15.6

    \[\leadsto \color{blue}{\left(\frac{{\left(\frac{\frac{1}{{k}^{2}}}{{t}^{1}}\right)}^{1}}{\frac{\sin k}{\ell}} \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right)} \cdot 2\]
  7. Using strategy rm

  8. Applied add-cube-cbrt15.8

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

  9. Applied add-cube-cbrt15.9

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

  10. Applied sqr-pow15.9

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

  11. Applied *-un-lft-identity15.9

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

  12. Applied times-frac15.7

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

  13. Applied times-frac11.1

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

  14. Applied unpow-prod-down11.1

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

  15. Applied times-frac4.5

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

  16. Applied associate-*l*2.4

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

  18. Applied add-cube-cbrt2.4

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

  19. Applied *-un-lft-identity2.4

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

  20. Applied times-frac2.4

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

  21. Applied times-frac2.4

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

  22. Applied unpow-prod-down2.4

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

  23. Applied times-frac1.5

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

  24. Final simplification1.5

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))