Average Error: 31.4 → 12.1
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\sqrt[3]{\frac{\sqrt{2}}{\sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\sin k}}}{\frac{t}{\ell}} \cdot \left(\left(\frac{\frac{\frac{\sqrt{2}}{\tan k}}{t}}{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \frac{\sqrt[3]{\frac{\sqrt{\sqrt{2}}}{\sin k}}}{\frac{t}{\ell}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{2}}}\right)\]
\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{\sqrt[3]{\frac{\sqrt{2}}{\sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\sin k}}}{\frac{t}{\ell}} \cdot \left(\left(\frac{\frac{\frac{\sqrt{2}}{\tan k}}{t}}{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \frac{\sqrt[3]{\frac{\sqrt{\sqrt{2}}}{\sin k}}}{\frac{t}{\ell}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{2}}}\right)
double f(double t, double l, double k) {
        double r2074208 = 2.0;
        double r2074209 = t;
        double r2074210 = 3.0;
        double r2074211 = pow(r2074209, r2074210);
        double r2074212 = l;
        double r2074213 = r2074212 * r2074212;
        double r2074214 = r2074211 / r2074213;
        double r2074215 = k;
        double r2074216 = sin(r2074215);
        double r2074217 = r2074214 * r2074216;
        double r2074218 = tan(r2074215);
        double r2074219 = r2074217 * r2074218;
        double r2074220 = 1.0;
        double r2074221 = r2074215 / r2074209;
        double r2074222 = pow(r2074221, r2074208);
        double r2074223 = r2074220 + r2074222;
        double r2074224 = r2074223 + r2074220;
        double r2074225 = r2074219 * r2074224;
        double r2074226 = r2074208 / r2074225;
        return r2074226;
}
↓
double f(double t, double l, double k) {
        double r2074227 = 2.0;
        double r2074228 = sqrt(r2074227);
        double r2074229 = k;
        double r2074230 = sin(r2074229);
        double r2074231 = r2074228 / r2074230;
        double r2074232 = cbrt(r2074231);
        double r2074233 = r2074232 * r2074232;
        double r2074234 = t;
        double r2074235 = l;
        double r2074236 = r2074234 / r2074235;
        double r2074237 = r2074233 / r2074236;
        double r2074238 = tan(r2074229);
        double r2074239 = r2074228 / r2074238;
        double r2074240 = r2074239 / r2074234;
        double r2074241 = r2074229 / r2074234;
        double r2074242 = r2074241 * r2074241;
        double r2074243 = r2074242 + r2074227;
        double r2074244 = r2074240 / r2074243;
        double r2074245 = sqrt(r2074228);
        double r2074246 = r2074245 / r2074230;
        double r2074247 = cbrt(r2074246);
        double r2074248 = r2074247 / r2074236;
        double r2074249 = r2074244 * r2074248;
        double r2074250 = cbrt(r2074245);
        double r2074251 = r2074249 * r2074250;
        double r2074252 = r2074237 * r2074251;
        return r2074252;
}

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 31.4

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

  2. Simplified20.4

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

  4. Applied associate-*r*18.6

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

  6. Applied *-un-lft-identity18.6

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

  7. Applied *-un-lft-identity18.6

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

  8. Applied add-sqr-sqrt18.7

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

  9. Applied times-frac18.7

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

  10. Applied times-frac18.4

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

  11. Applied times-frac17.0

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

  12. Simplified17.0

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

  14. Applied add-cube-cbrt17.2

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

  15. Applied times-frac14.2

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

  16. Applied associate-*l*12.2

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

  18. Applied *-un-lft-identity12.2

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

  19. Applied *-un-lft-identity12.2

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

  20. Applied add-sqr-sqrt12.2

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

  21. Applied sqrt-prod12.1

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

  22. Applied times-frac12.2

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

  23. Applied cbrt-prod12.1

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

  24. Applied times-frac12.1

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

  25. Applied associate-*l*12.1

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

  26. Final simplification12.1

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

Reproduce

herbie shell --seed 2019152 
(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))))