Average Error: 47.1 → 11.3
Time: 2.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}} \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}} \cdot \left(\frac{\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\tan k}}{\sqrt[3]{\frac{k}{t}}} \cdot \frac{\ell}{t}\right)\right)\]
\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)}
↓
\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}} \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}} \cdot \left(\frac{\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\tan k}}{\sqrt[3]{\frac{k}{t}}} \cdot \frac{\ell}{t}\right)\right)
double f(double t, double l, double k) {
        double r2436426 = 2.0;
        double r2436427 = t;
        double r2436428 = 3.0;
        double r2436429 = pow(r2436427, r2436428);
        double r2436430 = l;
        double r2436431 = r2436430 * r2436430;
        double r2436432 = r2436429 / r2436431;
        double r2436433 = k;
        double r2436434 = sin(r2436433);
        double r2436435 = r2436432 * r2436434;
        double r2436436 = tan(r2436433);
        double r2436437 = r2436435 * r2436436;
        double r2436438 = 1.0;
        double r2436439 = r2436433 / r2436427;
        double r2436440 = pow(r2436439, r2436426);
        double r2436441 = r2436438 + r2436440;
        double r2436442 = r2436441 - r2436438;
        double r2436443 = r2436437 * r2436442;
        double r2436444 = r2436426 / r2436443;
        return r2436444;
}
↓
double f(double t, double l, double k) {
        double r2436445 = l;
        double r2436446 = t;
        double r2436447 = r2436445 / r2436446;
        double r2436448 = k;
        double r2436449 = sin(r2436448);
        double r2436450 = r2436447 / r2436449;
        double r2436451 = r2436448 / r2436446;
        double r2436452 = r2436450 / r2436451;
        double r2436453 = 2.0;
        double r2436454 = sqrt(r2436453);
        double r2436455 = cbrt(r2436446);
        double r2436456 = r2436455 * r2436455;
        double r2436457 = r2436454 / r2436456;
        double r2436458 = cbrt(r2436451);
        double r2436459 = r2436458 * r2436458;
        double r2436460 = r2436457 / r2436459;
        double r2436461 = r2436454 / r2436455;
        double r2436462 = tan(r2436448);
        double r2436463 = r2436461 / r2436462;
        double r2436464 = r2436463 / r2436458;
        double r2436465 = r2436464 * r2436447;
        double r2436466 = r2436460 * r2436465;
        double r2436467 = r2436452 * r2436466;
        return r2436467;
}

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 47.1

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

  2. Simplified31.0

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

  4. Applied times-frac20.5

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

  6. Applied *-un-lft-identity20.5

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

  7. Applied *-un-lft-identity20.5

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

  8. Applied times-frac19.7

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

  9. Applied times-frac13.6

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

  10. Applied associate-*r*12.0

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

  12. Applied add-cube-cbrt12.2

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

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

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

  14. Applied add-cube-cbrt12.3

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

  15. Applied add-sqr-sqrt12.3

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

  16. Applied times-frac12.3

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

  17. Applied times-frac12.3

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

  18. Applied times-frac11.7

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

  19. Applied associate-*l*11.3

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

  20. Final simplification11.3

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

Reproduce

herbie shell --seed 2019153 +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))))