Average Error: 46.9 → 1.4
Time: 2.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\sqrt[3]{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}}}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}} \cdot \left(\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}}}{\sqrt[3]{\sqrt[3]{\frac{k}{\ell}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{k}{\ell}}} \cdot \sqrt[3]{\sqrt[3]{\frac{k}{\ell}}}\right)}\right)\]
\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)}
↓
\frac{\sqrt[3]{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}}}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}} \cdot \left(\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}}}}{\sqrt[3]{\sqrt[3]{\frac{k}{\ell}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{k}{\ell}}} \cdot \sqrt[3]{\sqrt[3]{\frac{k}{\ell}}}\right)}\right)
double f(double t, double l, double k) {
        double r5526572 = 2.0;
        double r5526573 = t;
        double r5526574 = 3.0;
        double r5526575 = pow(r5526573, r5526574);
        double r5526576 = l;
        double r5526577 = r5526576 * r5526576;
        double r5526578 = r5526575 / r5526577;
        double r5526579 = k;
        double r5526580 = sin(r5526579);
        double r5526581 = r5526578 * r5526580;
        double r5526582 = tan(r5526579);
        double r5526583 = r5526581 * r5526582;
        double r5526584 = 1.0;
        double r5526585 = r5526579 / r5526573;
        double r5526586 = pow(r5526585, r5526572);
        double r5526587 = r5526584 + r5526586;
        double r5526588 = r5526587 - r5526584;
        double r5526589 = r5526583 * r5526588;
        double r5526590 = r5526572 / r5526589;
        return r5526590;
}
↓
double f(double t, double l, double k) {
        double r5526591 = 2.0;
        double r5526592 = cbrt(r5526591);
        double r5526593 = t;
        double r5526594 = cbrt(r5526593);
        double r5526595 = r5526592 / r5526594;
        double r5526596 = k;
        double r5526597 = tan(r5526596);
        double r5526598 = cbrt(r5526597);
        double r5526599 = r5526595 / r5526598;
        double r5526600 = r5526599 * r5526599;
        double r5526601 = l;
        double r5526602 = r5526596 / r5526601;
        double r5526603 = r5526600 / r5526602;
        double r5526604 = cbrt(r5526603);
        double r5526605 = r5526604 * r5526604;
        double r5526606 = cbrt(r5526602);
        double r5526607 = r5526606 * r5526606;
        double r5526608 = r5526605 / r5526607;
        double r5526609 = sin(r5526596);
        double r5526610 = r5526599 / r5526609;
        double r5526611 = cbrt(r5526606);
        double r5526612 = r5526611 * r5526611;
        double r5526613 = r5526611 * r5526612;
        double r5526614 = r5526604 / r5526613;
        double r5526615 = r5526610 * r5526614;
        double r5526616 = r5526608 * r5526615;
        return r5526616;
}

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 46.9

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

  2. Simplified30.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\tan k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t}}}\]
  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. Simplified15.5

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

  7. Applied add-cube-cbrt15.6

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

  8. Applied add-cube-cbrt15.7

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

  9. Applied add-cube-cbrt15.9

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

  10. Applied times-frac15.9

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

  11. Applied times-frac15.9

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

  12. Applied times-frac15.4

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

  13. Simplified3.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\frac{\ell}{1}}}}{\frac{k}{\frac{\ell}{1}}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{\tan k}}}{\sin k}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt3.4

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

  16. Applied add-cube-cbrt3.4

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

  17. Applied times-frac3.4

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

  18. Applied associate-*l*1.3

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

  20. Applied add-cube-cbrt1.4

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

  21. Final simplification1.4

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

Reproduce

herbie shell --seed 2019163 +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))))