Average Error: 1.8 → 1.8
Time: 24.8s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(1 + \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}} \cdot \sqrt[3]{\left|\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}\right|}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right) \cdot \frac{1}{2}}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\left(1 + \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}} \cdot \sqrt[3]{\left|\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}\right|}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right) \cdot \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r46523 = 1.0;
        double r46524 = 2.0;
        double r46525 = r46523 / r46524;
        double r46526 = l;
        double r46527 = r46524 * r46526;
        double r46528 = Om;
        double r46529 = r46527 / r46528;
        double r46530 = pow(r46529, r46524);
        double r46531 = kx;
        double r46532 = sin(r46531);
        double r46533 = pow(r46532, r46524);
        double r46534 = ky;
        double r46535 = sin(r46534);
        double r46536 = pow(r46535, r46524);
        double r46537 = r46533 + r46536;
        double r46538 = r46530 * r46537;
        double r46539 = r46523 + r46538;
        double r46540 = sqrt(r46539);
        double r46541 = r46523 / r46540;
        double r46542 = r46523 + r46541;
        double r46543 = r46525 * r46542;
        double r46544 = sqrt(r46543);
        return r46544;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r46545 = 1.0;
        double r46546 = cbrt(r46545);
        double r46547 = 2.0;
        double r46548 = Om;
        double r46549 = l;
        double r46550 = r46548 / r46549;
        double r46551 = r46547 / r46550;
        double r46552 = pow(r46551, r46547);
        double r46553 = kx;
        double r46554 = sin(r46553);
        double r46555 = pow(r46554, r46547);
        double r46556 = ky;
        double r46557 = sin(r46556);
        double r46558 = pow(r46557, r46547);
        double r46559 = r46555 + r46558;
        double r46560 = r46552 * r46559;
        double r46561 = r46560 + r46545;
        double r46562 = cbrt(r46561);
        double r46563 = sqrt(r46562);
        double r46564 = cbrt(r46563);
        double r46565 = fabs(r46562);
        double r46566 = cbrt(r46565);
        double r46567 = r46564 * r46566;
        double r46568 = r46546 / r46567;
        double r46569 = r46546 * r46546;
        double r46570 = sqrt(r46561);
        double r46571 = cbrt(r46570);
        double r46572 = r46571 * r46571;
        double r46573 = r46569 / r46572;
        double r46574 = r46568 * r46573;
        double r46575 = r46545 + r46574;
        double r46576 = r46545 / r46547;
        double r46577 = r46575 * r46576;
        double r46578 = sqrt(r46577);
        return r46578;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.8

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  4. Applied add-cube-cbrt1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]

  5. Applied times-frac1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  6. Simplified1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]

  7. Simplified1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}}}\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\color{blue}{\left(\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1} \cdot \sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}\right) \cdot \sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}}}}\right)}\]

  10. Applied sqrt-prod1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\color{blue}{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1} \cdot \sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}}}}\right)}\]

  11. Applied cbrt-prod1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1} \cdot \sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}}}}\right)}\]

  12. Simplified1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\left|\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} + 1}\right|}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}}}\right)}\]

  13. Simplified1.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) + 1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\left|\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} + 1}\right|} \cdot \color{blue}{\sqrt[3]{\sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} + 1}}}}}\right)}\]

  14. Final simplification1.8

    \[\leadsto \sqrt{\left(1 + \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}} \cdot \sqrt[3]{\left|\sqrt[3]{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}\right|}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right) \cdot \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))