Average Error: 1.8 → 1.5
Time: 33.6s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}\right)\right)\right)} + \frac{1}{2}}\]
\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{\frac{\frac{1}{2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}\right)\right)\right)} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1250539 = 1.0;
        double r1250540 = 2.0;
        double r1250541 = r1250539 / r1250540;
        double r1250542 = l;
        double r1250543 = r1250540 * r1250542;
        double r1250544 = Om;
        double r1250545 = r1250543 / r1250544;
        double r1250546 = pow(r1250545, r1250540);
        double r1250547 = kx;
        double r1250548 = sin(r1250547);
        double r1250549 = pow(r1250548, r1250540);
        double r1250550 = ky;
        double r1250551 = sin(r1250550);
        double r1250552 = pow(r1250551, r1250540);
        double r1250553 = r1250549 + r1250552;
        double r1250554 = r1250546 * r1250553;
        double r1250555 = r1250539 + r1250554;
        double r1250556 = sqrt(r1250555);
        double r1250557 = r1250539 / r1250556;
        double r1250558 = r1250539 + r1250557;
        double r1250559 = r1250541 * r1250558;
        double r1250560 = sqrt(r1250559);
        return r1250560;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1250561 = 0.5;
        double r1250562 = 2.0;
        double r1250563 = l;
        double r1250564 = r1250562 * r1250563;
        double r1250565 = Om;
        double r1250566 = r1250564 / r1250565;
        double r1250567 = ky;
        double r1250568 = sin(r1250567);
        double r1250569 = r1250568 * r1250568;
        double r1250570 = kx;
        double r1250571 = sin(r1250570);
        double r1250572 = r1250571 * r1250571;
        double r1250573 = r1250569 + r1250572;
        double r1250574 = r1250566 * r1250573;
        double r1250575 = r1250574 * r1250566;
        double r1250576 = 1.0;
        double r1250577 = r1250575 + r1250576;
        double r1250578 = sqrt(r1250577);
        double r1250579 = cbrt(r1250578);
        double r1250580 = cbrt(r1250579);
        double r1250581 = r1250580 * r1250580;
        double r1250582 = r1250580 * r1250581;
        double r1250583 = r1250579 * r1250582;
        double r1250584 = r1250582 * r1250583;
        double r1250585 = r1250561 / r1250584;
        double r1250586 = r1250585 + r1250561;
        double r1250587 = sqrt(r1250586);
        return r1250587;
}

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. Simplified1.8

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

  4. Applied associate-*r*1.5

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

  6. Applied add-cube-cbrt1.5

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

  8. Applied add-cube-cbrt1.5

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

  10. Applied add-cube-cbrt1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2019142 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))