Average Error: 1.6 → 1.3
Time: 24.9s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]
\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{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r3093454 = 1.0;
        double r3093455 = 2.0;
        double r3093456 = r3093454 / r3093455;
        double r3093457 = l;
        double r3093458 = r3093455 * r3093457;
        double r3093459 = Om;
        double r3093460 = r3093458 / r3093459;
        double r3093461 = pow(r3093460, r3093455);
        double r3093462 = kx;
        double r3093463 = sin(r3093462);
        double r3093464 = pow(r3093463, r3093455);
        double r3093465 = ky;
        double r3093466 = sin(r3093465);
        double r3093467 = pow(r3093466, r3093455);
        double r3093468 = r3093464 + r3093467;
        double r3093469 = r3093461 * r3093468;
        double r3093470 = r3093454 + r3093469;
        double r3093471 = sqrt(r3093470);
        double r3093472 = r3093454 / r3093471;
        double r3093473 = r3093454 + r3093472;
        double r3093474 = r3093456 * r3093473;
        double r3093475 = sqrt(r3093474);
        return r3093475;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r3093476 = 1.0;
        double r3093477 = 2.0;
        double r3093478 = r3093476 / r3093477;
        double r3093479 = l;
        double r3093480 = r3093477 * r3093479;
        double r3093481 = Om;
        double r3093482 = r3093480 / r3093481;
        double r3093483 = 2.0;
        double r3093484 = r3093477 / r3093483;
        double r3093485 = pow(r3093482, r3093484);
        double r3093486 = kx;
        double r3093487 = sin(r3093486);
        double r3093488 = pow(r3093487, r3093477);
        double r3093489 = ky;
        double r3093490 = sin(r3093489);
        double r3093491 = pow(r3093490, r3093477);
        double r3093492 = r3093488 + r3093491;
        double r3093493 = r3093485 * r3093492;
        double r3093494 = r3093485 * r3093493;
        double r3093495 = r3093476 + r3093494;
        double r3093496 = sqrt(r3093495);
        double r3093497 = r3093476 / r3093496;
        double r3093498 = r3093476 + r3093497;
        double r3093499 = r3093478 * r3093498;
        double r3093500 = sqrt(r3093499);
        return r3093500;
}

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 1.6

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

  3. Applied sqr-pow1.6

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

  4. Applied associate-*l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019173 
(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))))))))))