Average Error: 1.7 → 1.4
Time: 34.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}} + \frac{1}{2}}\]
\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}}{\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}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1196392 = 1.0;
        double r1196393 = 2.0;
        double r1196394 = r1196392 / r1196393;
        double r1196395 = l;
        double r1196396 = r1196393 * r1196395;
        double r1196397 = Om;
        double r1196398 = r1196396 / r1196397;
        double r1196399 = pow(r1196398, r1196393);
        double r1196400 = kx;
        double r1196401 = sin(r1196400);
        double r1196402 = pow(r1196401, r1196393);
        double r1196403 = ky;
        double r1196404 = sin(r1196403);
        double r1196405 = pow(r1196404, r1196393);
        double r1196406 = r1196402 + r1196405;
        double r1196407 = r1196399 * r1196406;
        double r1196408 = r1196392 + r1196407;
        double r1196409 = sqrt(r1196408);
        double r1196410 = r1196392 / r1196409;
        double r1196411 = r1196392 + r1196410;
        double r1196412 = r1196394 * r1196411;
        double r1196413 = sqrt(r1196412);
        return r1196413;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1196414 = 0.5;
        double r1196415 = 2.0;
        double r1196416 = l;
        double r1196417 = r1196415 * r1196416;
        double r1196418 = Om;
        double r1196419 = r1196417 / r1196418;
        double r1196420 = ky;
        double r1196421 = sin(r1196420);
        double r1196422 = r1196421 * r1196421;
        double r1196423 = kx;
        double r1196424 = sin(r1196423);
        double r1196425 = r1196424 * r1196424;
        double r1196426 = r1196422 + r1196425;
        double r1196427 = r1196419 * r1196426;
        double r1196428 = r1196427 * r1196419;
        double r1196429 = 1.0;
        double r1196430 = r1196428 + r1196429;
        double r1196431 = sqrt(r1196430);
        double r1196432 = r1196414 / r1196431;
        double r1196433 = r1196432 + r1196414;
        double r1196434 = sqrt(r1196433);
        return r1196434;
}

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 1.7

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

  2. Simplified1.7

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

  4. Applied associate-*r*1.4

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

  5. Final simplification1.4

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

Reproduce

herbie shell --seed 2019130 
(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))))))))))