Average Error: 1.6 → 1.6
Time: 26.8s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}} + \frac{1}{2}}\]
\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(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r807357 = 1.0;
        double r807358 = 2.0;
        double r807359 = r807357 / r807358;
        double r807360 = l;
        double r807361 = r807358 * r807360;
        double r807362 = Om;
        double r807363 = r807361 / r807362;
        double r807364 = pow(r807363, r807358);
        double r807365 = kx;
        double r807366 = sin(r807365);
        double r807367 = pow(r807366, r807358);
        double r807368 = ky;
        double r807369 = sin(r807368);
        double r807370 = pow(r807369, r807358);
        double r807371 = r807367 + r807370;
        double r807372 = r807364 * r807371;
        double r807373 = r807357 + r807372;
        double r807374 = sqrt(r807373);
        double r807375 = r807357 / r807374;
        double r807376 = r807357 + r807375;
        double r807377 = r807359 * r807376;
        double r807378 = sqrt(r807377);
        return r807378;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r807379 = 0.5;
        double r807380 = ky;
        double r807381 = sin(r807380);
        double r807382 = r807381 * r807381;
        double r807383 = kx;
        double r807384 = sin(r807383);
        double r807385 = r807384 * r807384;
        double r807386 = r807382 + r807385;
        double r807387 = 2.0;
        double r807388 = l;
        double r807389 = r807387 * r807388;
        double r807390 = Om;
        double r807391 = r807389 / r807390;
        double r807392 = r807391 * r807391;
        double r807393 = r807386 * r807392;
        double r807394 = 1.0;
        double r807395 = r807393 + r807394;
        double r807396 = sqrt(r807395);
        double r807397 = r807379 / r807396;
        double r807398 = r807397 + r807379;
        double r807399 = sqrt(r807398);
        return r807399;
}

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. Simplified1.6

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

  3. Final simplification1.6

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

Reproduce

herbie shell --seed 2019133 
(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))))))))))