Average Error: 1.7 → 1.4
Time: 29.7s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\frac{\ell + \ell}{Om} \cdot \left(\frac{\ell + \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) + 1}} + \frac{1}{2}}\]
\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{\frac{\ell + \ell}{Om} \cdot \left(\frac{\ell + \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) + 1}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1710317 = 1.0;
        double r1710318 = 2.0;
        double r1710319 = r1710317 / r1710318;
        double r1710320 = l;
        double r1710321 = r1710318 * r1710320;
        double r1710322 = Om;
        double r1710323 = r1710321 / r1710322;
        double r1710324 = pow(r1710323, r1710318);
        double r1710325 = kx;
        double r1710326 = sin(r1710325);
        double r1710327 = pow(r1710326, r1710318);
        double r1710328 = ky;
        double r1710329 = sin(r1710328);
        double r1710330 = pow(r1710329, r1710318);
        double r1710331 = r1710327 + r1710330;
        double r1710332 = r1710324 * r1710331;
        double r1710333 = r1710317 + r1710332;
        double r1710334 = sqrt(r1710333);
        double r1710335 = r1710317 / r1710334;
        double r1710336 = r1710317 + r1710335;
        double r1710337 = r1710319 * r1710336;
        double r1710338 = sqrt(r1710337);
        return r1710338;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1710339 = 0.5;
        double r1710340 = l;
        double r1710341 = r1710340 + r1710340;
        double r1710342 = Om;
        double r1710343 = r1710341 / r1710342;
        double r1710344 = ky;
        double r1710345 = sin(r1710344);
        double r1710346 = r1710345 * r1710345;
        double r1710347 = kx;
        double r1710348 = sin(r1710347);
        double r1710349 = r1710348 * r1710348;
        double r1710350 = r1710346 + r1710349;
        double r1710351 = r1710343 * r1710350;
        double r1710352 = r1710343 * r1710351;
        double r1710353 = 1.0;
        double r1710354 = r1710352 + r1710353;
        double r1710355 = sqrt(r1710354);
        double r1710356 = r1710339 / r1710355;
        double r1710357 = r1710356 + r1710339;
        double r1710358 = sqrt(r1710357);
        return r1710358;
}

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

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

  4. Applied associate-*r*1.4

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

  5. Final simplification1.4

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

Reproduce

herbie shell --seed 2019162 
(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))))))))))