Average Error: 1.5 → 1.2
Time: 25.3s
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 r618441 = 1.0;
        double r618442 = 2.0;
        double r618443 = r618441 / r618442;
        double r618444 = l;
        double r618445 = r618442 * r618444;
        double r618446 = Om;
        double r618447 = r618445 / r618446;
        double r618448 = pow(r618447, r618442);
        double r618449 = kx;
        double r618450 = sin(r618449);
        double r618451 = pow(r618450, r618442);
        double r618452 = ky;
        double r618453 = sin(r618452);
        double r618454 = pow(r618453, r618442);
        double r618455 = r618451 + r618454;
        double r618456 = r618448 * r618455;
        double r618457 = r618441 + r618456;
        double r618458 = sqrt(r618457);
        double r618459 = r618441 / r618458;
        double r618460 = r618441 + r618459;
        double r618461 = r618443 * r618460;
        double r618462 = sqrt(r618461);
        return r618462;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r618463 = 0.5;
        double r618464 = 2.0;
        double r618465 = l;
        double r618466 = r618464 * r618465;
        double r618467 = Om;
        double r618468 = r618466 / r618467;
        double r618469 = ky;
        double r618470 = sin(r618469);
        double r618471 = r618470 * r618470;
        double r618472 = kx;
        double r618473 = sin(r618472);
        double r618474 = r618473 * r618473;
        double r618475 = r618471 + r618474;
        double r618476 = r618468 * r618475;
        double r618477 = r618476 * r618468;
        double r618478 = 1.0;
        double r618479 = r618477 + r618478;
        double r618480 = sqrt(r618479);
        double r618481 = r618463 / r618480;
        double r618482 = r618481 + r618463;
        double r618483 = sqrt(r618482);
        return r618483;
}

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 1.5

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

  2. Simplified1.5

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

  4. Applied associate-*r*1.2

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

  5. Taylor expanded around -inf 1.2

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

  6. Simplified1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2019128 
(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))))))))))