Average Error: 1.6 → 1.6
Time: 36.6s
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 r1339494 = 1.0;
        double r1339495 = 2.0;
        double r1339496 = r1339494 / r1339495;
        double r1339497 = l;
        double r1339498 = r1339495 * r1339497;
        double r1339499 = Om;
        double r1339500 = r1339498 / r1339499;
        double r1339501 = pow(r1339500, r1339495);
        double r1339502 = kx;
        double r1339503 = sin(r1339502);
        double r1339504 = pow(r1339503, r1339495);
        double r1339505 = ky;
        double r1339506 = sin(r1339505);
        double r1339507 = pow(r1339506, r1339495);
        double r1339508 = r1339504 + r1339507;
        double r1339509 = r1339501 * r1339508;
        double r1339510 = r1339494 + r1339509;
        double r1339511 = sqrt(r1339510);
        double r1339512 = r1339494 / r1339511;
        double r1339513 = r1339494 + r1339512;
        double r1339514 = r1339496 * r1339513;
        double r1339515 = sqrt(r1339514);
        return r1339515;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1339516 = 0.5;
        double r1339517 = ky;
        double r1339518 = sin(r1339517);
        double r1339519 = r1339518 * r1339518;
        double r1339520 = kx;
        double r1339521 = sin(r1339520);
        double r1339522 = r1339521 * r1339521;
        double r1339523 = r1339519 + r1339522;
        double r1339524 = 2.0;
        double r1339525 = l;
        double r1339526 = r1339524 * r1339525;
        double r1339527 = Om;
        double r1339528 = r1339526 / r1339527;
        double r1339529 = r1339528 * r1339528;
        double r1339530 = r1339523 * r1339529;
        double r1339531 = 1.0;
        double r1339532 = r1339530 + r1339531;
        double r1339533 = sqrt(r1339532);
        double r1339534 = r1339516 / r1339533;
        double r1339535 = r1339534 + r1339516;
        double r1339536 = sqrt(r1339535);
        return r1339536;
}

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 2019139 
(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))))))))))