Average Error: 1.8 → 1.4
Time: 2.3m
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\ell \cdot \frac{2}{Om}\right) \cdot \left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) + 1}} + \frac{1}{2}}\]
\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{\left(\ell \cdot \frac{2}{Om}\right) \cdot \left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) + 1}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r13230582 = 1.0;
        double r13230583 = 2.0;
        double r13230584 = r13230582 / r13230583;
        double r13230585 = l;
        double r13230586 = r13230583 * r13230585;
        double r13230587 = Om;
        double r13230588 = r13230586 / r13230587;
        double r13230589 = pow(r13230588, r13230583);
        double r13230590 = kx;
        double r13230591 = sin(r13230590);
        double r13230592 = pow(r13230591, r13230583);
        double r13230593 = ky;
        double r13230594 = sin(r13230593);
        double r13230595 = pow(r13230594, r13230583);
        double r13230596 = r13230592 + r13230595;
        double r13230597 = r13230589 * r13230596;
        double r13230598 = r13230582 + r13230597;
        double r13230599 = sqrt(r13230598);
        double r13230600 = r13230582 / r13230599;
        double r13230601 = r13230582 + r13230600;
        double r13230602 = r13230584 * r13230601;
        double r13230603 = sqrt(r13230602);
        return r13230603;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r13230604 = 0.5;
        double r13230605 = l;
        double r13230606 = 2.0;
        double r13230607 = Om;
        double r13230608 = r13230606 / r13230607;
        double r13230609 = r13230605 * r13230608;
        double r13230610 = ky;
        double r13230611 = sin(r13230610);
        double r13230612 = r13230611 * r13230611;
        double r13230613 = kx;
        double r13230614 = sin(r13230613);
        double r13230615 = r13230614 * r13230614;
        double r13230616 = r13230612 + r13230615;
        double r13230617 = r13230616 * r13230609;
        double r13230618 = r13230609 * r13230617;
        double r13230619 = 1.0;
        double r13230620 = r13230618 + r13230619;
        double r13230621 = sqrt(r13230620);
        double r13230622 = r13230604 / r13230621;
        double r13230623 = r13230622 + r13230604;
        double r13230624 = sqrt(r13230623);
        return r13230624;
}

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 1.8

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

  2. Simplified1.4

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

  3. Final simplification1.4

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

Reproduce

herbie shell --seed 2019124 
(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))))))))))