Average Error: 1.7 → 1.4
Time: 7.9s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]
\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{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r55654 = 1.0;
        double r55655 = 2.0;
        double r55656 = r55654 / r55655;
        double r55657 = l;
        double r55658 = r55655 * r55657;
        double r55659 = Om;
        double r55660 = r55658 / r55659;
        double r55661 = pow(r55660, r55655);
        double r55662 = kx;
        double r55663 = sin(r55662);
        double r55664 = pow(r55663, r55655);
        double r55665 = ky;
        double r55666 = sin(r55665);
        double r55667 = pow(r55666, r55655);
        double r55668 = r55664 + r55667;
        double r55669 = r55661 * r55668;
        double r55670 = r55654 + r55669;
        double r55671 = sqrt(r55670);
        double r55672 = r55654 / r55671;
        double r55673 = r55654 + r55672;
        double r55674 = r55656 * r55673;
        double r55675 = sqrt(r55674);
        return r55675;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r55676 = 1.0;
        double r55677 = 2.0;
        double r55678 = r55676 / r55677;
        double r55679 = l;
        double r55680 = r55677 * r55679;
        double r55681 = Om;
        double r55682 = r55680 / r55681;
        double r55683 = 2.0;
        double r55684 = r55677 / r55683;
        double r55685 = pow(r55682, r55684);
        double r55686 = kx;
        double r55687 = sin(r55686);
        double r55688 = pow(r55687, r55677);
        double r55689 = ky;
        double r55690 = sin(r55689);
        double r55691 = pow(r55690, r55677);
        double r55692 = r55688 + r55691;
        double r55693 = r55685 * r55692;
        double r55694 = r55685 * r55693;
        double r55695 = r55676 + r55694;
        double r55696 = sqrt(r55695);
        double r55697 = r55676 / r55696;
        double r55698 = r55676 + r55697;
        double r55699 = r55678 * r55698;
        double r55700 = sqrt(r55699);
        return r55700;
}

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. Using strategy rm

  3. Applied sqr-pow1.7

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

  4. Applied associate-*l*1.4

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

  5. Final simplification1.4

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))