Average Error: 1.5 → 1.3
Time: 20.9s
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 r45767 = 1.0;
        double r45768 = 2.0;
        double r45769 = r45767 / r45768;
        double r45770 = l;
        double r45771 = r45768 * r45770;
        double r45772 = Om;
        double r45773 = r45771 / r45772;
        double r45774 = pow(r45773, r45768);
        double r45775 = kx;
        double r45776 = sin(r45775);
        double r45777 = pow(r45776, r45768);
        double r45778 = ky;
        double r45779 = sin(r45778);
        double r45780 = pow(r45779, r45768);
        double r45781 = r45777 + r45780;
        double r45782 = r45774 * r45781;
        double r45783 = r45767 + r45782;
        double r45784 = sqrt(r45783);
        double r45785 = r45767 / r45784;
        double r45786 = r45767 + r45785;
        double r45787 = r45769 * r45786;
        double r45788 = sqrt(r45787);
        return r45788;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r45789 = 1.0;
        double r45790 = 2.0;
        double r45791 = r45789 / r45790;
        double r45792 = l;
        double r45793 = r45790 * r45792;
        double r45794 = Om;
        double r45795 = r45793 / r45794;
        double r45796 = 2.0;
        double r45797 = r45790 / r45796;
        double r45798 = pow(r45795, r45797);
        double r45799 = kx;
        double r45800 = sin(r45799);
        double r45801 = pow(r45800, r45790);
        double r45802 = ky;
        double r45803 = sin(r45802);
        double r45804 = pow(r45803, r45790);
        double r45805 = r45801 + r45804;
        double r45806 = r45798 * r45805;
        double r45807 = r45798 * r45806;
        double r45808 = r45789 + r45807;
        double r45809 = sqrt(r45808);
        double r45810 = r45789 / r45809;
        double r45811 = r45789 + r45810;
        double r45812 = r45791 * r45811;
        double r45813 = sqrt(r45812);
        return r45813;
}

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. Using strategy rm

  3. Applied sqr-pow1.5

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

  4. Applied associate-*l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019291 
(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))))))))))