Average Error: 1.7 → 1.4
Time: 31.5s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(\frac{1}{\sqrt{\left(\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)} + 1}} + 1\right) \cdot \frac{1}{2}}\]
\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{\left(\frac{1}{\sqrt{\left(\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)} + 1}} + 1\right) \cdot \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1323691 = 1.0;
        double r1323692 = 2.0;
        double r1323693 = r1323691 / r1323692;
        double r1323694 = l;
        double r1323695 = r1323692 * r1323694;
        double r1323696 = Om;
        double r1323697 = r1323695 / r1323696;
        double r1323698 = pow(r1323697, r1323692);
        double r1323699 = kx;
        double r1323700 = sin(r1323699);
        double r1323701 = pow(r1323700, r1323692);
        double r1323702 = ky;
        double r1323703 = sin(r1323702);
        double r1323704 = pow(r1323703, r1323692);
        double r1323705 = r1323701 + r1323704;
        double r1323706 = r1323698 * r1323705;
        double r1323707 = r1323691 + r1323706;
        double r1323708 = sqrt(r1323707);
        double r1323709 = r1323691 / r1323708;
        double r1323710 = r1323691 + r1323709;
        double r1323711 = r1323693 * r1323710;
        double r1323712 = sqrt(r1323711);
        return r1323712;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1323713 = 1.0;
        double r1323714 = kx;
        double r1323715 = sin(r1323714);
        double r1323716 = 2.0;
        double r1323717 = pow(r1323715, r1323716);
        double r1323718 = ky;
        double r1323719 = sin(r1323718);
        double r1323720 = pow(r1323719, r1323716);
        double r1323721 = r1323717 + r1323720;
        double r1323722 = l;
        double r1323723 = r1323722 * r1323716;
        double r1323724 = Om;
        double r1323725 = r1323723 / r1323724;
        double r1323726 = 2.0;
        double r1323727 = r1323716 / r1323726;
        double r1323728 = pow(r1323725, r1323727);
        double r1323729 = r1323721 * r1323728;
        double r1323730 = r1323729 * r1323728;
        double r1323731 = r1323730 + r1323713;
        double r1323732 = sqrt(r1323731);
        double r1323733 = r1323713 / r1323732;
        double r1323734 = r1323733 + r1323713;
        double r1323735 = r1323713 / r1323716;
        double r1323736 = r1323734 * r1323735;
        double r1323737 = sqrt(r1323736);
        return r1323737;
}

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

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

Reproduce

herbie shell --seed 2019200 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))