Average Error: 1.7 → 1.4
Time: 6.9s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{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({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r44745 = 1.0;
        double r44746 = 2.0;
        double r44747 = r44745 / r44746;
        double r44748 = l;
        double r44749 = r44746 * r44748;
        double r44750 = Om;
        double r44751 = r44749 / r44750;
        double r44752 = pow(r44751, r44746);
        double r44753 = kx;
        double r44754 = sin(r44753);
        double r44755 = pow(r44754, r44746);
        double r44756 = ky;
        double r44757 = sin(r44756);
        double r44758 = pow(r44757, r44746);
        double r44759 = r44755 + r44758;
        double r44760 = r44752 * r44759;
        double r44761 = r44745 + r44760;
        double r44762 = sqrt(r44761);
        double r44763 = r44745 / r44762;
        double r44764 = r44745 + r44763;
        double r44765 = r44747 * r44764;
        double r44766 = sqrt(r44765);
        return r44766;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r44767 = 1.0;
        double r44768 = 2.0;
        double r44769 = r44767 / r44768;
        double r44770 = l;
        double r44771 = r44768 * r44770;
        double r44772 = Om;
        double r44773 = r44771 / r44772;
        double r44774 = 2.0;
        double r44775 = r44768 / r44774;
        double r44776 = pow(r44773, r44775);
        double r44777 = kx;
        double r44778 = sin(r44777);
        double r44779 = pow(r44778, r44768);
        double r44780 = ky;
        double r44781 = sin(r44780);
        double r44782 = pow(r44781, r44768);
        double r44783 = r44779 + r44782;
        double r44784 = sqrt(r44783);
        double r44785 = r44776 * r44784;
        double r44786 = r44785 * r44785;
        double r44787 = r44767 + r44786;
        double r44788 = sqrt(r44787);
        double r44789 = r44767 / r44788;
        double r44790 = r44767 + r44789;
        double r44791 = r44769 * r44790;
        double r44792 = sqrt(r44791);
        return r44792;
}

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 add-sqr-sqrt1.7

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

  4. Applied sqr-pow1.7

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

  5. Applied unswap-sqr1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2020020 
(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))))))))))