Average Error: 1.7 → 1.7
Time: 18.9s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right)}\]
\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 + \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r39739 = 1.0;
        double r39740 = 2.0;
        double r39741 = r39739 / r39740;
        double r39742 = l;
        double r39743 = r39740 * r39742;
        double r39744 = Om;
        double r39745 = r39743 / r39744;
        double r39746 = pow(r39745, r39740);
        double r39747 = kx;
        double r39748 = sin(r39747);
        double r39749 = pow(r39748, r39740);
        double r39750 = ky;
        double r39751 = sin(r39750);
        double r39752 = pow(r39751, r39740);
        double r39753 = r39749 + r39752;
        double r39754 = r39746 * r39753;
        double r39755 = r39739 + r39754;
        double r39756 = sqrt(r39755);
        double r39757 = r39739 / r39756;
        double r39758 = r39739 + r39757;
        double r39759 = r39741 * r39758;
        double r39760 = sqrt(r39759);
        return r39760;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r39761 = 1.0;
        double r39762 = 2.0;
        double r39763 = r39761 / r39762;
        double r39764 = Om;
        double r39765 = r39762 / r39764;
        double r39766 = l;
        double r39767 = r39765 * r39766;
        double r39768 = pow(r39767, r39762);
        double r39769 = kx;
        double r39770 = sin(r39769);
        double r39771 = pow(r39770, r39762);
        double r39772 = ky;
        double r39773 = sin(r39772);
        double r39774 = pow(r39773, r39762);
        double r39775 = r39771 + r39774;
        double r39776 = fma(r39768, r39775, r39761);
        double r39777 = sqrt(r39776);
        double r39778 = r39761 / r39777;
        double r39779 = sqrt(r39778);
        double r39780 = r39779 * r39779;
        double r39781 = r39761 + r39780;
        double r39782 = r39763 * r39781;
        double r39783 = sqrt(r39782);
        return r39783;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

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

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

  4. Simplified1.7

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

  5. Simplified1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(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))))))))))