Average Error: 1.4 → 1.6
Time: 32.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\sqrt[3]{\log \left(e^{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right)} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right)\right)\right)}}\]
\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{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\sqrt[3]{\log \left(e^{\left(\sqrt[3]{\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[3]{\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) \cdot \sqrt[3]{\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)} \cdot \left(\sqrt[3]{\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 \left(\sqrt[3]{\sqrt[3]{\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 \left(\sqrt[3]{\sqrt[3]{\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[3]{\sqrt[3]{\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)\right)\right)}}
double f(double l, double Om, double kx, double ky) {
        double r1766682 = 1.0;
        double r1766683 = 2.0;
        double r1766684 = r1766682 / r1766683;
        double r1766685 = l;
        double r1766686 = r1766683 * r1766685;
        double r1766687 = Om;
        double r1766688 = r1766686 / r1766687;
        double r1766689 = pow(r1766688, r1766683);
        double r1766690 = kx;
        double r1766691 = sin(r1766690);
        double r1766692 = pow(r1766691, r1766683);
        double r1766693 = ky;
        double r1766694 = sin(r1766693);
        double r1766695 = pow(r1766694, r1766683);
        double r1766696 = r1766692 + r1766695;
        double r1766697 = r1766689 * r1766696;
        double r1766698 = r1766682 + r1766697;
        double r1766699 = sqrt(r1766698);
        double r1766700 = r1766682 / r1766699;
        double r1766701 = r1766682 + r1766700;
        double r1766702 = r1766684 * r1766701;
        double r1766703 = sqrt(r1766702);
        return r1766703;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1766704 = 1.0;
        double r1766705 = 2.0;
        double r1766706 = r1766704 / r1766705;
        double r1766707 = r1766704 * r1766706;
        double r1766708 = Om;
        double r1766709 = r1766705 / r1766708;
        double r1766710 = l;
        double r1766711 = r1766709 * r1766710;
        double r1766712 = pow(r1766711, r1766705);
        double r1766713 = ky;
        double r1766714 = sin(r1766713);
        double r1766715 = pow(r1766714, r1766705);
        double r1766716 = kx;
        double r1766717 = sin(r1766716);
        double r1766718 = pow(r1766717, r1766705);
        double r1766719 = r1766715 + r1766718;
        double r1766720 = fma(r1766712, r1766719, r1766704);
        double r1766721 = sqrt(r1766720);
        double r1766722 = cbrt(r1766721);
        double r1766723 = r1766722 * r1766722;
        double r1766724 = r1766723 * r1766722;
        double r1766725 = exp(r1766724);
        double r1766726 = log(r1766725);
        double r1766727 = cbrt(r1766726);
        double r1766728 = cbrt(r1766722);
        double r1766729 = r1766728 * r1766728;
        double r1766730 = r1766728 * r1766729;
        double r1766731 = r1766722 * r1766730;
        double r1766732 = r1766727 * r1766731;
        double r1766733 = r1766707 / r1766732;
        double r1766734 = r1766707 + r1766733;
        double r1766735 = sqrt(r1766734);
        return r1766735;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  4. Applied add-cube-cbrt1.4

    \[\leadsto \sqrt{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}}}\]
  5. Using strategy rm

  6. Applied add-log-exp1.5

    \[\leadsto \sqrt{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right)}}}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt1.6

    \[\leadsto \sqrt{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right)}}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt1.6

    \[\leadsto \sqrt{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right)\right) \cdot \sqrt[3]{\log \left(e^{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\ell \cdot \frac{2}{Om}\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}}\right)}}}\]

  11. Final simplification1.6

    \[\leadsto \sqrt{1 \cdot \frac{1}{2} + \frac{1 \cdot \frac{1}{2}}{\sqrt[3]{\log \left(e^{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right)} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{Om} \cdot \ell\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}}\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2019172 +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))))))))))