Average Error: 1.7 → 1.7
Time: 17.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{\frac{1}{\left|\left(\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right|}}{\sqrt{\sqrt[3]{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{\frac{1}{\left|\left(\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right|}}{\sqrt{\sqrt[3]{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r52877 = 1.0;
        double r52878 = 2.0;
        double r52879 = r52877 / r52878;
        double r52880 = l;
        double r52881 = r52878 * r52880;
        double r52882 = Om;
        double r52883 = r52881 / r52882;
        double r52884 = pow(r52883, r52878);
        double r52885 = kx;
        double r52886 = sin(r52885);
        double r52887 = pow(r52886, r52878);
        double r52888 = ky;
        double r52889 = sin(r52888);
        double r52890 = pow(r52889, r52878);
        double r52891 = r52887 + r52890;
        double r52892 = r52884 * r52891;
        double r52893 = r52877 + r52892;
        double r52894 = sqrt(r52893);
        double r52895 = r52877 / r52894;
        double r52896 = r52877 + r52895;
        double r52897 = r52879 * r52896;
        double r52898 = sqrt(r52897);
        return r52898;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r52899 = 1.0;
        double r52900 = 2.0;
        double r52901 = r52899 / r52900;
        double r52902 = kx;
        double r52903 = sin(r52902);
        double r52904 = pow(r52903, r52900);
        double r52905 = ky;
        double r52906 = sin(r52905);
        double r52907 = pow(r52906, r52900);
        double r52908 = r52904 + r52907;
        double r52909 = l;
        double r52910 = r52900 * r52909;
        double r52911 = Om;
        double r52912 = r52910 / r52911;
        double r52913 = pow(r52912, r52900);
        double r52914 = fma(r52908, r52913, r52899);
        double r52915 = cbrt(r52914);
        double r52916 = cbrt(r52915);
        double r52917 = r52916 * r52916;
        double r52918 = r52917 * r52916;
        double r52919 = cbrt(r52918);
        double r52920 = r52919 * r52916;
        double r52921 = r52920 * r52916;
        double r52922 = fabs(r52921);
        double r52923 = r52899 / r52922;
        double r52924 = r52913 * r52908;
        double r52925 = r52899 + r52924;
        double r52926 = cbrt(r52925);
        double r52927 = sqrt(r52926);
        double r52928 = r52923 / r52927;
        double r52929 = r52899 + r52928;
        double r52930 = r52901 * r52929;
        double r52931 = sqrt(r52930);
        return r52931;
}

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-cube-cbrt1.7

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

  4. Applied sqrt-prod1.7

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

  5. Applied associate-/r*1.7

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

  6. Simplified1.7

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

  8. Applied add-cube-cbrt1.7

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

  10. Applied add-cube-cbrt1.7

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

  11. Final simplification1.7

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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))))))))))