Average Error: 1.7 → 1.7
Time: 7.7s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{\frac{1}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\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}{\sqrt[3]{\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[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}{\sqrt[3]{\sqrt{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 r49948 = 1.0;
        double r49949 = 2.0;
        double r49950 = r49948 / r49949;
        double r49951 = l;
        double r49952 = r49949 * r49951;
        double r49953 = Om;
        double r49954 = r49952 / r49953;
        double r49955 = pow(r49954, r49949);
        double r49956 = kx;
        double r49957 = sin(r49956);
        double r49958 = pow(r49957, r49949);
        double r49959 = ky;
        double r49960 = sin(r49959);
        double r49961 = pow(r49960, r49949);
        double r49962 = r49958 + r49961;
        double r49963 = r49955 * r49962;
        double r49964 = r49948 + r49963;
        double r49965 = sqrt(r49964);
        double r49966 = r49948 / r49965;
        double r49967 = r49948 + r49966;
        double r49968 = r49950 * r49967;
        double r49969 = sqrt(r49968);
        return r49969;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r49970 = 1.0;
        double r49971 = 2.0;
        double r49972 = r49970 / r49971;
        double r49973 = l;
        double r49974 = r49971 * r49973;
        double r49975 = Om;
        double r49976 = r49974 / r49975;
        double r49977 = pow(r49976, r49971);
        double r49978 = kx;
        double r49979 = sin(r49978);
        double r49980 = pow(r49979, r49971);
        double r49981 = ky;
        double r49982 = sin(r49981);
        double r49983 = pow(r49982, r49971);
        double r49984 = r49980 + r49983;
        double r49985 = r49977 * r49984;
        double r49986 = r49970 + r49985;
        double r49987 = sqrt(r49986);
        double r49988 = cbrt(r49987);
        double r49989 = r49988 * r49988;
        double r49990 = r49970 / r49989;
        double r49991 = r49990 / r49988;
        double r49992 = r49970 + r49991;
        double r49993 = r49972 * r49992;
        double r49994 = sqrt(r49993);
        return r49994;
}

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-cube-cbrt1.7

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

  4. Applied associate-/r*1.7

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

  5. Final simplification1.7

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

Reproduce

herbie shell --seed 2020001 +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))))))))))