Average Error: 12.5 → 12.9
Time: 10.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)
double f(double kx, double ky, double th) {
        double r39881 = ky;
        double r39882 = sin(r39881);
        double r39883 = kx;
        double r39884 = sin(r39883);
        double r39885 = 2.0;
        double r39886 = pow(r39884, r39885);
        double r39887 = pow(r39882, r39885);
        double r39888 = r39886 + r39887;
        double r39889 = sqrt(r39888);
        double r39890 = r39882 / r39889;
        double r39891 = th;
        double r39892 = sin(r39891);
        double r39893 = r39890 * r39892;
        return r39893;
}
↓
double f(double kx, double ky, double th) {
        double r39894 = ky;
        double r39895 = sin(r39894);
        double r39896 = kx;
        double r39897 = sin(r39896);
        double r39898 = 2.0;
        double r39899 = pow(r39897, r39898);
        double r39900 = pow(r39895, r39898);
        double r39901 = r39899 + r39900;
        double r39902 = sqrt(r39901);
        double r39903 = r39895 / r39902;
        double r39904 = cbrt(r39903);
        double r39905 = r39904 * r39904;
        double r39906 = th;
        double r39907 = sin(r39906);
        double r39908 = r39904 * r39907;
        double r39909 = r39905 * r39908;
        return r39909;
}

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.5

    \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
  2. Using strategy rm

  3. Applied add-cube-cbrt12.9

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

  4. Applied associate-*l*12.9

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

  5. Final simplification12.9

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

Reproduce

herbie shell --seed 2019353 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))