Average Error: 12.4 → 12.5
Time: 33.8s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sin kx \cdot \sqrt[3]{\sin kx}\right)}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sin kx \cdot \sqrt[3]{\sin kx}\right)}}
double f(double kx, double ky, double th) {
        double r512894 = ky;
        double r512895 = sin(r512894);
        double r512896 = kx;
        double r512897 = sin(r512896);
        double r512898 = 2.0;
        double r512899 = pow(r512897, r512898);
        double r512900 = pow(r512895, r512898);
        double r512901 = r512899 + r512900;
        double r512902 = sqrt(r512901);
        double r512903 = r512895 / r512902;
        double r512904 = th;
        double r512905 = sin(r512904);
        double r512906 = r512903 * r512905;
        return r512906;
}
↓
double f(double kx, double ky, double th) {
        double r512907 = th;
        double r512908 = sin(r512907);
        double r512909 = ky;
        double r512910 = sin(r512909);
        double r512911 = r512910 * r512910;
        double r512912 = kx;
        double r512913 = sin(r512912);
        double r512914 = cbrt(r512913);
        double r512915 = r512914 * r512914;
        double r512916 = r512913 * r512914;
        double r512917 = r512915 * r512916;
        double r512918 = r512911 + r512917;
        double r512919 = sqrt(r512918);
        double r512920 = r512910 / r512919;
        double r512921 = r512908 * r512920;
        return r512921;
}

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 12.4

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

  2. Simplified12.4

    \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt12.5

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

  5. Applied associate-*l*12.5

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

  6. Final simplification12.5

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

Reproduce

herbie shell --seed 2019152 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))