Average Error: 12.3 → 9.3
Time: 15.3s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r62868 = ky;
        double r62869 = sin(r62868);
        double r62870 = kx;
        double r62871 = sin(r62870);
        double r62872 = 2.0;
        double r62873 = pow(r62871, r62872);
        double r62874 = pow(r62869, r62872);
        double r62875 = r62873 + r62874;
        double r62876 = sqrt(r62875);
        double r62877 = r62869 / r62876;
        double r62878 = th;
        double r62879 = sin(r62878);
        double r62880 = r62877 * r62879;
        return r62880;
}
↓
double f(double kx, double ky, double th) {
        double r62881 = ky;
        double r62882 = sin(r62881);
        double r62883 = cbrt(r62882);
        double r62884 = r62883 * r62883;
        double r62885 = kx;
        double r62886 = sin(r62885);
        double r62887 = 2.0;
        double r62888 = 2.0;
        double r62889 = r62887 / r62888;
        double r62890 = pow(r62886, r62889);
        double r62891 = pow(r62882, r62889);
        double r62892 = hypot(r62890, r62891);
        double r62893 = cbrt(r62892);
        double r62894 = r62893 * r62893;
        double r62895 = r62884 / r62894;
        double r62896 = r62883 / r62893;
        double r62897 = r62895 * r62896;
        double r62898 = th;
        double r62899 = sin(r62898);
        double r62900 = r62897 * r62899;
        return r62900;
}

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 12.3

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

  3. Applied sqr-pow12.3

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

  4. Applied sqr-pow12.3

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

  5. Applied hypot-def8.8

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \sin th\]
  6. Using strategy rm

  7. Applied add-cube-cbrt9.7

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

  8. Applied add-cube-cbrt9.3

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

  9. Applied times-frac9.3

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

  10. Final simplification9.3

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))