Average Error: 12.6 → 12.7
Time: 31.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sqrt[3]{\sin kx} \cdot \left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sin kx\right)}}\]
\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 + \sqrt[3]{\sin kx} \cdot \left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sin kx\right)}}
double f(double kx, double ky, double th) {
        double r564869 = ky;
        double r564870 = sin(r564869);
        double r564871 = kx;
        double r564872 = sin(r564871);
        double r564873 = 2.0;
        double r564874 = pow(r564872, r564873);
        double r564875 = pow(r564870, r564873);
        double r564876 = r564874 + r564875;
        double r564877 = sqrt(r564876);
        double r564878 = r564870 / r564877;
        double r564879 = th;
        double r564880 = sin(r564879);
        double r564881 = r564878 * r564880;
        return r564881;
}
↓
double f(double kx, double ky, double th) {
        double r564882 = th;
        double r564883 = sin(r564882);
        double r564884 = ky;
        double r564885 = sin(r564884);
        double r564886 = r564885 * r564885;
        double r564887 = kx;
        double r564888 = sin(r564887);
        double r564889 = cbrt(r564888);
        double r564890 = r564889 * r564889;
        double r564891 = r564890 * r564888;
        double r564892 = r564889 * r564891;
        double r564893 = r564886 + r564892;
        double r564894 = sqrt(r564893);
        double r564895 = r564885 / r564894;
        double r564896 = r564883 * r564895;
        return r564896;
}

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 12.6

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

  2. Simplified12.6

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

  4. Applied add-cube-cbrt12.7

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

  5. Applied associate-*r*12.7

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

  6. Final simplification12.7

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

Reproduce

herbie shell --seed 2019155 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))