Average Error: 12.5 → 12.6
Time: 10.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin ky \cdot \frac{\sin th}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}
double f(double kx, double ky, double th) {
        double r42874 = ky;
        double r42875 = sin(r42874);
        double r42876 = kx;
        double r42877 = sin(r42876);
        double r42878 = 2.0;
        double r42879 = pow(r42877, r42878);
        double r42880 = pow(r42875, r42878);
        double r42881 = r42879 + r42880;
        double r42882 = sqrt(r42881);
        double r42883 = r42875 / r42882;
        double r42884 = th;
        double r42885 = sin(r42884);
        double r42886 = r42883 * r42885;
        return r42886;
}
↓
double f(double kx, double ky, double th) {
        double r42887 = ky;
        double r42888 = sin(r42887);
        double r42889 = th;
        double r42890 = sin(r42889);
        double r42891 = kx;
        double r42892 = sin(r42891);
        double r42893 = 2.0;
        double r42894 = pow(r42892, r42893);
        double r42895 = pow(r42888, r42893);
        double r42896 = r42894 + r42895;
        double r42897 = sqrt(r42896);
        double r42898 = r42890 / r42897;
        double r42899 = r42888 * r42898;
        return r42899;
}

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 div-inv12.6

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

  4. Applied associate-*l*12.7

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

  5. Simplified12.6

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

  6. Final simplification12.6

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

Reproduce

herbie shell --seed 2020036 
(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)))