Average Error: 12.6 → 12.7
Time: 34.5s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky
double f(double kx, double ky, double th) {
        double r1264934 = ky;
        double r1264935 = sin(r1264934);
        double r1264936 = kx;
        double r1264937 = sin(r1264936);
        double r1264938 = 2.0;
        double r1264939 = pow(r1264937, r1264938);
        double r1264940 = pow(r1264935, r1264938);
        double r1264941 = r1264939 + r1264940;
        double r1264942 = sqrt(r1264941);
        double r1264943 = r1264935 / r1264942;
        double r1264944 = th;
        double r1264945 = sin(r1264944);
        double r1264946 = r1264943 * r1264945;
        return r1264946;
}
↓
double f(double kx, double ky, double th) {
        double r1264947 = th;
        double r1264948 = sin(r1264947);
        double r1264949 = ky;
        double r1264950 = sin(r1264949);
        double r1264951 = r1264950 * r1264950;
        double r1264952 = kx;
        double r1264953 = sin(r1264952);
        double r1264954 = r1264953 * r1264953;
        double r1264955 = r1264951 + r1264954;
        double r1264956 = sqrt(r1264955);
        double r1264957 = r1264948 / r1264956;
        double r1264958 = r1264957 * r1264950;
        return r1264958;
}

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. Using strategy rm

  3. Applied div-inv12.7

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

  4. Applied associate-*l*12.8

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

  5. Simplified12.7

    \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]

  6. Final simplification12.7

    \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky\]

Reproduce

herbie shell --seed 2019163 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))