Average Error: 12.4 → 12.5
Time: 13.5s
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 r47027 = ky;
        double r47028 = sin(r47027);
        double r47029 = kx;
        double r47030 = sin(r47029);
        double r47031 = 2.0;
        double r47032 = pow(r47030, r47031);
        double r47033 = pow(r47028, r47031);
        double r47034 = r47032 + r47033;
        double r47035 = sqrt(r47034);
        double r47036 = r47028 / r47035;
        double r47037 = th;
        double r47038 = sin(r47037);
        double r47039 = r47036 * r47038;
        return r47039;
}
↓
double f(double kx, double ky, double th) {
        double r47040 = ky;
        double r47041 = sin(r47040);
        double r47042 = th;
        double r47043 = sin(r47042);
        double r47044 = kx;
        double r47045 = sin(r47044);
        double r47046 = 2.0;
        double r47047 = pow(r47045, r47046);
        double r47048 = pow(r47041, r47046);
        double r47049 = r47047 + r47048;
        double r47050 = sqrt(r47049);
        double r47051 = r47043 / r47050;
        double r47052 = r47041 * r47051;
        return r47052;
}

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. Using strategy rm

  3. Applied div-inv12.5

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

  4. Applied associate-*l*12.5

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

  5. Simplified12.5

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

  6. Final simplification12.5

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

Reproduce

herbie shell --seed 2019318 
(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)))