Average Error: 12.6 → 12.7
Time: 36.5s
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 r1234140 = ky;
        double r1234141 = sin(r1234140);
        double r1234142 = kx;
        double r1234143 = sin(r1234142);
        double r1234144 = 2.0;
        double r1234145 = pow(r1234143, r1234144);
        double r1234146 = pow(r1234141, r1234144);
        double r1234147 = r1234145 + r1234146;
        double r1234148 = sqrt(r1234147);
        double r1234149 = r1234141 / r1234148;
        double r1234150 = th;
        double r1234151 = sin(r1234150);
        double r1234152 = r1234149 * r1234151;
        return r1234152;
}
↓
double f(double kx, double ky, double th) {
        double r1234153 = th;
        double r1234154 = sin(r1234153);
        double r1234155 = ky;
        double r1234156 = sin(r1234155);
        double r1234157 = r1234156 * r1234156;
        double r1234158 = kx;
        double r1234159 = sin(r1234158);
        double r1234160 = r1234159 * r1234159;
        double r1234161 = r1234157 + r1234160;
        double r1234162 = sqrt(r1234161);
        double r1234163 = r1234154 / r1234162;
        double r1234164 = r1234163 * r1234156;
        return r1234164;
}

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*12.7

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

  5. Simplified12.7

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

  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 2019168 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))