Average Error: 12.5 → 12.7
Time: 37.3s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2}}} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin ky}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2}}} \cdot \sin th
double f(double kx, double ky, double th) {
        double r1521302 = ky;
        double r1521303 = sin(r1521302);
        double r1521304 = kx;
        double r1521305 = sin(r1521304);
        double r1521306 = 2.0;
        double r1521307 = pow(r1521305, r1521306);
        double r1521308 = pow(r1521303, r1521306);
        double r1521309 = r1521307 + r1521308;
        double r1521310 = sqrt(r1521309);
        double r1521311 = r1521303 / r1521310;
        double r1521312 = th;
        double r1521313 = sin(r1521312);
        double r1521314 = r1521311 * r1521313;
        return r1521314;
}
↓
double f(double kx, double ky, double th) {
        double r1521315 = ky;
        double r1521316 = sin(r1521315);
        double r1521317 = 2.0;
        double r1521318 = pow(r1521316, r1521317);
        double r1521319 = kx;
        double r1521320 = sin(r1521319);
        double r1521321 = cbrt(r1521320);
        double r1521322 = r1521321 * r1521321;
        double r1521323 = pow(r1521322, r1521317);
        double r1521324 = pow(r1521321, r1521317);
        double r1521325 = r1521323 * r1521324;
        double r1521326 = r1521318 + r1521325;
        double r1521327 = sqrt(r1521326);
        double r1521328 = r1521316 / r1521327;
        double r1521329 = th;
        double r1521330 = sin(r1521329);
        double r1521331 = r1521328 * r1521330;
        return r1521331;
}

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 add-cube-cbrt12.7

    \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sqrt[3]{\sin kx}\right)}}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

  4. Applied unpow-prod-down12.7

    \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2}} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

  5. Final simplification12.7

    \[\leadsto \frac{\sin ky}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2}}} \cdot \sin th\]

Reproduce

herbie shell --seed 2019171 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))