Average Error: 12.3 → 12.7
Time: 36.2s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\frac{1}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}}}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\frac{1}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}}}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}} \cdot \sin th
double f(double kx, double ky, double th) {
        double r2037017 = ky;
        double r2037018 = sin(r2037017);
        double r2037019 = kx;
        double r2037020 = sin(r2037019);
        double r2037021 = 2.0;
        double r2037022 = pow(r2037020, r2037021);
        double r2037023 = pow(r2037018, r2037021);
        double r2037024 = r2037022 + r2037023;
        double r2037025 = sqrt(r2037024);
        double r2037026 = r2037018 / r2037025;
        double r2037027 = th;
        double r2037028 = sin(r2037027);
        double r2037029 = r2037026 * r2037028;
        return r2037029;
}
↓
double f(double kx, double ky, double th) {
        double r2037030 = 1.0;
        double r2037031 = kx;
        double r2037032 = sin(r2037031);
        double r2037033 = 2.0;
        double r2037034 = pow(r2037032, r2037033);
        double r2037035 = ky;
        double r2037036 = sin(r2037035);
        double r2037037 = pow(r2037036, r2037033);
        double r2037038 = r2037034 + r2037037;
        double r2037039 = sqrt(r2037038);
        double r2037040 = r2037039 / r2037036;
        double r2037041 = cbrt(r2037040);
        double r2037042 = r2037041 * r2037041;
        double r2037043 = r2037030 / r2037042;
        double r2037044 = r2037043 / r2037041;
        double r2037045 = th;
        double r2037046 = sin(r2037045);
        double r2037047 = r2037044 * r2037046;
        return r2037047;
}

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 12.3

    \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
  2. Using strategy rm

  3. Applied clear-num12.4

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

  5. Applied add-cube-cbrt12.7

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}\right) \cdot \sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}}} \cdot \sin th\]

  6. Applied associate-/r*12.7

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}}}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}}} \cdot \sin th\]

  7. Final simplification12.7

    \[\leadsto \frac{\frac{1}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}}}{\sqrt[3]{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}}} \cdot \sin th\]

Reproduce

herbie shell --seed 2019174 
(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)))