Average Error: 12.1 → 12.4
Time: 38.3s
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 r37282 = ky;
        double r37283 = sin(r37282);
        double r37284 = kx;
        double r37285 = sin(r37284);
        double r37286 = 2.0;
        double r37287 = pow(r37285, r37286);
        double r37288 = pow(r37283, r37286);
        double r37289 = r37287 + r37288;
        double r37290 = sqrt(r37289);
        double r37291 = r37283 / r37290;
        double r37292 = th;
        double r37293 = sin(r37292);
        double r37294 = r37291 * r37293;
        return r37294;
}
↓
double f(double kx, double ky, double th) {
        double r37295 = 1.0;
        double r37296 = kx;
        double r37297 = sin(r37296);
        double r37298 = 2.0;
        double r37299 = pow(r37297, r37298);
        double r37300 = ky;
        double r37301 = sin(r37300);
        double r37302 = pow(r37301, r37298);
        double r37303 = r37299 + r37302;
        double r37304 = sqrt(r37303);
        double r37305 = r37304 / r37301;
        double r37306 = cbrt(r37305);
        double r37307 = r37306 * r37306;
        double r37308 = r37295 / r37307;
        double r37309 = r37308 / r37306;
        double r37310 = th;
        double r37311 = sin(r37310);
        double r37312 = r37309 * r37311;
        return r37312;
}

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 12.1

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

  3. Applied clear-num12.1

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

  5. Applied add-cube-cbrt12.4

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

  6. Applied associate-/r*12.4

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

  7. Final simplification12.4

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

Reproduce

herbie shell --seed 2019303 
(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)))