Average Error: 12.4 → 9.5
Time: 36.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)\right)
double f(double kx, double ky, double th) {
        double r848586 = ky;
        double r848587 = sin(r848586);
        double r848588 = kx;
        double r848589 = sin(r848588);
        double r848590 = 2.0;
        double r848591 = pow(r848589, r848590);
        double r848592 = pow(r848587, r848590);
        double r848593 = r848591 + r848592;
        double r848594 = sqrt(r848593);
        double r848595 = r848587 / r848594;
        double r848596 = th;
        double r848597 = sin(r848596);
        double r848598 = r848595 * r848597;
        return r848598;
}
↓
double f(double kx, double ky, double th) {
        double r848599 = th;
        double r848600 = sin(r848599);
        double r848601 = ky;
        double r848602 = sin(r848601);
        double r848603 = kx;
        double r848604 = sin(r848603);
        double r848605 = hypot(r848602, r848604);
        double r848606 = r848602 / r848605;
        double r848607 = cbrt(r848606);
        double r848608 = r848607 * r848607;
        double r848609 = r848600 * r848608;
        double r848610 = cbrt(r848607);
        double r848611 = r848610 * r848610;
        double r848612 = r848610 * r848611;
        double r848613 = r848609 * r848612;
        return r848613;
}

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. Simplified9.0

    \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt9.4

    \[\leadsto \sin th \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)}\]

  5. Applied associate-*r*9.4

    \[\leadsto \color{blue}{\left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt9.5

    \[\leadsto \left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)}\]

  8. Final simplification9.5

    \[\leadsto \left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)\right)\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))