Average Error: 12.3 → 9.3
Time: 11.5s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r40611 = ky;
        double r40612 = sin(r40611);
        double r40613 = kx;
        double r40614 = sin(r40613);
        double r40615 = 2.0;
        double r40616 = pow(r40614, r40615);
        double r40617 = pow(r40612, r40615);
        double r40618 = r40616 + r40617;
        double r40619 = sqrt(r40618);
        double r40620 = r40612 / r40619;
        double r40621 = th;
        double r40622 = sin(r40621);
        double r40623 = r40620 * r40622;
        return r40623;
}
↓
double f(double kx, double ky, double th) {
        double r40624 = ky;
        double r40625 = sin(r40624);
        double r40626 = cbrt(r40625);
        double r40627 = r40626 * r40626;
        double r40628 = kx;
        double r40629 = sin(r40628);
        double r40630 = hypot(r40625, r40629);
        double r40631 = cbrt(r40630);
        double r40632 = r40631 * r40631;
        double r40633 = r40627 / r40632;
        double r40634 = r40626 / r40631;
        double r40635 = r40633 * r40634;
        double r40636 = th;
        double r40637 = sin(r40636);
        double r40638 = r40635 * r40637;
        return r40638;
}

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. Taylor expanded around inf 12.3

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

  3. Simplified8.8

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

  5. Applied add-cube-cbrt9.7

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

  6. Applied add-cube-cbrt9.3

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

  7. Applied times-frac9.3

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

  8. Final simplification9.3

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))