Average Error: 12.5 → 9.2
Time: 10.7s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \left(\sqrt[3]{\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 \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)
double f(double kx, double ky, double th) {
        double r44626 = ky;
        double r44627 = sin(r44626);
        double r44628 = kx;
        double r44629 = sin(r44628);
        double r44630 = 2.0;
        double r44631 = pow(r44629, r44630);
        double r44632 = pow(r44627, r44630);
        double r44633 = r44631 + r44632;
        double r44634 = sqrt(r44633);
        double r44635 = r44627 / r44634;
        double r44636 = th;
        double r44637 = sin(r44636);
        double r44638 = r44635 * r44637;
        return r44638;
}
↓
double f(double kx, double ky, double th) {
        double r44639 = ky;
        double r44640 = sin(r44639);
        double r44641 = kx;
        double r44642 = sin(r44641);
        double r44643 = hypot(r44640, r44642);
        double r44644 = r44640 / r44643;
        double r44645 = cbrt(r44644);
        double r44646 = cbrt(r44640);
        double r44647 = r44646 * r44646;
        double r44648 = cbrt(r44643);
        double r44649 = r44648 * r44648;
        double r44650 = r44647 / r44649;
        double r44651 = cbrt(r44650);
        double r44652 = r44646 / r44648;
        double r44653 = cbrt(r44652);
        double r44654 = r44651 * r44653;
        double r44655 = r44645 * r44654;
        double r44656 = th;
        double r44657 = sin(r44656);
        double r44658 = r44645 * r44657;
        double r44659 = r44655 * r44658;
        return r44659;
}

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. Taylor expanded around inf 12.5

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

  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-cbrt9.2

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

  6. Applied associate-*l*9.2

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

  8. Applied add-cube-cbrt9.2

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

  9. Applied add-cube-cbrt9.2

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

  10. Applied times-frac9.2

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

  11. Applied cbrt-prod9.2

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

  12. Final simplification9.2

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

Reproduce

herbie shell --seed 2019353 +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)))