Average Error: 12.3 → 8.7
Time: 27.6s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\left(\sin ky\right), \left(\sin kx\right)\right)}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\left(\sin ky\right), \left(\sin kx\right)\right)}
double f(double kx, double ky, double th) {
        double r351733 = ky;
        double r351734 = sin(r351733);
        double r351735 = kx;
        double r351736 = sin(r351735);
        double r351737 = 2.0;
        double r351738 = pow(r351736, r351737);
        double r351739 = pow(r351734, r351737);
        double r351740 = r351738 + r351739;
        double r351741 = sqrt(r351740);
        double r351742 = r351734 / r351741;
        double r351743 = th;
        double r351744 = sin(r351743);
        double r351745 = r351742 * r351744;
        return r351745;
}
↓
double f(double kx, double ky, double th) {
        double r351746 = ky;
        double r351747 = sin(r351746);
        double r351748 = th;
        double r351749 = sin(r351748);
        double r351750 = kx;
        double r351751 = sin(r351750);
        double r351752 = hypot(r351747, r351751);
        double r351753 = r351749 / r351752;
        double r351754 = r351747 * r351753;
        return r351754;
}

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. Simplified11.3

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

  4. Applied associate-/l*8.7

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

  5. Taylor expanded around inf 12.3

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

  6. Simplified8.7

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

  8. Applied associate-/r/8.7

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

  9. Final simplification8.7

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

Reproduce

herbie shell --seed 2019128 +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)))