Average Error: 12.6 → 9.0
Time: 10.1s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r37877 = ky;
        double r37878 = sin(r37877);
        double r37879 = kx;
        double r37880 = sin(r37879);
        double r37881 = 2.0;
        double r37882 = pow(r37880, r37881);
        double r37883 = pow(r37878, r37881);
        double r37884 = r37882 + r37883;
        double r37885 = sqrt(r37884);
        double r37886 = r37878 / r37885;
        double r37887 = th;
        double r37888 = sin(r37887);
        double r37889 = r37886 * r37888;
        return r37889;
}
↓
double f(double kx, double ky, double th) {
        double r37890 = ky;
        double r37891 = sin(r37890);
        double r37892 = 1.0;
        double r37893 = kx;
        double r37894 = sin(r37893);
        double r37895 = hypot(r37891, r37894);
        double r37896 = r37892 / r37895;
        double r37897 = r37891 * r37896;
        double r37898 = th;
        double r37899 = sin(r37898);
        double r37900 = r37897 * r37899;
        return r37900;
}

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 12.6

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

  2. Taylor expanded around inf 12.6

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

  3. Simplified8.9

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

  5. Applied div-inv9.0

    \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} \cdot \sin th\]

  6. Final simplification9.0

    \[\leadsto \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2020089 +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)))