Average Error: 13.0 → 9.2
Time: 18.0s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\]
\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(\sin kx, \sin ky\right)}
double f(double kx, double ky, double th) {
        double r38891 = ky;
        double r38892 = sin(r38891);
        double r38893 = kx;
        double r38894 = sin(r38893);
        double r38895 = 2.0;
        double r38896 = pow(r38894, r38895);
        double r38897 = pow(r38892, r38895);
        double r38898 = r38896 + r38897;
        double r38899 = sqrt(r38898);
        double r38900 = r38892 / r38899;
        double r38901 = th;
        double r38902 = sin(r38901);
        double r38903 = r38900 * r38902;
        return r38903;
}
↓
double f(double kx, double ky, double th) {
        double r38904 = ky;
        double r38905 = sin(r38904);
        double r38906 = th;
        double r38907 = sin(r38906);
        double r38908 = kx;
        double r38909 = sin(r38908);
        double r38910 = hypot(r38909, r38905);
        double r38911 = r38907 / r38910;
        double r38912 = r38905 * r38911;
        return r38912;
}

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 13.0

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

  2. Taylor expanded around inf 13.0

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

  3. Simplified9.1

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

  5. Applied div-inv9.2

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

  6. Applied associate-*l*9.3

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

  7. Simplified9.2

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

  8. Final simplification9.2

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

Reproduce

herbie shell --seed 2020047 +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)))