Average Error: 13.0 → 9.2
Time: 19.8s
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 r41898 = ky;
        double r41899 = sin(r41898);
        double r41900 = kx;
        double r41901 = sin(r41900);
        double r41902 = 2.0;
        double r41903 = pow(r41901, r41902);
        double r41904 = pow(r41899, r41902);
        double r41905 = r41903 + r41904;
        double r41906 = sqrt(r41905);
        double r41907 = r41899 / r41906;
        double r41908 = th;
        double r41909 = sin(r41908);
        double r41910 = r41907 * r41909;
        return r41910;
}
↓
double f(double kx, double ky, double th) {
        double r41911 = ky;
        double r41912 = sin(r41911);
        double r41913 = th;
        double r41914 = sin(r41913);
        double r41915 = kx;
        double r41916 = sin(r41915);
        double r41917 = hypot(r41916, r41912);
        double r41918 = r41914 / r41917;
        double r41919 = r41912 * r41918;
        return r41919;
}

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)))