Average Error: 12.7 → 8.9
Time: 1.1m
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\]
\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 ky, \sin kx\right)}
double f(double kx, double ky, double th) {
        double r1163711 = ky;
        double r1163712 = sin(r1163711);
        double r1163713 = kx;
        double r1163714 = sin(r1163713);
        double r1163715 = 2.0;
        double r1163716 = pow(r1163714, r1163715);
        double r1163717 = pow(r1163712, r1163715);
        double r1163718 = r1163716 + r1163717;
        double r1163719 = sqrt(r1163718);
        double r1163720 = r1163712 / r1163719;
        double r1163721 = th;
        double r1163722 = sin(r1163721);
        double r1163723 = r1163720 * r1163722;
        return r1163723;
}
↓
double f(double kx, double ky, double th) {
        double r1163724 = ky;
        double r1163725 = sin(r1163724);
        double r1163726 = th;
        double r1163727 = sin(r1163726);
        double r1163728 = kx;
        double r1163729 = sin(r1163728);
        double r1163730 = hypot(r1163725, r1163729);
        double r1163731 = r1163727 / r1163730;
        double r1163732 = r1163725 * r1163731;
        return r1163732;
}

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 12.7

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

  2. Taylor expanded around inf 12.7

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

  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. Applied associate-*l*9.0

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

  7. Simplified8.9

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

  8. Final simplification8.9

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))