Average Error: 12.6 → 8.9
Time: 36.0s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\]
\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 r36276 = ky;
        double r36277 = sin(r36276);
        double r36278 = kx;
        double r36279 = sin(r36278);
        double r36280 = 2.0;
        double r36281 = pow(r36279, r36280);
        double r36282 = pow(r36277, r36280);
        double r36283 = r36281 + r36282;
        double r36284 = sqrt(r36283);
        double r36285 = r36277 / r36284;
        double r36286 = th;
        double r36287 = sin(r36286);
        double r36288 = r36285 * r36287;
        return r36288;
}
↓
double f(double kx, double ky, double th) {
        double r36289 = ky;
        double r36290 = sin(r36289);
        double r36291 = th;
        double r36292 = sin(r36291);
        double r36293 = kx;
        double r36294 = sin(r36293);
        double r36295 = hypot(r36290, r36294);
        double r36296 = r36292 / r36295;
        double r36297 = r36290 * r36296;
        return r36297;
}

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. Simplified8.8

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

  5. Applied div-inv8.9

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

  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 2019326 +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)))