Average Error: 12.2 → 8.7
Time: 31.3s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}
double f(double kx, double ky, double th) {
        double r48252 = ky;
        double r48253 = sin(r48252);
        double r48254 = kx;
        double r48255 = sin(r48254);
        double r48256 = 2.0;
        double r48257 = pow(r48255, r48256);
        double r48258 = pow(r48253, r48256);
        double r48259 = r48257 + r48258;
        double r48260 = sqrt(r48259);
        double r48261 = r48253 / r48260;
        double r48262 = th;
        double r48263 = sin(r48262);
        double r48264 = r48261 * r48263;
        return r48264;
}
↓
double f(double kx, double ky, double th) {
        double r48265 = th;
        double r48266 = sin(r48265);
        double r48267 = kx;
        double r48268 = sin(r48267);
        double r48269 = 2.0;
        double r48270 = 2.0;
        double r48271 = r48269 / r48270;
        double r48272 = pow(r48268, r48271);
        double r48273 = ky;
        double r48274 = sin(r48273);
        double r48275 = pow(r48274, r48271);
        double r48276 = hypot(r48272, r48275);
        double r48277 = r48276 / r48274;
        double r48278 = r48266 / r48277;
        return r48278;
}

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 12.2

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

  3. Applied sqr-pow12.2

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

  4. Applied sqr-pow12.2

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

  5. Applied hypot-def8.7

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \sin th\]
  6. Using strategy rm

  7. Applied clear-num8.7

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}} \cdot \sin th\]
  8. Using strategy rm

  9. Applied associate-*l/8.7

    \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}}\]

  10. Simplified8.7

    \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}\]

  11. Final simplification8.7

    \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}\]

Reproduce

herbie shell --seed 2019194 +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)))