Average Error: 12.5 → 8.9
Time: 12.1s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sqrt{1}\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sqrt{1}\right)
double f(double kx, double ky, double th) {
        double r53228 = ky;
        double r53229 = sin(r53228);
        double r53230 = kx;
        double r53231 = sin(r53230);
        double r53232 = 2.0;
        double r53233 = pow(r53231, r53232);
        double r53234 = pow(r53229, r53232);
        double r53235 = r53233 + r53234;
        double r53236 = sqrt(r53235);
        double r53237 = r53229 / r53236;
        double r53238 = th;
        double r53239 = sin(r53238);
        double r53240 = r53237 * r53239;
        return r53240;
}
↓
double f(double kx, double ky, double th) {
        double r53241 = th;
        double r53242 = sin(r53241);
        double r53243 = ky;
        double r53244 = sin(r53243);
        double r53245 = kx;
        double r53246 = sin(r53245);
        double r53247 = 2.0;
        double r53248 = 2.0;
        double r53249 = r53247 / r53248;
        double r53250 = pow(r53246, r53249);
        double r53251 = pow(r53244, r53249);
        double r53252 = hypot(r53250, r53251);
        double r53253 = r53244 / r53252;
        double r53254 = 1.0;
        double r53255 = sqrt(r53254);
        double r53256 = r53253 * r53255;
        double r53257 = r53242 * r53256;
        return r53257;
}

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 12.5

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

  3. Applied sqr-pow12.5

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

  4. Applied sqr-pow12.5

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

  5. Applied hypot-def8.9

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

  7. Applied add-sqr-sqrt9.2

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

  8. Applied *-un-lft-identity9.2

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

  9. Applied times-frac9.2

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

  11. Applied *-un-lft-identity9.2

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

  12. Applied add-sqr-sqrt9.2

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

  13. Applied times-frac9.2

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

  14. Applied associate-*l*9.2

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

  15. Simplified8.9

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

  16. Final simplification8.9

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

Reproduce

herbie shell --seed 2020060 +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)))