Average Error: 13.0 → 9.6
Time: 17.1s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r49336 = ky;
        double r49337 = sin(r49336);
        double r49338 = kx;
        double r49339 = sin(r49338);
        double r49340 = 2.0;
        double r49341 = pow(r49339, r49340);
        double r49342 = pow(r49337, r49340);
        double r49343 = r49341 + r49342;
        double r49344 = sqrt(r49343);
        double r49345 = r49337 / r49344;
        double r49346 = th;
        double r49347 = sin(r49346);
        double r49348 = r49345 * r49347;
        return r49348;
}
↓
double f(double kx, double ky, double th) {
        double r49349 = ky;
        double r49350 = sin(r49349);
        double r49351 = cbrt(r49350);
        double r49352 = r49351 * r49351;
        double r49353 = kx;
        double r49354 = sin(r49353);
        double r49355 = hypot(r49354, r49350);
        double r49356 = cbrt(r49355);
        double r49357 = r49356 * r49356;
        double r49358 = r49352 / r49357;
        double r49359 = cbrt(r49358);
        double r49360 = r49359 * r49359;
        double r49361 = r49351 / r49356;
        double r49362 = cbrt(r49361);
        double r49363 = r49362 * r49362;
        double r49364 = r49360 * r49363;
        double r49365 = r49350 / r49355;
        double r49366 = cbrt(r49365);
        double r49367 = r49364 * r49366;
        double r49368 = th;
        double r49369 = sin(r49368);
        double r49370 = r49367 * r49369;
        return r49370;
}

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 add-cube-cbrt9.5

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right)} \cdot \sin th\]
  6. Using strategy rm

  7. Applied add-cube-cbrt9.5

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  8. Applied add-cube-cbrt9.5

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right) \cdot \sqrt[3]{\sin ky}}}{\left(\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  9. Applied times-frac9.5

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \sqrt[3]{\color{blue}{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  10. Applied cbrt-prod9.5

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  11. Applied add-cube-cbrt9.6

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  12. Applied add-cube-cbrt9.6

    \[\leadsto \left(\left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right) \cdot \sqrt[3]{\sin ky}}}{\left(\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  13. Applied times-frac9.6

    \[\leadsto \left(\left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  14. Applied cbrt-prod9.6

    \[\leadsto \left(\left(\color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  15. Applied swap-sqr9.6

    \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right)} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

  16. Final simplification9.6

    \[\leadsto \left(\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}\right)\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2020045 +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)))