Average Error: 12.5 → 12.9
Time: 39.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sin th \cdot \frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sqrt[3]{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right) \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sin th \cdot \frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sqrt[3]{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right) \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}
double f(double kx, double ky, double th) {
        double r808461 = ky;
        double r808462 = sin(r808461);
        double r808463 = kx;
        double r808464 = sin(r808463);
        double r808465 = 2.0;
        double r808466 = pow(r808464, r808465);
        double r808467 = pow(r808462, r808465);
        double r808468 = r808466 + r808467;
        double r808469 = sqrt(r808468);
        double r808470 = r808462 / r808469;
        double r808471 = th;
        double r808472 = sin(r808471);
        double r808473 = r808470 * r808472;
        return r808473;
}
↓
double f(double kx, double ky, double th) {
        double r808474 = th;
        double r808475 = sin(r808474);
        double r808476 = ky;
        double r808477 = sin(r808476);
        double r808478 = cbrt(r808477);
        double r808479 = r808478 * r808478;
        double r808480 = kx;
        double r808481 = sin(r808480);
        double r808482 = r808481 * r808481;
        double r808483 = r808477 * r808477;
        double r808484 = r808482 + r808483;
        double r808485 = sqrt(r808484);
        double r808486 = cbrt(r808485);
        double r808487 = r808486 * r808486;
        double r808488 = r808479 / r808487;
        double r808489 = r808475 * r808488;
        double r808490 = r808478 / r808486;
        double r808491 = r808489 * r808490;
        return r808491;
}

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. Simplified12.5

    \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt13.3

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

  5. Applied add-cube-cbrt12.9

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

  6. Applied times-frac12.9

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

  7. Applied associate-*r*12.9

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

  8. Final simplification12.9

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

Reproduce

herbie shell --seed 2019144 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))