Average Error: 12.1 → 12.4
Time: 40.2s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx} \cdot \sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sqrt{\sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}}\right) \cdot \sin th\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx} \cdot \sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sqrt{\sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}}\right) \cdot \sin th\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}
double f(double kx, double ky, double th) {
        double r786444 = ky;
        double r786445 = sin(r786444);
        double r786446 = kx;
        double r786447 = sin(r786446);
        double r786448 = 2.0;
        double r786449 = pow(r786447, r786448);
        double r786450 = pow(r786445, r786448);
        double r786451 = r786449 + r786450;
        double r786452 = sqrt(r786451);
        double r786453 = r786445 / r786452;
        double r786454 = th;
        double r786455 = sin(r786454);
        double r786456 = r786453 * r786455;
        return r786456;
}
↓
double f(double kx, double ky, double th) {
        double r786457 = ky;
        double r786458 = sin(r786457);
        double r786459 = r786458 * r786458;
        double r786460 = kx;
        double r786461 = sin(r786460);
        double r786462 = r786461 * r786461;
        double r786463 = r786459 + r786462;
        double r786464 = sqrt(r786463);
        double r786465 = r786458 / r786464;
        double r786466 = cbrt(r786465);
        double r786467 = cbrt(r786463);
        double r786468 = r786467 * r786467;
        double r786469 = sqrt(r786468);
        double r786470 = sqrt(r786467);
        double r786471 = r786469 * r786470;
        double r786472 = r786458 / r786471;
        double r786473 = cbrt(r786472);
        double r786474 = r786466 * r786473;
        double r786475 = th;
        double r786476 = sin(r786475);
        double r786477 = r786474 * r786476;
        double r786478 = r786477 * r786466;
        return r786478;
}

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 12.1

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

  2. Simplified12.1

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

  4. Applied add-cube-cbrt12.4

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

  5. Applied associate-*r*12.4

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

  7. Applied add-cube-cbrt12.4

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

  8. Applied sqrt-prod12.4

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

  9. Final simplification12.4

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

Reproduce

herbie shell --seed 2019149 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))