Average Error: 12.0 → 12.3
Time: 41.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)\right)}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)\right)}}
double f(double kx, double ky, double th) {
        double r1059347 = ky;
        double r1059348 = sin(r1059347);
        double r1059349 = kx;
        double r1059350 = sin(r1059349);
        double r1059351 = 2.0;
        double r1059352 = pow(r1059350, r1059351);
        double r1059353 = pow(r1059348, r1059351);
        double r1059354 = r1059352 + r1059353;
        double r1059355 = sqrt(r1059354);
        double r1059356 = r1059348 / r1059355;
        double r1059357 = th;
        double r1059358 = sin(r1059357);
        double r1059359 = r1059356 * r1059358;
        return r1059359;
}
↓
double f(double kx, double ky, double th) {
        double r1059360 = th;
        double r1059361 = sin(r1059360);
        double r1059362 = ky;
        double r1059363 = sin(r1059362);
        double r1059364 = r1059363 * r1059363;
        double r1059365 = kx;
        double r1059366 = sin(r1059365);
        double r1059367 = cbrt(r1059366);
        double r1059368 = r1059367 * r1059367;
        double r1059369 = r1059368 * r1059368;
        double r1059370 = r1059368 * r1059369;
        double r1059371 = r1059364 + r1059370;
        double r1059372 = sqrt(r1059371);
        double r1059373 = r1059363 / r1059372;
        double r1059374 = r1059361 * r1059373;
        return r1059374;
}

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 12.0

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

  2. Simplified12.0

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

  4. Applied add-cube-cbrt12.1

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

  5. Applied add-cube-cbrt12.3

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

  6. Applied swap-sqr12.3

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

  7. Final simplification12.3

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

Reproduce

herbie shell --seed 2019151 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))