Average Error: 12.2 → 12.4
Time: 36.3s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \sqrt[3]{\sin kx \cdot \sin kx} \cdot \left(\sqrt[3]{\sin kx \cdot \sin kx} \cdot \sqrt[3]{\sin kx \cdot \sin kx}\right)}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin ky \cdot \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \sqrt[3]{\sin kx \cdot \sin kx} \cdot \left(\sqrt[3]{\sin kx \cdot \sin kx} \cdot \sqrt[3]{\sin kx \cdot \sin kx}\right)}}
double f(double kx, double ky, double th) {
        double r766333 = ky;
        double r766334 = sin(r766333);
        double r766335 = kx;
        double r766336 = sin(r766335);
        double r766337 = 2.0;
        double r766338 = pow(r766336, r766337);
        double r766339 = pow(r766334, r766337);
        double r766340 = r766338 + r766339;
        double r766341 = sqrt(r766340);
        double r766342 = r766334 / r766341;
        double r766343 = th;
        double r766344 = sin(r766343);
        double r766345 = r766342 * r766344;
        return r766345;
}
↓
double f(double kx, double ky, double th) {
        double r766346 = ky;
        double r766347 = sin(r766346);
        double r766348 = th;
        double r766349 = sin(r766348);
        double r766350 = r766347 * r766347;
        double r766351 = kx;
        double r766352 = sin(r766351);
        double r766353 = r766352 * r766352;
        double r766354 = cbrt(r766353);
        double r766355 = r766354 * r766354;
        double r766356 = r766354 * r766355;
        double r766357 = r766350 + r766356;
        double r766358 = sqrt(r766357);
        double r766359 = r766349 / r766358;
        double r766360 = r766347 * r766359;
        return r766360;
}

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 12.2

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

  3. Applied div-inv12.3

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

  4. Applied associate-*l*12.3

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

  5. Simplified12.3

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

  7. Applied add-cube-cbrt12.4

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

  8. Final simplification12.4

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

Reproduce

herbie shell --seed 2019134 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))