Average Error: 12.6 → 12.9
Time: 31.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
double f(double kx, double ky, double th) {
        double r32594 = ky;
        double r32595 = sin(r32594);
        double r32596 = kx;
        double r32597 = sin(r32596);
        double r32598 = 2.0;
        double r32599 = pow(r32597, r32598);
        double r32600 = pow(r32595, r32598);
        double r32601 = r32599 + r32600;
        double r32602 = sqrt(r32601);
        double r32603 = r32595 / r32602;
        double r32604 = th;
        double r32605 = sin(r32604);
        double r32606 = r32603 * r32605;
        return r32606;
}
↓
double f(double kx, double ky, double th) {
        double r32607 = ky;
        double r32608 = sin(r32607);
        double r32609 = kx;
        double r32610 = sin(r32609);
        double r32611 = cbrt(r32610);
        double r32612 = r32611 * r32611;
        double r32613 = 2.0;
        double r32614 = pow(r32612, r32613);
        double r32615 = pow(r32611, r32613);
        double r32616 = r32614 * r32615;
        double r32617 = pow(r32608, r32613);
        double r32618 = r32616 + r32617;
        double r32619 = sqrt(r32618);
        double r32620 = r32608 / r32619;
        double r32621 = th;
        double r32622 = sin(r32621);
        double r32623 = r32620 * r32622;
        return r32623;
}

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 12.6

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

  3. Applied add-cube-cbrt12.9

    \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sqrt[3]{\sin kx}\right)}}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

  4. Applied unpow-prod-down12.9

    \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2}} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

  5. Final simplification12.9

    \[\leadsto \frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

Reproduce

herbie shell --seed 2019326 
(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)))