Average Error: 12.8 → 13.1
Time: 9.8s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\left(\sqrt[3]{\sqrt[3]{\sin kx}} \cdot \sqrt[3]{\sqrt[3]{\sin kx}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\sin kx}} \cdot \sqrt[3]{\sqrt[3]{\sin kx}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin kx}}}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
\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(\left(\sqrt[3]{\sqrt[3]{\sin kx}} \cdot \sqrt[3]{\sqrt[3]{\sin kx}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\sin kx}} \cdot \sqrt[3]{\sqrt[3]{\sin kx}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin kx}}}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
double f(double kx, double ky, double th) {
        double r35594 = ky;
        double r35595 = sin(r35594);
        double r35596 = kx;
        double r35597 = sin(r35596);
        double r35598 = 2.0;
        double r35599 = pow(r35597, r35598);
        double r35600 = pow(r35595, r35598);
        double r35601 = r35599 + r35600;
        double r35602 = sqrt(r35601);
        double r35603 = r35595 / r35602;
        double r35604 = th;
        double r35605 = sin(r35604);
        double r35606 = r35603 * r35605;
        return r35606;
}
↓
double f(double kx, double ky, double th) {
        double r35607 = ky;
        double r35608 = sin(r35607);
        double r35609 = kx;
        double r35610 = sin(r35609);
        double r35611 = cbrt(r35610);
        double r35612 = r35611 * r35611;
        double r35613 = 2.0;
        double r35614 = pow(r35612, r35613);
        double r35615 = cbrt(r35611);
        double r35616 = r35615 * r35615;
        double r35617 = r35616 * r35615;
        double r35618 = cbrt(r35617);
        double r35619 = r35616 * r35618;
        double r35620 = pow(r35619, r35613);
        double r35621 = r35614 * r35620;
        double r35622 = pow(r35608, r35613);
        double r35623 = r35621 + r35622;
        double r35624 = sqrt(r35623);
        double r35625 = r35608 / r35624;
        double r35626 = th;
        double r35627 = sin(r35626);
        double r35628 = r35625 * r35627;
        return r35628;
}

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 12.8

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

  3. Applied add-cube-cbrt13.0

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

  4. Applied unpow-prod-down13.0

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

  6. Applied add-cube-cbrt13.0

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

  8. Applied add-cube-cbrt13.1

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

  9. Final simplification13.1

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

Reproduce

herbie shell --seed 2019362 
(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)))