Average Error: 12.7 → 12.9
Time: 10.0s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{1}{\frac{\sqrt{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{1}{\frac{\sqrt{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sin th
double f(double kx, double ky, double th) {
        double r35788 = ky;
        double r35789 = sin(r35788);
        double r35790 = kx;
        double r35791 = sin(r35790);
        double r35792 = 2.0;
        double r35793 = pow(r35791, r35792);
        double r35794 = pow(r35789, r35792);
        double r35795 = r35793 + r35794;
        double r35796 = sqrt(r35795);
        double r35797 = r35789 / r35796;
        double r35798 = th;
        double r35799 = sin(r35798);
        double r35800 = r35797 * r35799;
        return r35800;
}
↓
double f(double kx, double ky, double th) {
        double r35801 = 1.0;
        double r35802 = kx;
        double r35803 = sin(r35802);
        double r35804 = 2.0;
        double r35805 = pow(r35803, r35804);
        double r35806 = cbrt(r35805);
        double r35807 = r35806 * r35806;
        double r35808 = r35807 * r35806;
        double r35809 = ky;
        double r35810 = sin(r35809);
        double r35811 = pow(r35810, r35804);
        double r35812 = r35808 + r35811;
        double r35813 = sqrt(r35812);
        double r35814 = r35813 / r35810;
        double r35815 = r35801 / r35814;
        double r35816 = th;
        double r35817 = sin(r35816);
        double r35818 = r35815 * r35817;
        return r35818;
}

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 12.7

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

  3. Applied clear-num12.8

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

  4. Taylor expanded around inf 12.8

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

  6. Applied add-cube-cbrt12.9

    \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sin th\]

  7. Final simplification12.9

    \[\leadsto \frac{1}{\frac{\sqrt{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sin th\]

Reproduce

herbie shell --seed 2019356 
(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)))