Average Error: 12.3 → 12.8
Time: 31.0s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\sqrt[3]{\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky} \cdot \sqrt[3]{\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky}\right)\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\sqrt[3]{\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky} \cdot \sqrt[3]{\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky}\right)\right)
double f(double kx, double ky, double th) {
        double r34833 = ky;
        double r34834 = sin(r34833);
        double r34835 = kx;
        double r34836 = sin(r34835);
        double r34837 = 2.0;
        double r34838 = pow(r34836, r34837);
        double r34839 = pow(r34834, r34837);
        double r34840 = r34838 + r34839;
        double r34841 = sqrt(r34840);
        double r34842 = r34834 / r34841;
        double r34843 = th;
        double r34844 = sin(r34843);
        double r34845 = r34842 * r34844;
        return r34845;
}
↓
double f(double kx, double ky, double th) {
        double r34846 = th;
        double r34847 = sin(r34846);
        double r34848 = ky;
        double r34849 = sin(r34848);
        double r34850 = cbrt(r34849);
        double r34851 = kx;
        double r34852 = sin(r34851);
        double r34853 = 2.0;
        double r34854 = pow(r34852, r34853);
        double r34855 = pow(r34849, r34853);
        double r34856 = r34854 + r34855;
        double r34857 = sqrt(r34856);
        double r34858 = cbrt(r34857);
        double r34859 = r34850 / r34858;
        double r34860 = 1.0;
        double r34861 = r34860 / r34856;
        double r34862 = sqrt(r34861);
        double r34863 = r34862 * r34849;
        double r34864 = cbrt(r34863);
        double r34865 = r34864 * r34864;
        double r34866 = r34859 * r34865;
        double r34867 = r34847 * r34866;
        return r34867;
}

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 12.3

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

  2. Taylor expanded around inf 12.6

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

  4. Applied add-cube-cbrt12.8

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

  5. Simplified12.8

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

  6. Simplified12.8

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

  8. Applied sqrt-div12.8

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

  9. Applied associate-*r/12.8

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

  10. Applied cbrt-div12.8

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

  11. Simplified12.8

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

  12. Final simplification12.8

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

Reproduce

herbie shell --seed 2019174 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))