Average Error: 12.4 → 12.5
Time: 35.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sin kx \cdot \sqrt[3]{\sin kx}\right)}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \left(\sin kx \cdot \sqrt[3]{\sin kx}\right)}}
double f(double kx, double ky, double th) {
        double r866920 = ky;
        double r866921 = sin(r866920);
        double r866922 = kx;
        double r866923 = sin(r866922);
        double r866924 = 2.0;
        double r866925 = pow(r866923, r866924);
        double r866926 = pow(r866921, r866924);
        double r866927 = r866925 + r866926;
        double r866928 = sqrt(r866927);
        double r866929 = r866921 / r866928;
        double r866930 = th;
        double r866931 = sin(r866930);
        double r866932 = r866929 * r866931;
        return r866932;
}
↓
double f(double kx, double ky, double th) {
        double r866933 = th;
        double r866934 = sin(r866933);
        double r866935 = ky;
        double r866936 = sin(r866935);
        double r866937 = r866936 * r866936;
        double r866938 = kx;
        double r866939 = sin(r866938);
        double r866940 = cbrt(r866939);
        double r866941 = r866940 * r866940;
        double r866942 = r866939 * r866940;
        double r866943 = r866941 * r866942;
        double r866944 = r866937 + r866943;
        double r866945 = sqrt(r866944);
        double r866946 = r866936 / r866945;
        double r866947 = r866934 * r866946;
        return r866947;
}

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 12.4

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

  2. Simplified12.4

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

  4. Applied add-cube-cbrt12.5

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

  5. Applied associate-*l*12.5

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

  6. Final simplification12.5

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

Reproduce

herbie shell --seed 2019152 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))