Average Error: 12.5 → 12.9
Time: 10.8s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)
double f(double kx, double ky, double th) {
        double r43039 = ky;
        double r43040 = sin(r43039);
        double r43041 = kx;
        double r43042 = sin(r43041);
        double r43043 = 2.0;
        double r43044 = pow(r43042, r43043);
        double r43045 = pow(r43040, r43043);
        double r43046 = r43044 + r43045;
        double r43047 = sqrt(r43046);
        double r43048 = r43040 / r43047;
        double r43049 = th;
        double r43050 = sin(r43049);
        double r43051 = r43048 * r43050;
        return r43051;
}
↓
double f(double kx, double ky, double th) {
        double r43052 = ky;
        double r43053 = sin(r43052);
        double r43054 = cbrt(r43053);
        double r43055 = r43054 * r43054;
        double r43056 = kx;
        double r43057 = sin(r43056);
        double r43058 = 2.0;
        double r43059 = pow(r43057, r43058);
        double r43060 = pow(r43053, r43058);
        double r43061 = r43059 + r43060;
        double r43062 = sqrt(r43061);
        double r43063 = cbrt(r43062);
        double r43064 = r43063 * r43063;
        double r43065 = r43055 / r43064;
        double r43066 = r43054 / r43063;
        double r43067 = th;
        double r43068 = sin(r43067);
        double r43069 = r43066 * r43068;
        double r43070 = r43065 * r43069;
        return r43070;
}

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 12.5

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

  3. Applied add-cube-cbrt13.3

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

  4. Applied add-cube-cbrt12.9

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

  5. Applied times-frac12.9

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

  6. Applied associate-*l*12.9

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

  7. Final simplification12.9

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

Reproduce

herbie shell --seed 2020049 
(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)))