Average Error: 12.1 → 12.4
Time: 49.6s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)\right) \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sin th \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)\right) \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}
double f(double kx, double ky, double th) {
        double r1276006 = ky;
        double r1276007 = sin(r1276006);
        double r1276008 = kx;
        double r1276009 = sin(r1276008);
        double r1276010 = 2.0;
        double r1276011 = pow(r1276009, r1276010);
        double r1276012 = pow(r1276007, r1276010);
        double r1276013 = r1276011 + r1276012;
        double r1276014 = sqrt(r1276013);
        double r1276015 = r1276007 / r1276014;
        double r1276016 = th;
        double r1276017 = sin(r1276016);
        double r1276018 = r1276015 * r1276017;
        return r1276018;
}
↓
double f(double kx, double ky, double th) {
        double r1276019 = th;
        double r1276020 = sin(r1276019);
        double r1276021 = ky;
        double r1276022 = sin(r1276021);
        double r1276023 = r1276022 * r1276022;
        double r1276024 = kx;
        double r1276025 = sin(r1276024);
        double r1276026 = r1276025 * r1276025;
        double r1276027 = r1276023 + r1276026;
        double r1276028 = sqrt(r1276027);
        double r1276029 = r1276022 / r1276028;
        double r1276030 = cbrt(r1276029);
        double r1276031 = r1276030 * r1276030;
        double r1276032 = r1276020 * r1276031;
        double r1276033 = cbrt(r1276022);
        double r1276034 = cbrt(r1276028);
        double r1276035 = r1276033 / r1276034;
        double r1276036 = r1276032 * r1276035;
        return r1276036;
}

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 12.1

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

  2. Simplified12.1

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

  4. Applied add-cube-cbrt12.4

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

  5. Applied associate-*r*12.4

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

  7. Applied cbrt-div12.4

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

  8. Final simplification12.4

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

Reproduce

herbie shell --seed 2019104 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))