Average Error: 12.9 → 9.4
Time: 10.5s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)
double f(double kx, double ky, double th) {
        double r37085 = ky;
        double r37086 = sin(r37085);
        double r37087 = kx;
        double r37088 = sin(r37087);
        double r37089 = 2.0;
        double r37090 = pow(r37088, r37089);
        double r37091 = pow(r37086, r37089);
        double r37092 = r37090 + r37091;
        double r37093 = sqrt(r37092);
        double r37094 = r37086 / r37093;
        double r37095 = th;
        double r37096 = sin(r37095);
        double r37097 = r37094 * r37096;
        return r37097;
}
↓
double f(double kx, double ky, double th) {
        double r37098 = ky;
        double r37099 = sin(r37098);
        double r37100 = kx;
        double r37101 = sin(r37100);
        double r37102 = hypot(r37099, r37101);
        double r37103 = r37099 / r37102;
        double r37104 = cbrt(r37103);
        double r37105 = cbrt(r37099);
        double r37106 = r37104 * r37105;
        double r37107 = cbrt(r37102);
        double r37108 = r37106 / r37107;
        double r37109 = th;
        double r37110 = sin(r37109);
        double r37111 = r37104 * r37110;
        double r37112 = r37108 * r37111;
        return r37112;
}

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 12.9

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

  2. Taylor expanded around inf 12.9

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

  3. Simplified9.1

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\]
  4. Using strategy rm

  5. Applied add-cube-cbrt9.5

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

  6. Applied associate-*l*9.5

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

  8. Applied cbrt-div9.4

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

  9. Applied associate-*r/9.4

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

  10. Final simplification9.4

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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)))