Average Error: 12.5 → 9.3
Time: 32.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\sin th \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\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(\sin th \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)
double f(double kx, double ky, double th) {
        double r1247963 = ky;
        double r1247964 = sin(r1247963);
        double r1247965 = kx;
        double r1247966 = sin(r1247965);
        double r1247967 = 2.0;
        double r1247968 = pow(r1247966, r1247967);
        double r1247969 = pow(r1247964, r1247967);
        double r1247970 = r1247968 + r1247969;
        double r1247971 = sqrt(r1247970);
        double r1247972 = r1247964 / r1247971;
        double r1247973 = th;
        double r1247974 = sin(r1247973);
        double r1247975 = r1247972 * r1247974;
        return r1247975;
}
↓
double f(double kx, double ky, double th) {
        double r1247976 = ky;
        double r1247977 = sin(r1247976);
        double r1247978 = kx;
        double r1247979 = sin(r1247978);
        double r1247980 = hypot(r1247977, r1247979);
        double r1247981 = r1247977 / r1247980;
        double r1247982 = cbrt(r1247981);
        double r1247983 = r1247982 * r1247982;
        double r1247984 = th;
        double r1247985 = sin(r1247984);
        double r1247986 = r1247985 * r1247982;
        double r1247987 = r1247983 * r1247986;
        return r1247987;
}

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. Taylor expanded around inf 12.5

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

  3. Simplified8.9

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

  5. Applied add-cube-cbrt9.2

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

  6. Applied associate-*l*9.3

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

  8. Applied *-un-lft-identity9.3

    \[\leadsto \color{blue}{\left(1 \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\right)} \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)\]

  9. Final simplification9.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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)))