Average Error: 4.1 → 0.2
Time: 11.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}
double f(double kx, double ky, double th) {
        double r46872 = ky;
        double r46873 = sin(r46872);
        double r46874 = kx;
        double r46875 = sin(r46874);
        double r46876 = 2.0;
        double r46877 = pow(r46875, r46876);
        double r46878 = pow(r46873, r46876);
        double r46879 = r46877 + r46878;
        double r46880 = sqrt(r46879);
        double r46881 = r46873 / r46880;
        double r46882 = th;
        double r46883 = sin(r46882);
        double r46884 = r46881 * r46883;
        return r46884;
}
↓
double f(double kx, double ky, double th) {
        double r46885 = th;
        double r46886 = sin(r46885);
        double r46887 = kx;
        double r46888 = sin(r46887);
        double r46889 = 2.0;
        double r46890 = 2.0;
        double r46891 = r46889 / r46890;
        double r46892 = pow(r46888, r46891);
        double r46893 = ky;
        double r46894 = sin(r46893);
        double r46895 = pow(r46894, r46891);
        double r46896 = hypot(r46892, r46895);
        double r46897 = r46896 / r46894;
        double r46898 = r46886 / r46897;
        return r46898;
}

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 4.1

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

  3. Applied sqr-pow4.1

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

  4. Applied sqr-pow4.1

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

  5. Applied hypot-def0.2

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \sin th\]
  6. Using strategy rm

  7. Applied *-commutative0.2

    \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}}\]
  8. Using strategy rm

  9. Applied clear-num0.3

    \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}}\]
  10. Using strategy rm

  11. Applied un-div-inv0.2

    \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}}\]

  12. Final simplification0.2

    \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}\]

Reproduce

herbie shell --seed 2020062 +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)))