Average Error: 3.8 → 0.2
Time: 11.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\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)}\right)\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r49965 = ky;
        double r49966 = sin(r49965);
        double r49967 = kx;
        double r49968 = sin(r49967);
        double r49969 = 2.0;
        double r49970 = pow(r49968, r49969);
        double r49971 = pow(r49966, r49969);
        double r49972 = r49970 + r49971;
        double r49973 = sqrt(r49972);
        double r49974 = r49966 / r49973;
        double r49975 = th;
        double r49976 = sin(r49975);
        double r49977 = r49974 * r49976;
        return r49977;
}
↓
double f(double kx, double ky, double th) {
        double r49978 = ky;
        double r49979 = sin(r49978);
        double r49980 = kx;
        double r49981 = sin(r49980);
        double r49982 = 2.0;
        double r49983 = 2.0;
        double r49984 = r49982 / r49983;
        double r49985 = pow(r49981, r49984);
        double r49986 = pow(r49979, r49984);
        double r49987 = hypot(r49985, r49986);
        double r49988 = r49979 / r49987;
        double r49989 = expm1(r49988);
        double r49990 = log1p(r49989);
        double r49991 = th;
        double r49992 = sin(r49991);
        double r49993 = r49990 * r49992;
        return r49993;
}

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 3.8

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

  3. Applied sqr-pow3.8

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

  4. Applied sqr-pow3.8

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

  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 log1p-expm1-u0.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right)} \cdot \sin th\]

  8. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2020057 +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)))