Average Error: 12.2 → 8.5
Time: 11.0s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{\sin th}{1 \cdot \mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin ky \cdot \frac{\sin th}{1 \cdot \mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}
double f(double kx, double ky, double th) {
        double r48948 = ky;
        double r48949 = sin(r48948);
        double r48950 = kx;
        double r48951 = sin(r48950);
        double r48952 = 2.0;
        double r48953 = pow(r48951, r48952);
        double r48954 = pow(r48949, r48952);
        double r48955 = r48953 + r48954;
        double r48956 = sqrt(r48955);
        double r48957 = r48949 / r48956;
        double r48958 = th;
        double r48959 = sin(r48958);
        double r48960 = r48957 * r48959;
        return r48960;
}
↓
double f(double kx, double ky, double th) {
        double r48961 = ky;
        double r48962 = sin(r48961);
        double r48963 = th;
        double r48964 = sin(r48963);
        double r48965 = 1.0;
        double r48966 = kx;
        double r48967 = sin(r48966);
        double r48968 = 2.0;
        double r48969 = 2.0;
        double r48970 = r48968 / r48969;
        double r48971 = pow(r48967, r48970);
        double r48972 = pow(r48962, r48970);
        double r48973 = hypot(r48971, r48972);
        double r48974 = r48965 * r48973;
        double r48975 = r48964 / r48974;
        double r48976 = r48962 * r48975;
        return r48976;
}

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 12.2

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

  3. Applied sqr-pow12.2

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

  4. Applied sqr-pow12.2

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

  5. Applied hypot-def8.5

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

  7. Applied div-inv8.6

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

  8. Applied associate-*l*8.6

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

  9. Simplified8.5

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

  11. Applied *-un-lft-identity8.5

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

  12. Final simplification8.5

    \[\leadsto \sin ky \cdot \frac{\sin th}{1 \cdot \mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\]

Reproduce

herbie shell --seed 2019347 +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)))