Average Error: 12.7 → 12.7
Time: 2.6m
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}
double f(double kx, double ky, double th) {
        double r4293891 = ky;
        double r4293892 = sin(r4293891);
        double r4293893 = kx;
        double r4293894 = sin(r4293893);
        double r4293895 = 2.0;
        double r4293896 = pow(r4293894, r4293895);
        double r4293897 = pow(r4293892, r4293895);
        double r4293898 = r4293896 + r4293897;
        double r4293899 = sqrt(r4293898);
        double r4293900 = r4293892 / r4293899;
        double r4293901 = th;
        double r4293902 = sin(r4293901);
        double r4293903 = r4293900 * r4293902;
        return r4293903;
}
↓
double f(double kx, double ky, double th) {
        double r4293904 = th;
        double r4293905 = sin(r4293904);
        double r4293906 = 1.0;
        double r4293907 = kx;
        double r4293908 = sin(r4293907);
        double r4293909 = r4293908 * r4293908;
        double r4293910 = ky;
        double r4293911 = sin(r4293910);
        double r4293912 = r4293911 * r4293911;
        double r4293913 = r4293909 + r4293912;
        double r4293914 = sqrt(r4293913);
        double r4293915 = r4293914 / r4293911;
        double r4293916 = r4293906 / r4293915;
        double r4293917 = r4293905 * r4293916;
        return r4293917;
}

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 12.7

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

  2. Simplified12.7

    \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity12.7

    \[\leadsto \sin th \cdot \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\]

  5. Applied associate-/l*12.7

    \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}}\]

  6. Taylor expanded around inf 12.7

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

  7. Simplified12.7

    \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}}\]

  8. Final simplification12.7

    \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}{\sin ky}}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))