Average Error: 12.3 → 12.7
Time: 1.1m
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\right)
double f(double kx, double ky, double th) {
        double r2186793 = ky;
        double r2186794 = sin(r2186793);
        double r2186795 = kx;
        double r2186796 = sin(r2186795);
        double r2186797 = 2.0;
        double r2186798 = pow(r2186796, r2186797);
        double r2186799 = pow(r2186794, r2186797);
        double r2186800 = r2186798 + r2186799;
        double r2186801 = sqrt(r2186800);
        double r2186802 = r2186794 / r2186801;
        double r2186803 = th;
        double r2186804 = sin(r2186803);
        double r2186805 = r2186802 * r2186804;
        return r2186805;
}
↓
double f(double kx, double ky, double th) {
        double r2186806 = th;
        double r2186807 = sin(r2186806);
        double r2186808 = ky;
        double r2186809 = sin(r2186808);
        double r2186810 = 1.0;
        double r2186811 = kx;
        double r2186812 = sin(r2186811);
        double r2186813 = r2186812 * r2186812;
        double r2186814 = r2186809 * r2186809;
        double r2186815 = r2186813 + r2186814;
        double r2186816 = r2186810 / r2186815;
        double r2186817 = sqrt(r2186816);
        double r2186818 = r2186809 * r2186817;
        double r2186819 = r2186807 * r2186818;
        return r2186819;
}

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 12.3

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

  2. Simplified12.3

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

  3. Taylor expanded around inf 14.0

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

  4. Simplified12.7

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

  5. Final simplification12.7

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

Reproduce

herbie shell --seed 2019128 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))