Average Error: 12.5 → 12.7
Time: 11.1s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \frac{1}{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin th}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin ky \cdot \frac{1}{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin th}}
double f(double kx, double ky, double th) {
        double r38759 = ky;
        double r38760 = sin(r38759);
        double r38761 = kx;
        double r38762 = sin(r38761);
        double r38763 = 2.0;
        double r38764 = pow(r38762, r38763);
        double r38765 = pow(r38760, r38763);
        double r38766 = r38764 + r38765;
        double r38767 = sqrt(r38766);
        double r38768 = r38760 / r38767;
        double r38769 = th;
        double r38770 = sin(r38769);
        double r38771 = r38768 * r38770;
        return r38771;
}
↓
double f(double kx, double ky, double th) {
        double r38772 = ky;
        double r38773 = sin(r38772);
        double r38774 = 1.0;
        double r38775 = kx;
        double r38776 = sin(r38775);
        double r38777 = 2.0;
        double r38778 = pow(r38776, r38777);
        double r38779 = pow(r38773, r38777);
        double r38780 = r38778 + r38779;
        double r38781 = sqrt(r38780);
        double r38782 = th;
        double r38783 = sin(r38782);
        double r38784 = r38781 / r38783;
        double r38785 = r38774 / r38784;
        double r38786 = r38773 * r38785;
        return r38786;
}

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 12.5

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

  3. Applied div-inv12.6

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

  4. Applied associate-*l*12.7

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

  5. Simplified12.6

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

  7. Applied clear-num12.7

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

  8. Final simplification12.7

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

Reproduce

herbie shell --seed 2020060 
(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)))