Average Error: 12.2 → 12.6
Time: 1.0m
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\right)
double f(double kx, double ky, double th) {
        double r1906604 = ky;
        double r1906605 = sin(r1906604);
        double r1906606 = kx;
        double r1906607 = sin(r1906606);
        double r1906608 = 2.0;
        double r1906609 = pow(r1906607, r1906608);
        double r1906610 = pow(r1906605, r1906608);
        double r1906611 = r1906609 + r1906610;
        double r1906612 = sqrt(r1906611);
        double r1906613 = r1906605 / r1906612;
        double r1906614 = th;
        double r1906615 = sin(r1906614);
        double r1906616 = r1906613 * r1906615;
        return r1906616;
}
↓
double f(double kx, double ky, double th) {
        double r1906617 = ky;
        double r1906618 = sin(r1906617);
        double r1906619 = th;
        double r1906620 = sin(r1906619);
        double r1906621 = 1.0;
        double r1906622 = kx;
        double r1906623 = sin(r1906622);
        double r1906624 = r1906623 * r1906623;
        double r1906625 = r1906618 * r1906618;
        double r1906626 = r1906624 + r1906625;
        double r1906627 = r1906621 / r1906626;
        double r1906628 = sqrt(r1906627);
        double r1906629 = r1906620 * r1906628;
        double r1906630 = r1906618 * r1906629;
        return r1906630;
}

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. Simplified12.2

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

  3. Taylor expanded around inf 13.9

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

  4. Simplified12.6

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

  5. Final simplification12.6

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

Reproduce

herbie shell --seed 2019107 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))