Average Error: 12.7 → 9.0
Time: 45.6s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\left(\sin kx\right), \left(\sin ky\right)\right)}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\left(\sin kx\right), \left(\sin ky\right)\right)}
double f(double kx, double ky, double th) {
        double r1257556 = ky;
        double r1257557 = sin(r1257556);
        double r1257558 = kx;
        double r1257559 = sin(r1257558);
        double r1257560 = 2.0;
        double r1257561 = pow(r1257559, r1257560);
        double r1257562 = pow(r1257557, r1257560);
        double r1257563 = r1257561 + r1257562;
        double r1257564 = sqrt(r1257563);
        double r1257565 = r1257557 / r1257564;
        double r1257566 = th;
        double r1257567 = sin(r1257566);
        double r1257568 = r1257565 * r1257567;
        return r1257568;
}
↓
double f(double kx, double ky, double th) {
        double r1257569 = th;
        double r1257570 = sin(r1257569);
        double r1257571 = ky;
        double r1257572 = sin(r1257571);
        double r1257573 = kx;
        double r1257574 = sin(r1257573);
        double r1257575 = hypot(r1257574, r1257572);
        double r1257576 = r1257572 / r1257575;
        double r1257577 = r1257570 * r1257576;
        return r1257577;
}

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. Simplified11.3

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

  4. Applied *-un-lft-identity11.3

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

  5. Applied times-frac9.0

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

  6. Simplified9.0

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

  8. Applied *-commutative9.0

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

  9. Final simplification9.0

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

Reproduce

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