Average Error: 12.1 → 8.8
Time: 34.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\left(\sin kx\right), \left(\sin ky\right)\right)}{\sin ky}}\right)\right)\right)\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\left(\sin kx\right), \left(\sin ky\right)\right)}{\sin ky}}\right)\right)\right)\right)
double f(double kx, double ky, double th) {
        double r738723 = ky;
        double r738724 = sin(r738723);
        double r738725 = kx;
        double r738726 = sin(r738725);
        double r738727 = 2.0;
        double r738728 = pow(r738726, r738727);
        double r738729 = pow(r738724, r738727);
        double r738730 = r738728 + r738729;
        double r738731 = sqrt(r738730);
        double r738732 = r738724 / r738731;
        double r738733 = th;
        double r738734 = sin(r738733);
        double r738735 = r738732 * r738734;
        return r738735;
}
↓
double f(double kx, double ky, double th) {
        double r738736 = th;
        double r738737 = sin(r738736);
        double r738738 = kx;
        double r738739 = sin(r738738);
        double r738740 = ky;
        double r738741 = sin(r738740);
        double r738742 = hypot(r738739, r738741);
        double r738743 = r738742 / r738741;
        double r738744 = r738737 / r738743;
        double r738745 = log1p(r738744);
        double r738746 = expm1(r738745);
        return r738746;
}

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 12.1

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

  2. Simplified8.8

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

  4. Applied clear-num8.8

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

  6. Applied un-div-inv8.8

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

  8. Applied expm1-log1p-u8.8

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\left(\sin kx\right), \left(\sin ky\right)\right)}{\sin ky}}\right)\right)\right)\right)}\]

  9. Final simplification8.8

    \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\left(\sin kx\right), \left(\sin ky\right)\right)}{\sin ky}}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019130 +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)))