Average Error: 12.1 → 8.8
Time: 37.7s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right)\right)}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right)\right)}
double f(double kx, double ky, double th) {
        double r535737 = ky;
        double r535738 = sin(r535737);
        double r535739 = kx;
        double r535740 = sin(r535739);
        double r535741 = 2.0;
        double r535742 = pow(r535740, r535741);
        double r535743 = pow(r535738, r535741);
        double r535744 = r535742 + r535743;
        double r535745 = sqrt(r535744);
        double r535746 = r535738 / r535745;
        double r535747 = th;
        double r535748 = sin(r535747);
        double r535749 = r535746 * r535748;
        return r535749;
}
↓
double f(double kx, double ky, double th) {
        double r535750 = th;
        double r535751 = sin(r535750);
        double r535752 = ky;
        double r535753 = sin(r535752);
        double r535754 = kx;
        double r535755 = sin(r535754);
        double r535756 = hypot(r535753, r535755);
        double r535757 = log1p(r535756);
        double r535758 = expm1(r535757);
        double r535759 = r535753 / r535758;
        double r535760 = r535751 * r535759;
        return r535760;
}

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. Simplified8.7

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

  4. Applied expm1-log1p-u8.8

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

  5. Final simplification8.8

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

Reproduce

herbie shell --seed 2019149 +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)))