Average Error: 12.5 → 8.9
Time: 11.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r48065 = ky;
        double r48066 = sin(r48065);
        double r48067 = kx;
        double r48068 = sin(r48067);
        double r48069 = 2.0;
        double r48070 = pow(r48068, r48069);
        double r48071 = pow(r48066, r48069);
        double r48072 = r48070 + r48071;
        double r48073 = sqrt(r48072);
        double r48074 = r48066 / r48073;
        double r48075 = th;
        double r48076 = sin(r48075);
        double r48077 = r48074 * r48076;
        return r48077;
}
↓
double f(double kx, double ky, double th) {
        double r48078 = ky;
        double r48079 = sin(r48078);
        double r48080 = kx;
        double r48081 = sin(r48080);
        double r48082 = 2.0;
        double r48083 = 2.0;
        double r48084 = r48082 / r48083;
        double r48085 = pow(r48081, r48084);
        double r48086 = pow(r48079, r48084);
        double r48087 = hypot(r48085, r48086);
        double r48088 = r48079 / r48087;
        double r48089 = expm1(r48088);
        double r48090 = log1p(r48089);
        double r48091 = th;
        double r48092 = sin(r48091);
        double r48093 = r48090 * r48092;
        return r48093;
}

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 sqr-pow12.5

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

  4. Applied sqr-pow12.5

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

  5. Applied hypot-def8.9

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

  7. Applied log1p-expm1-u8.9

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right)} \cdot \sin th\]

  8. Final simplification8.9

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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)))