Average Error: 12.9 → 9.1
Time: 10.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r42119 = ky;
        double r42120 = sin(r42119);
        double r42121 = kx;
        double r42122 = sin(r42121);
        double r42123 = 2.0;
        double r42124 = pow(r42122, r42123);
        double r42125 = pow(r42120, r42123);
        double r42126 = r42124 + r42125;
        double r42127 = sqrt(r42126);
        double r42128 = r42120 / r42127;
        double r42129 = th;
        double r42130 = sin(r42129);
        double r42131 = r42128 * r42130;
        return r42131;
}
↓
double f(double kx, double ky, double th) {
        double r42132 = ky;
        double r42133 = sin(r42132);
        double r42134 = 1.0;
        double r42135 = kx;
        double r42136 = sin(r42135);
        double r42137 = hypot(r42133, r42136);
        double r42138 = r42134 / r42137;
        double r42139 = r42133 * r42138;
        double r42140 = th;
        double r42141 = sin(r42140);
        double r42142 = r42139 * r42141;
        return r42142;
}

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 12.9

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

  2. Taylor expanded around inf 12.9

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

  3. Simplified9.0

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

  5. Applied div-inv9.1

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

  6. Final simplification9.1

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

Reproduce

herbie shell --seed 2020002 +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)))