Average Error: 13.1 → 13.2
Time: 36.5s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin th}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky
double f(double kx, double ky, double th) {
        double r1447263 = ky;
        double r1447264 = sin(r1447263);
        double r1447265 = kx;
        double r1447266 = sin(r1447265);
        double r1447267 = 2.0;
        double r1447268 = pow(r1447266, r1447267);
        double r1447269 = pow(r1447264, r1447267);
        double r1447270 = r1447268 + r1447269;
        double r1447271 = sqrt(r1447270);
        double r1447272 = r1447264 / r1447271;
        double r1447273 = th;
        double r1447274 = sin(r1447273);
        double r1447275 = r1447272 * r1447274;
        return r1447275;
}
↓
double f(double kx, double ky, double th) {
        double r1447276 = th;
        double r1447277 = sin(r1447276);
        double r1447278 = kx;
        double r1447279 = sin(r1447278);
        double r1447280 = 2.0;
        double r1447281 = pow(r1447279, r1447280);
        double r1447282 = ky;
        double r1447283 = sin(r1447282);
        double r1447284 = pow(r1447283, r1447280);
        double r1447285 = r1447281 + r1447284;
        double r1447286 = sqrt(r1447285);
        double r1447287 = r1447277 / r1447286;
        double r1447288 = r1447287 * r1447283;
        return r1447288;
}

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 13.1

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

  3. Applied div-inv13.2

    \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)} \cdot \sin th\]

  4. Applied associate-*l*13.2

    \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)}\]

  5. Simplified13.2

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

  6. Final simplification13.2

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

Reproduce

herbie shell --seed 2019169 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))