Average Error: 12.2 → 12.5
Time: 10.3s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r37254 = ky;
        double r37255 = sin(r37254);
        double r37256 = kx;
        double r37257 = sin(r37256);
        double r37258 = 2.0;
        double r37259 = pow(r37257, r37258);
        double r37260 = pow(r37255, r37258);
        double r37261 = r37259 + r37260;
        double r37262 = sqrt(r37261);
        double r37263 = r37255 / r37262;
        double r37264 = th;
        double r37265 = sin(r37264);
        double r37266 = r37263 * r37265;
        return r37266;
}
↓
double f(double kx, double ky, double th) {
        double r37267 = ky;
        double r37268 = sin(r37267);
        double r37269 = kx;
        double r37270 = sin(r37269);
        double r37271 = 2.0;
        double r37272 = pow(r37270, r37271);
        double r37273 = pow(r37268, r37271);
        double r37274 = r37272 + r37273;
        double r37275 = sqrt(r37274);
        double r37276 = r37268 / r37275;
        double r37277 = cbrt(r37276);
        double r37278 = 2.0;
        double r37279 = pow(r37270, r37278);
        double r37280 = pow(r37268, r37278);
        double r37281 = r37279 + r37280;
        double r37282 = sqrt(r37281);
        double r37283 = r37268 / r37282;
        double r37284 = cbrt(r37283);
        double r37285 = r37277 * r37284;
        double r37286 = r37285 * r37284;
        double r37287 = th;
        double r37288 = sin(r37287);
        double r37289 = r37286 * r37288;
        return r37289;
}

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 12.2

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

  3. Applied add-cube-cbrt12.5

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\]

  4. Taylor expanded around inf 12.5

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\color{blue}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sin th\]

  5. Final simplification12.5

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2019347 
(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)))