Average Error: 12.3 → 12.7
Time: 11.2s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{1}{\sqrt{1}} \cdot \left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{1}{\sqrt{1}} \cdot \left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\right)
double f(double kx, double ky, double th) {
        double r39226 = ky;
        double r39227 = sin(r39226);
        double r39228 = kx;
        double r39229 = sin(r39228);
        double r39230 = 2.0;
        double r39231 = pow(r39229, r39230);
        double r39232 = pow(r39227, r39230);
        double r39233 = r39231 + r39232;
        double r39234 = sqrt(r39233);
        double r39235 = r39227 / r39234;
        double r39236 = th;
        double r39237 = sin(r39236);
        double r39238 = r39235 * r39237;
        return r39238;
}
↓
double f(double kx, double ky, double th) {
        double r39239 = 1.0;
        double r39240 = sqrt(r39239);
        double r39241 = r39239 / r39240;
        double r39242 = ky;
        double r39243 = sin(r39242);
        double r39244 = cbrt(r39243);
        double r39245 = r39244 * r39244;
        double r39246 = kx;
        double r39247 = sin(r39246);
        double r39248 = 2.0;
        double r39249 = pow(r39247, r39248);
        double r39250 = pow(r39243, r39248);
        double r39251 = r39249 + r39250;
        double r39252 = sqrt(r39251);
        double r39253 = cbrt(r39252);
        double r39254 = r39253 * r39253;
        double r39255 = r39245 / r39254;
        double r39256 = r39244 / r39253;
        double r39257 = th;
        double r39258 = sin(r39257);
        double r39259 = r39256 * r39258;
        double r39260 = r39255 * r39259;
        double r39261 = r39241 * r39260;
        return r39261;
}

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 12.3

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

  3. Applied *-un-lft-identity12.3

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

  4. Applied sqrt-prod12.3

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

  5. Applied *-un-lft-identity12.3

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

  6. Applied times-frac12.3

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

  7. Applied associate-*l*12.3

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

  9. Applied add-cube-cbrt13.1

    \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\frac{\sin ky}{\color{blue}{\left(\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}} \cdot \sin th\right)\]

  10. Applied add-cube-cbrt12.7

    \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right) \cdot \sqrt[3]{\sin ky}}}{\left(\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]

  11. Applied times-frac12.7

    \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\right)\]

  12. Applied associate-*l*12.7

    \[\leadsto \frac{1}{\sqrt{1}} \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\right)}\]

  13. Final simplification12.7

    \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\right)\]

Reproduce

herbie shell --seed 2020001 
(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)))