Average Error: 12.9 → 9.7
Time: 1.1m
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \cdot \sin th\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \cdot \sin th\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)
double f(double kx, double ky, double th) {
        double r1108186 = ky;
        double r1108187 = sin(r1108186);
        double r1108188 = kx;
        double r1108189 = sin(r1108188);
        double r1108190 = 2.0;
        double r1108191 = pow(r1108189, r1108190);
        double r1108192 = pow(r1108187, r1108190);
        double r1108193 = r1108191 + r1108192;
        double r1108194 = sqrt(r1108193);
        double r1108195 = r1108187 / r1108194;
        double r1108196 = th;
        double r1108197 = sin(r1108196);
        double r1108198 = r1108195 * r1108197;
        return r1108198;
}
↓
double f(double kx, double ky, double th) {
        double r1108199 = ky;
        double r1108200 = sin(r1108199);
        double r1108201 = cbrt(r1108200);
        double r1108202 = kx;
        double r1108203 = sin(r1108202);
        double r1108204 = hypot(r1108200, r1108203);
        double r1108205 = cbrt(r1108204);
        double r1108206 = r1108201 / r1108205;
        double r1108207 = cbrt(r1108206);
        double r1108208 = r1108206 * r1108206;
        double r1108209 = cbrt(r1108208);
        double r1108210 = r1108207 * r1108209;
        double r1108211 = th;
        double r1108212 = sin(r1108211);
        double r1108213 = r1108210 * r1108212;
        double r1108214 = r1108200 / r1108204;
        double r1108215 = cbrt(r1108214);
        double r1108216 = r1108215 * r1108215;
        double r1108217 = r1108213 * r1108216;
        return r1108217;
}

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. Simplified9.3

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

  5. Applied add-cube-cbrt9.7

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} \cdot \sin th\]

  6. Applied associate-*l*9.7

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt9.7

    \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}} \cdot \sin th\right)\]

  9. Applied add-cube-cbrt9.7

    \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right) \cdot \sqrt[3]{\sin ky}}}{\left(\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sin th\right)\]

  10. Applied times-frac9.7

    \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}} \cdot \sin th\right)\]

  11. Applied cbrt-prod9.7

    \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right)} \cdot \sin th\right)\]

  12. Simplified9.7

    \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right) \cdot \left(\left(\color{blue}{\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \cdot \sin th\right)\]

  13. Final simplification9.7

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \cdot \sin th\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sqrt[3]{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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)))