Average Error: 12.5 → 12.9
Time: 45.7s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\frac{\sin ky}{\sqrt{\sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}}}{\sqrt{\sqrt{\sqrt{\sqrt[3]{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \left|\sqrt[3]{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right|}}} \cdot \left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\frac{\sin ky}{\sqrt{\sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}}}{\sqrt{\sqrt{\sqrt{\sqrt[3]{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \left|\sqrt[3]{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right|}}} \cdot \left(\sin th \cdot \frac{1}{\sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right)
double f(double kx, double ky, double th) {
        double r888615 = ky;
        double r888616 = sin(r888615);
        double r888617 = kx;
        double r888618 = sin(r888617);
        double r888619 = 2.0;
        double r888620 = pow(r888618, r888619);
        double r888621 = pow(r888616, r888619);
        double r888622 = r888620 + r888621;
        double r888623 = sqrt(r888622);
        double r888624 = r888616 / r888623;
        double r888625 = th;
        double r888626 = sin(r888625);
        double r888627 = r888624 * r888626;
        return r888627;
}
↓
double f(double kx, double ky, double th) {
        double r888628 = ky;
        double r888629 = sin(r888628);
        double r888630 = kx;
        double r888631 = sin(r888630);
        double r888632 = r888631 * r888631;
        double r888633 = r888629 * r888629;
        double r888634 = r888632 + r888633;
        double r888635 = sqrt(r888634);
        double r888636 = sqrt(r888635);
        double r888637 = sqrt(r888636);
        double r888638 = r888629 / r888637;
        double r888639 = cbrt(r888634);
        double r888640 = sqrt(r888639);
        double r888641 = fabs(r888639);
        double r888642 = r888640 * r888641;
        double r888643 = sqrt(r888642);
        double r888644 = sqrt(r888643);
        double r888645 = r888638 / r888644;
        double r888646 = th;
        double r888647 = sin(r888646);
        double r888648 = 1.0;
        double r888649 = r888648 / r888636;
        double r888650 = r888647 * r888649;
        double r888651 = r888645 * r888650;
        return r888651;
}

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 12.5

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

  2. Simplified12.5

    \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt12.5

    \[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky} \cdot \sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}}\]

  5. Applied sqrt-prod12.8

    \[\leadsto \sin th \cdot \frac{\sin ky}{\color{blue}{\sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}}\]

  6. Applied *-un-lft-identity12.8

    \[\leadsto \sin th \cdot \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sqrt{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\]

  7. Applied times-frac12.8

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

  8. Applied associate-*r*12.8

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

  10. Applied add-sqr-sqrt12.9

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

  11. Applied associate-/r*12.9

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

  13. Applied add-cube-cbrt12.9

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

  14. Applied sqrt-prod12.9

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

  15. Simplified12.9

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

  16. Final simplification12.9

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

Reproduce

herbie shell --seed 2019133 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))