Average Error: 12.6 → 13.0
Time: 11.8s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{\frac{\sin th}{\sqrt{\sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}}{\sqrt{\sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin ky}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{\frac{\sin th}{\sqrt{\sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}}{\sqrt{\sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}
double f(double kx, double ky, double th) {
        double r44991 = ky;
        double r44992 = sin(r44991);
        double r44993 = kx;
        double r44994 = sin(r44993);
        double r44995 = 2.0;
        double r44996 = pow(r44994, r44995);
        double r44997 = pow(r44992, r44995);
        double r44998 = r44996 + r44997;
        double r44999 = sqrt(r44998);
        double r45000 = r44992 / r44999;
        double r45001 = th;
        double r45002 = sin(r45001);
        double r45003 = r45000 * r45002;
        return r45003;
}
↓
double f(double kx, double ky, double th) {
        double r45004 = ky;
        double r45005 = sin(r45004);
        double r45006 = kx;
        double r45007 = sin(r45006);
        double r45008 = 2.0;
        double r45009 = pow(r45007, r45008);
        double r45010 = pow(r45005, r45008);
        double r45011 = r45009 + r45010;
        double r45012 = sqrt(r45011);
        double r45013 = sqrt(r45012);
        double r45014 = r45005 / r45013;
        double r45015 = th;
        double r45016 = sin(r45015);
        double r45017 = cbrt(r45011);
        double r45018 = r45017 * r45017;
        double r45019 = sqrt(r45018);
        double r45020 = sqrt(r45019);
        double r45021 = r45016 / r45020;
        double r45022 = sqrt(r45017);
        double r45023 = sqrt(r45022);
        double r45024 = r45021 / r45023;
        double r45025 = r45014 * r45024;
        return r45025;
}

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 12.6

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

  3. Applied div-inv12.7

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

  4. Applied associate-*l*12.7

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

  5. Simplified12.6

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

  7. Applied add-sqr-sqrt12.6

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

  8. Applied sqrt-prod12.9

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

  9. Applied *-un-lft-identity12.9

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

  10. Applied times-frac12.9

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

  11. Applied associate-*r*12.9

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

  12. Simplified12.8

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

  14. Applied add-cube-cbrt12.9

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

  15. Applied sqrt-prod12.9

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

  16. Applied sqrt-prod12.9

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

  17. Applied associate-/r*13.0

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

  18. Final simplification13.0

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

Reproduce

herbie shell --seed 2020089 
(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)))