Average Error: 0.5 → 0.5
Time: 1.6m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right) \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{1}{\cos th}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right) \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{1}{\cos th}}
double f(double a1, double a2, double th) {
        double r4139707 = th;
        double r4139708 = cos(r4139707);
        double r4139709 = 2.0;
        double r4139710 = sqrt(r4139709);
        double r4139711 = r4139708 / r4139710;
        double r4139712 = a1;
        double r4139713 = r4139712 * r4139712;
        double r4139714 = r4139711 * r4139713;
        double r4139715 = a2;
        double r4139716 = r4139715 * r4139715;
        double r4139717 = r4139711 * r4139716;
        double r4139718 = r4139714 + r4139717;
        return r4139718;
}
↓
double f(double a1, double a2, double th) {
        double r4139719 = a1;
        double r4139720 = r4139719 * r4139719;
        double r4139721 = a2;
        double r4139722 = r4139721 * r4139721;
        double r4139723 = r4139720 + r4139722;
        double r4139724 = sqrt(r4139723);
        double r4139725 = 2.0;
        double r4139726 = sqrt(r4139725);
        double r4139727 = cbrt(r4139726);
        double r4139728 = cbrt(r4139727);
        double r4139729 = r4139728 * r4139727;
        double r4139730 = r4139728 * r4139729;
        double r4139731 = r4139724 / r4139730;
        double r4139732 = 1.0;
        double r4139733 = r4139732 / r4139728;
        double r4139734 = r4139731 * r4139733;
        double r4139735 = r4139724 / r4139727;
        double r4139736 = th;
        double r4139737 = cos(r4139736);
        double r4139738 = r4139732 / r4139737;
        double r4139739 = r4139735 / r4139738;
        double r4139740 = r4139734 * r4139739;
        return r4139740;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.4

    \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\color{blue}{1 \cdot \cos th}}}\]

  5. Applied add-cube-cbrt0.4

    \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}{1 \cdot \cos th}}\]

  6. Applied times-frac0.4

    \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\color{blue}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{1} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\cos th}}}\]

  7. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{1}}}{\frac{\sqrt[3]{\sqrt{2}}}{\cos th}}}\]

  8. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}{\frac{\sqrt[3]{\sqrt{2}}}{\cos th}}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt{2}}}{\color{blue}{1 \cdot \cos th}}}\]

  11. Applied add-cube-cbrt0.4

    \[\leadsto \frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}}{1 \cdot \cos th}}\]

  12. Applied times-frac0.5

    \[\leadsto \frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\color{blue}{\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{1} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\cos th}}}\]

  13. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{1} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\cos th}}\]

  14. Applied times-frac0.6

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}}{\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{1} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\cos th}}\]

  15. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{1}} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\cos th}}}\]

  16. Simplified0.4

    \[\leadsto \color{blue}{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\cos th}}\]
  17. Using strategy rm

  18. Applied div-inv0.5

    \[\leadsto \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\color{blue}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \frac{1}{\cos th}}}\]

  19. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\color{blue}{1 \cdot \sqrt[3]{\sqrt{2}}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \frac{1}{\cos th}}\]

  20. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\frac{\color{blue}{1 \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}}{1 \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \frac{1}{\cos th}}\]

  21. Applied times-frac0.5

    \[\leadsto \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \frac{1}{\cos th}}\]

  22. Applied times-frac0.5

    \[\leadsto \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}} \cdot \color{blue}{\left(\frac{\frac{1}{1}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{1}{\cos th}}\right)}\]

  23. Applied associate-*r*0.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right) \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{1}{\cos th}}}\]

  24. Simplified0.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{1}{\cos th}}\]

  25. Final simplification0.5

    \[\leadsto \left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right) \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}{\frac{1}{\cos th}}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))