Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 32.3s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r2154766 = th;
        double r2154767 = cos(r2154766);
        double r2154768 = 2.0;
        double r2154769 = sqrt(r2154768);
        double r2154770 = r2154767 / r2154769;
        double r2154771 = a1;
        double r2154772 = r2154771 * r2154771;
        double r2154773 = r2154770 * r2154772;
        double r2154774 = a2;
        double r2154775 = r2154774 * r2154774;
        double r2154776 = r2154770 * r2154775;
        double r2154777 = r2154773 + r2154776;
        return r2154777;
}

↓

double f(double a1, double a2, double th) {
        double r2154778 = 1.0;
        double r2154779 = 2.0;
        double r2154780 = sqrt(r2154779);
        double r2154781 = sqrt(r2154780);
        double r2154782 = r2154778 / r2154781;
        double r2154783 = cbrt(r2154780);
        double r2154784 = sqrt(r2154783);
        double r2154785 = r2154782 / r2154784;
        double r2154786 = a2;
        double r2154787 = r2154786 * r2154786;
        double r2154788 = r2154785 * r2154787;
        double r2154789 = th;
        double r2154790 = cos(r2154789);
        double r2154791 = r2154783 * r2154783;
        double r2154792 = sqrt(r2154791);
        double r2154793 = r2154790 / r2154792;
        double r2154794 = r2154788 * r2154793;
        double r2154795 = a1;
        double r2154796 = r2154783 / r2154790;
        double r2154797 = r2154795 / r2154796;
        double r2154798 = r2154795 / r2154791;
        double r2154799 = r2154797 * r2154798;
        double r2154800 = r2154794 + r2154799;
        return r2154800;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

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. Taylor expanded around inf 0.5

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

3. Simplified 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied add-cube-cbrt 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied div-inv 0.5

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

12. Applied times-frac 0.5

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

13. Applied associate-*l* 0.4

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

15. Applied *-un-lft-identity 0.4

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

16. Applied add-cube-cbrt 0.4

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

17. Applied times-frac 0.5

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

18. Applied times-frac 0.4

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))