Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 44.6s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r1913843 = th;
        double r1913844 = cos(r1913843);
        double r1913845 = 2.0;
        double r1913846 = sqrt(r1913845);
        double r1913847 = r1913844 / r1913846;
        double r1913848 = a1;
        double r1913849 = r1913848 * r1913848;
        double r1913850 = r1913847 * r1913849;
        double r1913851 = a2;
        double r1913852 = r1913851 * r1913851;
        double r1913853 = r1913847 * r1913852;
        double r1913854 = r1913850 + r1913853;
        return r1913854;
}

↓

double f(double a1, double a2, double th) {
        double r1913855 = th;
        double r1913856 = cos(r1913855);
        double r1913857 = a2;
        double r1913858 = r1913857 * r1913857;
        double r1913859 = r1913856 * r1913858;
        double r1913860 = 2.0;
        double r1913861 = sqrt(r1913860);
        double r1913862 = r1913859 / r1913861;
        double r1913863 = a1;
        double r1913864 = r1913863 * r1913863;
        double r1913865 = r1913864 * r1913856;
        double r1913866 = sqrt(r1913861);
        double r1913867 = r1913865 / r1913866;
        double r1913868 = 1.0;
        double r1913869 = sqrt(r1913866);
        double r1913870 = r1913868 / r1913869;
        double r1913871 = r1913870 / r1913869;
        double r1913872 = r1913867 * r1913871;
        double r1913873 = r1913862 + r1913872;
        return r1913873;
}

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. Using strategy rm

3. Applied associate-*l/ 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied *-un-lft-identity 0.5

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

14. Applied times-frac 0.5

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

15. Applied times-frac 0.5

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

16. Applied associate-*l* 0.5

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019138 +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))))