Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 12.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r106844 = th;
        double r106845 = cos(r106844);
        double r106846 = 2.0;
        double r106847 = sqrt(r106846);
        double r106848 = r106845 / r106847;
        double r106849 = a1;
        double r106850 = r106849 * r106849;
        double r106851 = r106848 * r106850;
        double r106852 = a2;
        double r106853 = r106852 * r106852;
        double r106854 = r106848 * r106853;
        double r106855 = r106851 + r106854;
        return r106855;
}

↓

double f(double a1, double a2, double th) {
        double r106856 = 1.0;
        double r106857 = 2.0;
        double r106858 = sqrt(r106857);
        double r106859 = sqrt(r106858);
        double r106860 = sqrt(r106859);
        double r106861 = r106856 / r106860;
        double r106862 = r106861 / r106860;
        double r106863 = a1;
        double r106864 = th;
        double r106865 = cos(r106864);
        double r106866 = r106865 / r106859;
        double r106867 = r106866 * r106863;
        double r106868 = r106863 * r106867;
        double r106869 = r106862 * r106868;
        double r106870 = sqrt(r106856);
        double r106871 = r106856 / r106870;
        double r106872 = a2;
        double r106873 = r106865 * r106872;
        double r106874 = r106873 / r106858;
        double r106875 = r106871 * r106874;
        double r106876 = r106875 * r106872;
        double r106877 = r106869 + r106876;
        return r106877;
}

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-*r* 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied *-un-lft-identity 0.5

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

12. Applied times-frac 0.5

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

13. Applied associate-*l* 0.5

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

14. Simplified 0.5

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

16. Applied add-sqr-sqrt 0.5

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

17. Applied sqrt-prod 0.5

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

18. Applied sqrt-prod 0.5

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

19. Applied add-sqr-sqrt 0.5

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

20. Applied sqrt-prod 0.5

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

21. Applied sqrt-prod 0.5

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

22. Applied *-un-lft-identity 0.5

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

23. Applied times-frac 0.5

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

24. Applied times-frac 0.5

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

25. Applied associate-*l* 0.5

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

26. Simplified 0.4

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

27. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))