Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 59.4s

Precision: 64

Math

\[\frac{\cos th}{\sqrt{2.0}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2.0}} \cdot \left(a2 \cdot a2\right)\]

↓

\[\frac{a1}{\left(\sqrt{\sqrt{2.0}} \cdot \sqrt{\sqrt{\sqrt{2.0}}}\right) \cdot \frac{\sqrt{\sqrt{\sqrt{2.0}}}}{a1}} \cdot \cos th + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

C

double f(double a1, double a2, double th) {
        double r2918948 = th;
        double r2918949 = cos(r2918948);
        double r2918950 = 2.0;
        double r2918951 = sqrt(r2918950);
        double r2918952 = r2918949 / r2918951;
        double r2918953 = a1;
        double r2918954 = r2918953 * r2918953;
        double r2918955 = r2918952 * r2918954;
        double r2918956 = a2;
        double r2918957 = r2918956 * r2918956;
        double r2918958 = r2918952 * r2918957;
        double r2918959 = r2918955 + r2918958;
        return r2918959;
}

↓

double f(double a1, double a2, double th) {
        double r2918960 = a1;
        double r2918961 = 2.0;
        double r2918962 = sqrt(r2918961);
        double r2918963 = sqrt(r2918962);
        double r2918964 = sqrt(r2918963);
        double r2918965 = r2918963 * r2918964;
        double r2918966 = r2918964 / r2918960;
        double r2918967 = r2918965 * r2918966;
        double r2918968 = r2918960 / r2918967;
        double r2918969 = th;
        double r2918970 = cos(r2918969);
        double r2918971 = r2918968 * r2918970;
        double r2918972 = r2918970 / r2918963;
        double r2918973 = r2918972 / r2918963;
        double r2918974 = a2;
        double r2918975 = r2918974 * r2918974;
        double r2918976 = r2918973 * r2918975;
        double r2918977 = r2918971 + r2918976;
        return r2918977;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

1. Initial program 0.5

   \[\frac{\cos th}{\sqrt{2.0}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2.0}} \cdot \left(a2 \cdot a2\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.5

   \[\leadsto \frac{\cos th}{\sqrt{2.0}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{\color{blue}{\sqrt{2.0} \cdot \sqrt{2.0}}}} \cdot \left(a2 \cdot a2\right)\]

4. Applied sqrt-prod 0.5

   \[\leadsto \frac{\cos th}{\sqrt{2.0}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\color{blue}{\sqrt{\sqrt{2.0}} \cdot \sqrt{\sqrt{2.0}}}} \cdot \left(a2 \cdot a2\right)\]

5. Applied associate-/r* 0.5

   \[\leadsto \frac{\cos th}{\sqrt{2.0}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}}} \cdot \left(a2 \cdot a2\right)\]
6. Using strategy rm

7. Applied div-inv 0.5

   \[\leadsto \color{blue}{\left(\cos th \cdot \frac{1}{\sqrt{2.0}}\right)} \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

8. Applied associate-*l* 0.5

   \[\leadsto \color{blue}{\cos th \cdot \left(\frac{1}{\sqrt{2.0}} \cdot \left(a1 \cdot a1\right)\right)} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

9. Simplified 0.5

   \[\leadsto \cos th \cdot \color{blue}{\frac{a1}{\frac{\sqrt{2.0}}{a1}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]
10. Using strategy rm

11. Applied *-un-lft-identity 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\frac{\sqrt{2.0}}{\color{blue}{1 \cdot a1}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

12. Applied add-sqr-sqrt 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\frac{\sqrt{\color{blue}{\sqrt{2.0} \cdot \sqrt{2.0}}}}{1 \cdot a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

13. Applied sqrt-prod 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\frac{\color{blue}{\sqrt{\sqrt{2.0}} \cdot \sqrt{\sqrt{2.0}}}}{1 \cdot a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

14. Applied times-frac 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\color{blue}{\frac{\sqrt{\sqrt{2.0}}}{1} \cdot \frac{\sqrt{\sqrt{2.0}}}{a1}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

15. Simplified 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\color{blue}{\sqrt{\sqrt{2.0}}} \cdot \frac{\sqrt{\sqrt{2.0}}}{a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]
16. Using strategy rm

17. Applied *-un-lft-identity 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\sqrt{\sqrt{2.0}} \cdot \frac{\sqrt{\sqrt{2.0}}}{\color{blue}{1 \cdot a1}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

18. Applied add-sqr-sqrt 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\sqrt{\sqrt{2.0}} \cdot \frac{\sqrt{\sqrt{\color{blue}{\sqrt{2.0} \cdot \sqrt{2.0}}}}}{1 \cdot a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

19. Applied sqrt-prod 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\sqrt{\sqrt{2.0}} \cdot \frac{\sqrt{\color{blue}{\sqrt{\sqrt{2.0}} \cdot \sqrt{\sqrt{2.0}}}}}{1 \cdot a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

20. Applied sqrt-prod 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\sqrt{\sqrt{2.0}} \cdot \frac{\color{blue}{\sqrt{\sqrt{\sqrt{2.0}}} \cdot \sqrt{\sqrt{\sqrt{2.0}}}}}{1 \cdot a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

21. Applied times-frac 0.5

    \[\leadsto \cos th \cdot \frac{a1}{\sqrt{\sqrt{2.0}} \cdot \color{blue}{\left(\frac{\sqrt{\sqrt{\sqrt{2.0}}}}{1} \cdot \frac{\sqrt{\sqrt{\sqrt{2.0}}}}{a1}\right)}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

22. Applied associate-*r* 0.4

    \[\leadsto \cos th \cdot \frac{a1}{\color{blue}{\left(\sqrt{\sqrt{2.0}} \cdot \frac{\sqrt{\sqrt{\sqrt{2.0}}}}{1}\right) \cdot \frac{\sqrt{\sqrt{\sqrt{2.0}}}}{a1}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

23. Simplified 0.4

    \[\leadsto \cos th \cdot \frac{a1}{\color{blue}{\left(\sqrt{\sqrt{\sqrt{2.0}}} \cdot \sqrt{\sqrt{2.0}}\right)} \cdot \frac{\sqrt{\sqrt{\sqrt{2.0}}}}{a1}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

24. Final simplification 0.4

    \[\leadsto \frac{a1}{\left(\sqrt{\sqrt{2.0}} \cdot \sqrt{\sqrt{\sqrt{2.0}}}\right) \cdot \frac{\sqrt{\sqrt{\sqrt{2.0}}}}{a1}} \cdot \cos th + \frac{\frac{\cos th}{\sqrt{\sqrt{2.0}}}}{\sqrt{\sqrt{2.0}}} \cdot \left(a2 \cdot a2\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))