Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 45.5s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r3097873 = th;
        double r3097874 = cos(r3097873);
        double r3097875 = 2.0;
        double r3097876 = sqrt(r3097875);
        double r3097877 = r3097874 / r3097876;
        double r3097878 = a1;
        double r3097879 = r3097878 * r3097878;
        double r3097880 = r3097877 * r3097879;
        double r3097881 = a2;
        double r3097882 = r3097881 * r3097881;
        double r3097883 = r3097877 * r3097882;
        double r3097884 = r3097880 + r3097883;
        return r3097884;
}

↓

double f(double a1, double a2, double th) {
        double r3097885 = 1.0;
        double r3097886 = 2.0;
        double r3097887 = sqrt(r3097886);
        double r3097888 = sqrt(r3097887);
        double r3097889 = r3097885 / r3097888;
        double r3097890 = a2;
        double r3097891 = r3097890 * r3097890;
        double r3097892 = a1;
        double r3097893 = r3097892 * r3097892;
        double r3097894 = r3097891 + r3097893;
        double r3097895 = sqrt(r3097888);
        double r3097896 = th;
        double r3097897 = cos(r3097896);
        double r3097898 = r3097895 / r3097897;
        double r3097899 = r3097894 / r3097898;
        double r3097900 = r3097899 / r3097895;
        double r3097901 = r3097889 * r3097900;
        return r3097901;
}

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. Simplified 0.5

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.6

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

6. Applied associate-/l* 0.6

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

8. Applied associate-/r/ 0.6

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

9. Applied associate-/r* 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied *-un-lft-identity 0.5

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

15. Applied add-sqr-sqrt 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied sqrt-prod 0.5

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

18. Applied times-frac 0.5

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

19. Applied *-un-lft-identity 0.5

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

20. Applied times-frac 0.5

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

21. Applied times-frac 0.7

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

22. Simplified 0.5

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

23. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))