Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 1.0m

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 r2418812 = th;
        double r2418813 = cos(r2418812);
        double r2418814 = 2.0;
        double r2418815 = sqrt(r2418814);
        double r2418816 = r2418813 / r2418815;
        double r2418817 = a1;
        double r2418818 = r2418817 * r2418817;
        double r2418819 = r2418816 * r2418818;
        double r2418820 = a2;
        double r2418821 = r2418820 * r2418820;
        double r2418822 = r2418816 * r2418821;
        double r2418823 = r2418819 + r2418822;
        return r2418823;
}

↓

double f(double a1, double a2, double th) {
        double r2418824 = a1;
        double r2418825 = 2.0;
        double r2418826 = sqrt(r2418825);
        double r2418827 = sqrt(r2418826);
        double r2418828 = sqrt(r2418827);
        double r2418829 = r2418827 * r2418828;
        double r2418830 = r2418828 / r2418824;
        double r2418831 = r2418829 * r2418830;
        double r2418832 = r2418824 / r2418831;
        double r2418833 = th;
        double r2418834 = cos(r2418833);
        double r2418835 = r2418832 * r2418834;
        double r2418836 = r2418834 / r2418827;
        double r2418837 = r2418836 / r2418827;
        double r2418838 = a2;
        double r2418839 = r2418838 * r2418838;
        double r2418840 = r2418837 * r2418839;
        double r2418841 = r2418835 + r2418840;
        return r2418841;
}

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

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

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