Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 2.2m

Precision: 64

Math

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

↓

\[a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right) + \left(\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{1}{\sqrt{\sqrt{2}}}\right) \cdot \frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right)\]

C

double f(double a1, double a2, double th) {
        double r3268753 = th;
        double r3268754 = cos(r3268753);
        double r3268755 = 2.0;
        double r3268756 = sqrt(r3268755);
        double r3268757 = r3268754 / r3268756;
        double r3268758 = a1;
        double r3268759 = r3268758 * r3268758;
        double r3268760 = r3268757 * r3268759;
        double r3268761 = a2;
        double r3268762 = r3268761 * r3268761;
        double r3268763 = r3268757 * r3268762;
        double r3268764 = r3268760 + r3268763;
        return r3268764;
}

↓

double f(double a1, double a2, double th) {
        double r3268765 = a2;
        double r3268766 = th;
        double r3268767 = cos(r3268766);
        double r3268768 = 2.0;
        double r3268769 = sqrt(r3268768);
        double r3268770 = r3268767 / r3268769;
        double r3268771 = r3268765 * r3268770;
        double r3268772 = r3268765 * r3268771;
        double r3268773 = cbrt(r3268769);
        double r3268774 = sqrt(r3268773);
        double r3268775 = r3268767 / r3268774;
        double r3268776 = 1.0;
        double r3268777 = sqrt(r3268769);
        double r3268778 = r3268776 / r3268777;
        double r3268779 = r3268775 * r3268778;
        double r3268780 = fabs(r3268773);
        double r3268781 = r3268776 / r3268780;
        double r3268782 = r3268779 * r3268781;
        double r3268783 = a1;
        double r3268784 = r3268783 * r3268783;
        double r3268785 = r3268782 * r3268784;
        double r3268786 = r3268772 + r3268785;
        return r3268786;
}

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 add-sqr-sqrt 0.5

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied clear-num 0.5

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

9. Applied associate-*r* 0.5

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

11. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

12. Applied sqrt-prod 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\color{blue}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

13. Applied *-un-lft-identity 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\frac{\color{blue}{1 \cdot \cos th}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

14. Applied times-frac 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\color{blue}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

15. Applied *-un-lft-identity 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{1 \cdot \sqrt{2}}}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

16. Applied sqrt-prod 0.7

    \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{2}}}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

17. Applied times-frac 0.5

    \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{1}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

18. Applied add-sqr-sqrt 0.5

    \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{\sqrt{1}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

19. Applied times-frac 0.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt{1}}{\frac{\sqrt{1}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}} \cdot \frac{\sqrt{1}}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}\right)} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

20. Simplified 0.5

    \[\leadsto \left(\color{blue}{\frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\sqrt{1}}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}\right) \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

21. Simplified 0.5

    \[\leadsto \left(\frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right)}\right) \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

22. Final simplification 0.5

    \[\leadsto a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right) + \left(\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{1}{\sqrt{\sqrt{2}}}\right) \cdot \frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right)\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))