Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 39.1s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r2487818 = th;
        double r2487819 = cos(r2487818);
        double r2487820 = 2.0;
        double r2487821 = sqrt(r2487820);
        double r2487822 = r2487819 / r2487821;
        double r2487823 = a1;
        double r2487824 = r2487823 * r2487823;
        double r2487825 = r2487822 * r2487824;
        double r2487826 = a2;
        double r2487827 = r2487826 * r2487826;
        double r2487828 = r2487822 * r2487827;
        double r2487829 = r2487825 + r2487828;
        return r2487829;
}

↓

double f(double a1, double a2, double th) {
        double r2487830 = 1.0;
        double r2487831 = 2.0;
        double r2487832 = sqrt(r2487831);
        double r2487833 = sqrt(r2487832);
        double r2487834 = sqrt(r2487833);
        double r2487835 = r2487830 / r2487834;
        double r2487836 = th;
        double r2487837 = cos(r2487836);
        double r2487838 = r2487837 / r2487833;
        double r2487839 = r2487834 / r2487838;
        double r2487840 = r2487835 / r2487839;
        double r2487841 = a2;
        double r2487842 = r2487841 * r2487841;
        double r2487843 = r2487840 * r2487842;
        double r2487844 = a1;
        double r2487845 = r2487838 / r2487833;
        double r2487846 = r2487844 * r2487845;
        double r2487847 = r2487844 * r2487846;
        double r2487848 = r2487843 + r2487847;
        return r2487848;
}

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

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied associate-/r* 0.5

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

13. Applied add-sqr-sqrt 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied *-un-lft-identity 0.5

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

17. Applied times-frac 0.5

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

18. Applied associate-/l* 0.4

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019146 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))