Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 6.9s

Precision: 64

Math

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

↓

\[\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt{2}}}{\cos th}}\]

C

double code(double a1, double a2, double th) {
	return ((double) (((double) (((double) (((double) cos(th)) / ((double) sqrt(2.0)))) * ((double) (a1 * a1)))) + ((double) (((double) (((double) cos(th)) / ((double) sqrt(2.0)))) * ((double) (a2 * a2))))));
}

↓

double code(double a1, double a2, double th) {
	return ((double) (((double) sqrt(((double) (((double) (a1 * a1)) + ((double) (a2 * a2)))))) * ((double) (((double) (((double) sqrt(((double) (((double) (a1 * a1)) + ((double) (a2 * a2)))))) / ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))))) / ((double) (((double) cbrt(((double) sqrt(2.0)))) / ((double) cos(th))))))));
}

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

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

4. Applied clear-num 0.5

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

5. Applied associate-*l/ 0.5

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

6. Simplified 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied times-frac 0.5

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

12. Applied add-sqr-sqrt 0.5

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

13. Applied times-frac 0.5

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

14. Simplified 0.5

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

16. Applied *-un-lft-identity 0.5

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

17. Applied add-cube-cbrt 0.5

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

18. Applied times-frac 0.5

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

19. Applied associate-/r* 0.5

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

20. Simplified 0.5

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

21. Final simplification 0.5

    \[\leadsto \sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\frac{\sqrt[3]{\sqrt{2}}}{\cos th}}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))