Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 9.7s

Precision: binary64

Math

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

↓

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

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) (((double) (((double) sqrt(((double) (((double) (a1 * a1)) + ((double) (a2 * a2)))))) / ((double) sqrt(((double) sqrt(2.0)))))) * ((double) cos(th)))) * ((double) (((double) (((double) sqrt(((double) (((double) (a1 * a1)) + ((double) (a2 * a2)))))) / ((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) / ((double) sqrt(((double) sqrt(((double) sqrt(2.0))))))))));
}

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

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.6

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

6. Applied associate-/r* 0.5

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied add-sqr-sqrt 0.5

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

15. Applied times-frac 0.6

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

16. Applied times-frac 0.8

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

17. Applied associate-*r* 0.8

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

18. Simplified 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))