Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 9.4s

Precision: 64

Math

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

↓

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

C

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

↓

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

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

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied associate-*r* 0.5

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

9. Simplified 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied associate-/r* 0.5

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

15. Applied add-sqr-sqrt 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied sqrt-prod 0.5

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

18. Applied associate-/r* 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020066 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))