Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 9.9s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r103444 = th;
        double r103445 = cos(r103444);
        double r103446 = 2.0;
        double r103447 = sqrt(r103446);
        double r103448 = r103445 / r103447;
        double r103449 = a1;
        double r103450 = r103449 * r103449;
        double r103451 = r103448 * r103450;
        double r103452 = a2;
        double r103453 = r103452 * r103452;
        double r103454 = r103448 * r103453;
        double r103455 = r103451 + r103454;
        return r103455;
}

↓

double f(double a1, double a2, double th) {
        double r103456 = 1.0;
        double r103457 = 2.0;
        double r103458 = sqrt(r103457);
        double r103459 = sqrt(r103458);
        double r103460 = sqrt(r103459);
        double r103461 = r103456 / r103460;
        double r103462 = r103461 / r103460;
        double r103463 = th;
        double r103464 = cos(r103463);
        double r103465 = r103464 / r103459;
        double r103466 = a1;
        double r103467 = r103465 * r103466;
        double r103468 = r103462 * r103467;
        double r103469 = r103468 * r103466;
        double r103470 = r103465 / r103459;
        double r103471 = a2;
        double r103472 = r103471 * r103471;
        double r103473 = r103470 * r103472;
        double r103474 = r103469 + r103473;
        return r103474;
}

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

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied sqrt-prod 0.5

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

12. Applied add-sqr-sqrt 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied *-un-lft-identity 0.5

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

16. Applied times-frac 0.5

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

17. Applied times-frac 0.5

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

18. Applied associate-*l* 0.6

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

19. Simplified 0.5

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

21. Applied add-sqr-sqrt 0.5

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

22. Applied sqrt-prod 0.4

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

23. Applied associate-/r* 0.4

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

24. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020062 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))