Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 15.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r117982 = th;
        double r117983 = cos(r117982);
        double r117984 = 2.0;
        double r117985 = sqrt(r117984);
        double r117986 = r117983 / r117985;
        double r117987 = a1;
        double r117988 = r117987 * r117987;
        double r117989 = r117986 * r117988;
        double r117990 = a2;
        double r117991 = r117990 * r117990;
        double r117992 = r117986 * r117991;
        double r117993 = r117989 + r117992;
        return r117993;
}

↓

double f(double a1, double a2, double th) {
        double r117994 = th;
        double r117995 = cos(r117994);
        double r117996 = a1;
        double r117997 = 2.0;
        double r117998 = pow(r117996, r117997);
        double r117999 = 2.0;
        double r118000 = sqrt(r117999);
        double r118001 = r117998 / r118000;
        double r118002 = r117995 * r118001;
        double r118003 = a2;
        double r118004 = pow(r118003, r117997);
        double r118005 = sqrt(r118000);
        double r118006 = sqrt(r118005);
        double r118007 = r118004 / r118006;
        double r118008 = r118005 / r117995;
        double r118009 = r118006 * r118008;
        double r118010 = r118007 / r118009;
        double r118011 = r118002 + r118010;
        return r118011;
}

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 div-inv 0.5

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

4. Applied associate-*l* 0.5

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

5. Simplified 0.5

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied sqrt-prod 0.5

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

9. Applied associate-/r* 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied add-sqr-sqrt 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied *-un-lft-identity 0.5

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

18. Applied times-frac 0.5

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

19. Applied times-frac 0.5

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

20. Applied associate-*l* 0.6

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

21. Simplified 0.4

    \[\leadsto \cos th \cdot \frac{{a1}^{2}}{\sqrt{2}} + \frac{\frac{1}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \color{blue}{\frac{a2 \cdot a2}{\frac{\sqrt{\sqrt{2}}}{\cos th}}}\]
22. Using strategy rm

23. Applied frac-times 0.4

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

24. Simplified 0.5

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

25. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020042 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))