Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 15.9s

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 r113015 = th;
        double r113016 = cos(r113015);
        double r113017 = 2.0;
        double r113018 = sqrt(r113017);
        double r113019 = r113016 / r113018;
        double r113020 = a1;
        double r113021 = r113020 * r113020;
        double r113022 = r113019 * r113021;
        double r113023 = a2;
        double r113024 = r113023 * r113023;
        double r113025 = r113019 * r113024;
        double r113026 = r113022 + r113025;
        return r113026;
}

↓

double f(double a1, double a2, double th) {
        double r113027 = th;
        double r113028 = cos(r113027);
        double r113029 = a1;
        double r113030 = 2.0;
        double r113031 = pow(r113029, r113030);
        double r113032 = 2.0;
        double r113033 = sqrt(r113032);
        double r113034 = r113031 / r113033;
        double r113035 = r113028 * r113034;
        double r113036 = a2;
        double r113037 = pow(r113036, r113030);
        double r113038 = sqrt(r113033);
        double r113039 = sqrt(r113038);
        double r113040 = r113037 / r113039;
        double r113041 = r113038 / r113028;
        double r113042 = r113039 * r113041;
        double r113043 = r113040 / r113042;
        double r113044 = r113035 + r113043;
        return r113044;
}

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))))