Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 13.2s

Precision: 64

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

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r96810 = th;
        double r96811 = cos(r96810);
        double r96812 = 2.0;
        double r96813 = sqrt(r96812);
        double r96814 = r96811 / r96813;
        double r96815 = a1;
        double r96816 = r96815 * r96815;
        double r96817 = r96814 * r96816;
        double r96818 = a2;
        double r96819 = r96818 * r96818;
        double r96820 = r96814 * r96819;
        double r96821 = r96817 + r96820;
        return r96821;
}

↓

double f(double a1, double a2, double th) {
        double r96822 = a2;
        double r96823 = th;
        double r96824 = cos(r96823);
        double r96825 = r96824 * r96822;
        double r96826 = 2.0;
        double r96827 = sqrt(r96826);
        double r96828 = r96825 / r96827;
        double r96829 = r96822 * r96828;
        double r96830 = sqrt(r96827);
        double r96831 = sqrt(r96830);
        double r96832 = r96824 / r96831;
        double r96833 = a1;
        double r96834 = 2.0;
        double r96835 = pow(r96833, r96834);
        double r96836 = r96835 / r96830;
        double r96837 = r96836 / r96831;
        double r96838 = r96832 * r96837;
        double r96839 = r96829 + r96838;
        return r96839;
}

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

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied *-un-lft-identity 0.5

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

11. Applied times-frac 0.5

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

12. Applied associate-*l* 0.5

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

13. Simplified 0.5

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

15. Applied add-sqr-sqrt 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied sqrt-prod 0.5

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

18. Applied div-inv 0.5

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

19. Applied times-frac 0.5

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

20. Applied associate-*l* 0.5

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

21. Simplified 0.5

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

22. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))