Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 1.3m

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r1553879 = th;
        double r1553880 = cos(r1553879);
        double r1553881 = 2.0;
        double r1553882 = sqrt(r1553881);
        double r1553883 = r1553880 / r1553882;
        double r1553884 = a1;
        double r1553885 = r1553884 * r1553884;
        double r1553886 = r1553883 * r1553885;
        double r1553887 = a2;
        double r1553888 = r1553887 * r1553887;
        double r1553889 = r1553883 * r1553888;
        double r1553890 = r1553886 + r1553889;
        return r1553890;
}

↓

double f(double a1, double a2, double th) {
        double r1553891 = a2;
        double r1553892 = th;
        double r1553893 = cos(r1553892);
        double r1553894 = 2.0;
        double r1553895 = sqrt(r1553894);
        double r1553896 = r1553893 / r1553895;
        double r1553897 = r1553891 * r1553896;
        double r1553898 = r1553891 * r1553897;
        double r1553899 = a1;
        double r1553900 = r1553899 * r1553899;
        double r1553901 = 1.0;
        double r1553902 = sqrt(r1553895);
        double r1553903 = sqrt(r1553902);
        double r1553904 = r1553901 / r1553903;
        double r1553905 = r1553893 / r1553902;
        double r1553906 = r1553903 / r1553905;
        double r1553907 = r1553904 / r1553906;
        double r1553908 = r1553900 * r1553907;
        double r1553909 = r1553898 + r1553908;
        return r1553909;
}

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

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied associate-/l* 0.5

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

10. Applied associate-*r* 0.5

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

12. Applied *-un-lft-identity 0.5

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

13. Applied add-sqr-sqrt 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied times-frac 0.5

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

16. Applied associate-/r* 0.5

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

17. Simplified 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))