Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.5s

Precision: 64

Math

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

↓

\[\cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)\right)\]

C

double f(double a1, double a2, double th) {
        double r100914 = th;
        double r100915 = cos(r100914);
        double r100916 = 2.0;
        double r100917 = sqrt(r100916);
        double r100918 = r100915 / r100917;
        double r100919 = a1;
        double r100920 = r100919 * r100919;
        double r100921 = r100918 * r100920;
        double r100922 = a2;
        double r100923 = r100922 * r100922;
        double r100924 = r100918 * r100923;
        double r100925 = r100921 + r100924;
        return r100925;
}

↓

double f(double a1, double a2, double th) {
        double r100926 = th;
        double r100927 = cos(r100926);
        double r100928 = a2;
        double r100929 = 2.0;
        double r100930 = pow(r100928, r100929);
        double r100931 = a1;
        double r100932 = pow(r100931, r100929);
        double r100933 = r100930 + r100932;
        double r100934 = sqrt(r100933);
        double r100935 = 2.0;
        double r100936 = sqrt(r100935);
        double r100937 = cbrt(r100936);
        double r100938 = r100937 * r100937;
        double r100939 = r100934 / r100938;
        double r100940 = 1.0;
        double r100941 = cbrt(r100937);
        double r100942 = r100941 * r100941;
        double r100943 = r100940 / r100942;
        double r100944 = r100934 / r100941;
        double r100945 = r100943 * r100944;
        double r100946 = r100939 * r100945;
        double r100947 = r100927 * r100946;
        return r100947;
}

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. Simplified 0.5

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

4. Applied div-inv 0.6

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

5. Applied associate-*l* 0.6

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

6. Simplified 0.5

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

8. Applied add-cube-cbrt 0.5

   \[\leadsto \cos th \cdot \frac{{a2}^{2} + {a1}^{2}}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}\]

9. Applied add-sqr-sqrt 0.5

   \[\leadsto \cos th \cdot \frac{\color{blue}{\sqrt{{a2}^{2} + {a1}^{2}} \cdot \sqrt{{a2}^{2} + {a1}^{2}}}}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}\]

10. Applied times-frac 0.5

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}}}\right)}\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.5

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)\]

13. Applied *-un-lft-identity 0.5

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{\color{blue}{1 \cdot \left({a2}^{2} + {a1}^{2}\right)}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)\]

14. Applied sqrt-prod 0.5

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{{a2}^{2} + {a1}^{2}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)\]

15. Applied times-frac 0.5

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \color{blue}{\left(\frac{\sqrt{1}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)}\right)\]

16. Simplified 0.5

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \left(\color{blue}{\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}} \cdot \frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)\right)\]

17. Final simplification 0.5

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{{a2}^{2} + {a1}^{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)\right)\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))